Electronic Journal of Qualitative Theory of Differential Equations 2010, No. 31, 1-11; http://www.math.u-szeged.hu/ejqtde/

Homoclinic solutions for a class of non-periodic second order Hamiltonian systems^{1}

Jian Ding, Junxiang Xu, Fubao Zhang

Department of Mathematics, Southeast University, Nanjing 210018, People’s Republic of China. E-mail: df2001101@126.com

Abstract: We study the existence of homoclinic solutions for the second order Hamiltonian system ü + V_{u}(t, u) = f(t). Let V (t, u) = −K(t, u) + W (t, u) ∈ C^{1}(ℝ × ℝ^{n}, ℝ) be T -periodic in t, where K is a quadratic growth function and W may be asymptotically quadratic or super-quadratic at infinity. One homoclinic solution is obtained as a limit of solutions of a sequence of periodic second order differential equations. Key words: Hamiltonian systems; Homoclinic solutions; Condition(C); Asymptotically quadratic; Super-quadratic. AMS Subject Classification (2000): 37K05 .

1 Introduction and the main result

In this paper, we consider the existence of homoclinic solutions for the second order Hamiltonian system:

ü(t) + V_{u}(t, u(t)) = f(t),  (1.1)

where f ∈ L^{2}(ℝ, ℝ^{n}) is continuous and bounded, V (t, u) = −K(t, u) + W (t, u) ∈ C^{1}(ℝ × ℝ^{n}, ℝ) is T -periodic in t, T > 0.

Let us recall that a solution u(t) of (1.1) is homoclinic to 0 if u(t) ≢ 0, u(t) → 0 and u̇(t) → 0 as t → ±∞.

In recent years, the existence of homoclinic solutions for (1.1) has been studied extensively by variational methods(see, for instance, [6,7,11-13,15] ). Most of them considered (1.1) with W satisfying the Ambrosetti-Rabinowitz condition, that is, there exists μ > 2 such that

0 < μW (t, u) ≤ (W_{u}(t, u), u) for all t ∈ ℝ and u ∈ ℝ^{n} \ {0}.

It is well known that the major difficulty is to check the Palais-Smale (P S) condition (a compactness condition) when one considers (1.1) on the whole space ℝ by variational methods. Recall that a sequence {u_{n}} is said to be a (P S) sequence of φ provided that {φ(u_{n})} is bounded and φ′(u_{n}) → 0. φ satisfies the (P S) condition if any (P S) sequence possesses a convergent subsequence. Using the Ambrosetti-Rabinowitz condition, one can easily establish the boundedness of (P S) sequences, which is crucial to check the (P S) condition. Later some authors managed to weaken this condition (see, e.g., [4] and [10]). Many authors also treated some new growth conditions. For example, [1,9] considered the sub-quadratic case and [2, 14] dealt with the asymptotically quadratic case.

If L(t) and W (t, u) are either independent of t or periodic in t, the problem seems a little simple and there are many results. In [8,10], the authors considered

ü(t) + V_{u}(t, u(t)) = 0

1This work was supported by the Natural Science Foundation of China.

with V (t, u) = −\frac{1}{2}(Lu, u) + W (u) being independent of t, i.e., the system is autonomous. They obtained one homoclinic solution as a limit of solutions of a certain sequence of periodic systems. By this method, [5] considered the case that L(t) and W (t, u) are periodic in t. It also assumed that L(t) is positive definite and symmetric, W (t, u) satisfies the Ambrosetti-Rabinowitz growth condition. Without periodicity condition, the problem is quite different from the ones just described for the lack of compactness of the Sobolev embedding. Using the method in [5, 8, 10], [3] considered (1.1). The authors replaced \frac{1}{2}(L(t)u, u) by K(t, u) which satisfies the following condition

∃b_{1}, b_{2} > 0 such that for all (t, u) ∈ ℝ × ℝ^{n},

b_{1}|u|^{2} ≤ K(t, u) ≤ b_{2}|u|^{2}

and

K(t, u) ≤ (u, K_{u}(t, u)) ≤ 2K(t, u).

Given W satisfied the Ambrosetti-Rabinowitz condition, they obtained one homoclinic solution.

We note that the nonlinearity W in all the papers mentioned above are nonnegative, which ensures the variational functional possesses some good properties. If the nonlinearity is allowed to be negative, new difficulties arise and there haven’t been too many results. In this paper, we consider this case. we study homoclinic solutions of (1.1) on the whole space ℝ. (1.1) is not periodic in nature. We will consider some new nonlinearity W. Precisely, W is allowed to be negative near the origin and satisfies asymptotically quadratic or super-quadratic growth condition at infinity. It is obvious that W doesn’t satisfy the Ambrosetti-Rabinowitz condition. We will approximate a solution of (1.1) by a limit of solutions of a sequence of periodic systems.

We make the following assumptions: (A_{1}) V (t, u) = −K(t, u) + W (t, u) is a C^{1} function on ℝ × ℝ^{n}, T -periodic in t, T > 0, and

V_{u}(t, u) → 0 as |u| → 0 uniformly in t ∈ ℝ; (A_{2}) For all (t, u) ∈ ℝ × ℝ^{n}, 0 ≤ (u, K_{u}(t, u)) ≤ 2K(t, u), and there exist 0 < \underline{b} ≤ \overline{b} such that

\underline{b}|u|^{2} ≤ K(t, u) ≤ \overline{b}|u|^{2}; (A_{3}) There exist constants d_{1} > 0, r ≥ 2 such that W (t, u) ≤ d_{1}|u|^{r} for all (t, u) ∈ ℝ × ℝ^{n}; (A_{4}) There exist constants d_{2} > 0, μ with r ≥ μ > r − 1 and β ∈ L^{1}(ℝ, ℝ^{+}) such that

(W_{u}(t, u), u) − 2W (t, u) ≥ d_{2}|u|^{μ} − β(t)
for all (t, u) ∈ ℝ × ℝ^{n};

(A_{5}) There exist positive constants R, ρ_{0} > 0 such that \frac{W (t,u)}{|u|^{2}} ≥ \frac{2π^{2}}{T^{2}} as |u| > R, t ∈ [−T, T] and W (t, u) ≤ 0 as |u| ≤ ρ_{0}, t ∈ [−T, T].

For each k ∈ ℕ, let L_{2kT}^{2} be the Hilbert space of 2kT -periodic functions on ℝ with values in ℝ^{n} equipped with the norm

∥u∥_{L_{2kT}^{2}} = ( ∫_{−kT}^{kT} |u(t)|^{2}dt)^{\frac{1}{2}} ,

and L_{2kT}^{∞} be the space of 2kT -periodic essentially bounded functions from ℝ into ℝ^{n} equipped with the norm

∥u∥_{L_{2kT}^{∞}} := ess sup {|u(t)| : t ∈ [−kT, kT]}.

Denote E_{k} := W_{2kT}^{1,2} be the Hilbert space of 2kT -periodic functions on ℝ with values in ℝ^{n} under the norm

∥u∥_{E_{k}} := ( ∫_{−kT}^{kT} (|u̇(t)|^{2} + |u(t)|^{2}) dt)^{\frac{1}{2}} .

By Rabinowitz in [5], we have

Proposition 1 There is a constant C_{1} > 0 such that for each k ∈ ℕ and u ∈ E_{k} the following inequality holds:

∥u∥_{L_{2kT}^{∞}} ≤ C_{1}∥u∥_{E_{k}}.

We also make the following assumption: (A_{6}) f is nonzero continuous and bounded, there is a constant C_{0} such that ∥f∥_{L^{2}(ℝ,ℝ^{n})} =

( ∫_{ℝ} |f(t)|^{2} dt)^{\frac{1}{2}} ≤ C_{0} := min\{\frac{1}{2}, \underline{b}\}\frac{ρ_{0}}{2C_{1}} .

Our main result reads as follows.

Theorem 1 If assumptions (A_{1}) − (A_{6}) are satisfied, then (1.1) possesses at least one homoclinic solution u ∈ W^{1,2}(ℝ, ℝ^{n}).

Remark 1.1 (A_{5}) shows that W may be either asymptotically quadratic or super-quadratic growth at infinity.

Remark 1.2 There are functions which satisfy assumptions (A_{1}) − (A_{5}). For example,

K(t, x) = \{ \begin{matrix}(1 + \frac{1}{1+x^{2}} )x^{2} x ≥ 0, \\ (1 + \frac{2}{1+x^{2}} )x^{2} x < 0 \end{matrix}

and

W (t, x) = −2x^{2} + x^{4} (the super − quadratic case)

or

W (t, x) = x^{2} − 2x^{\frac{4}{3}} (the asymptotically quadratic case)

Let f_{k} : ℝ → ℝ^{n} be a 2kT -periodic extension of the restriction of f to the interval [−kT, kT] and u_{k} be a 2kT -periodic solution of

ü(t) + V_{u}(t, u(t)) = f_{k}(t)  (1.2)

obtained by the Mountain Pass Theorem. We will show that the sequence {u_{k}} possesses a subsequence which converges to a homoclinic solution of (1.1).

The main difficulties in treating (1.1) are caused by the fact that in order to get appropriate convergence of the sequence of approximative functions {u_{k}} we need the sequence {∥u_{k}∥_{E_{k}}} to be bounded uniformly in k ∈ ℕ and the constants ρ and α appearing in the condition (3) of the Mountain Pass Theorem (see Theorem 2) to be independent of k.

2 Proof of the main result

For each k ∈ ℕ, we consider the second order system (1.2) on E_{k}.

Define

φ_{k}(u) = ∫_{−kT}^{kT} [\frac{1}{2}|u̇(t)|^{2} − V (t, u(t)) + (f_{k}(t), u(t))]dt.

It is clear that φ_{k} ∈ C^{1}(E_{k}, ℝ) and

φ′_{k}(u)v = ∫_{−kT}^{kT} [(u̇(t), v̇(t)) − (V_{u}(t, u(t)), v(t)) + (f_{k}(t), v(t))]dt.

We all know that critical points of φ_{k} are classical 2kT -periodic solutions of (1.2).

Lemma 2.1 Under (A_{1}) − (A_{6}), for each k ∈ ℕ, (1.2) possesses a 2kT -periodic solution.

We will prove this lemma via the Mountain Pass Theorem by Rabinowitz in [14]. We state this theorem as follows.

Theorem 2 Let E be a real Banach space and φ : E → ℝ be a C^{1} function. If φ satisfies the following conditions:

(1) φ(0) = 0;

(2) φ satisfies the (PS) condition on E;

(3) There exist constants ρ, α > 0 such that φ|_{S_{ρ}} ≥ α;

(4) There exists e ∈ E\B_{ρ} such that φ(e) ≤ 0, where B_{ρ} is a closed ball in E of radius ρ centred at 0 and S_{ρ} is the boundary of B_{ρ}. Then φ possesses a critical value c ≥ α given by

c = inf_{g∈Γ} max_{s∈[0,1]} φ(g(s)),

where

Γ = {g ∈ C([0, 1], E) : g(0) = 0, g(1) = e}.

Instead of the (P S) condition, we use condition (C). Recall a function φ satisfies condition (C) on E if any sequence {u_{j}} ⊂ E such that {φ(u_{j})} is bounded and (1 + ∥u_{j}∥)∥φ′(u_{j})∥ → 0 has a convergent subsequence. The Mountain Pass Theorem still holds true under condition (C).

Proof of Lemma 2.1 From our assumptions, it is easy to see that φ_{k}(0) = 0. Step 1. φ_{k} satisfies condition (C).

Suppose {u_{j}} ⊂ E_{k}, {φ_{k}(u_{j})} is bounded and (1 + ∥u_{j}∥_{E_{k}})∥φ′_{k}(u_{j})∥ → 0 as j → ∞. Then there is a constant M_{k} > 0 such that

φ_{k}(u_{j}) ≤ M_{k}, (1 + ∥u_{j}∥_{E_{k}})∥φ′_{k}(u_{j})∥ ≤ M_{k}  (2.3)

for all j ∈ ℕ.

By (2.3), (A_{2}), and (A_{4}),

3M_{k} ≥ 2φ_{k}(u_{j}) − φ′_{k}(u_{j})u_{j}
= ∫_{−kT}^{kT} [(V_{u}(t, u_{j}(t)), u_{j}(t)) − 2V (t, u_{j}(t))] dt + ∫_{−kT}^{kT} (f_{k}(t), u_{j}(t)) dt

= ∫_{−kT}^{kT} [2K(t, u_{j}(t)) − (K_{u}(t, u_{j}(t)), u_{j}(t))] dt
+ ∫_{−kT}^{kT} [(W_{u}(t, u_{j}(t)), u_{j}(t)) − 2W (t, u_{j}(t))] dt + ∫_{−kT}^{kT} (f_{k}(t), u_{j}(t)) dt
≥ d_{2} ∫_{−kT}^{kT} |u_{j}(t)|^{μ} dt − ∫_{−kT}^{kT} β(t) dt − ∥f_{k}∥_{L_{2kT}^{2}} ∥u_{j}∥_{E_{k}}
≥ d_{2} ∫_{−kT}^{kT} |u_{j}(t)|^{μ} dt − β_{0} − C_{0}∥u_{j}∥_{E_{k}},

where β_{0} = ∫_{−kT}^{kT} β(t) dt.

Therefore,

∫_{−kT}^{kT} |u_{j}(t)|^{μ} dt ≤ \frac{1}{d_{2}} (3M_{k} + β_{0} + C_{0}∥u_{j}∥_{E_{k}}).  (2.4)

By (A_{2}) and (A_{3}),

\frac{1}{2}∥u̇_{j}∥_{L_{2kT}^{2}}^{2} = φ_{k}(u_{j}) − ∫_{−kT}^{kT} K(t, u_{j}(t)) dt + ∫_{−kT}^{kT} W (t, u_{j}(t)) dt
− ∫_{−kT}^{kT} (f_{k}(t), u_{j}(t)) dt
≤ M_{k} − \underline{b} ∫_{−kT}^{kT} |u_{j}(t)|^{2} dt + d_{1} ∫_{−kT}^{kT} |u_{j}(t)|^{r} dt
+C_{0}∥u_{j}∥_{E_{k}}.

Then one has

\frac{1}{2}∥u̇_{j}∥_{L_{2kT}^{2}}^{2} + \underline{b}∥u_{j}∥_{L_{2kT}^{2}}^{2} ≤ M_{k} + d_{1} ∫_{−kT}^{kT} |u_{j}(t)|^{r} dt + C_{0}∥u_{j}∥_{E_{k}}.

Therefore, by (2.4),

min\{\frac{1}{2}, \underline{b}\}∥u_{j}∥_{E_{k}}^{2} ≤ M_{k} + d_{1} ∫_{−kT}^{kT} |u_{j}(t)|^{r} dt + C_{0}∥u_{j}∥_{E_{k}}
≤ M_{k} + d_{1}∥u_{j}∥_{L_{2kT}^{∞}}^{r−μ} ∫_{−kT}^{kT} |u_{j}(t)|^{μ} dt + C_{0}∥u_{j}∥_{E_{k}}
≤ M_{k} + d_{1}C_{1}^{r−μ}∥u_{j}∥_{E_{k}}^{r−μ} ∫_{−kT}^{kT} |u_{j}(t)|^{μ} dt + C_{0}∥u_{j}∥_{E_{k}}
≤ M_{k} + \frac{d_{1}}{d_{2}} C_{1}^{r−μ}∥u_{j}∥_{E_{k}}^{r−μ}(3M_{k} + β_{0} + C_{0}∥u_{j}∥_{E_{k}}) + C_{0}∥u_{j}∥_{E_{k}}.

Since r − μ < 1, we get {∥u_{j}∥_{E_{k}}} is bounded. Going if necessary to a subsequence, we can assume that there exists u ∈ E_{k} such that u_{j} ⇀ u in E_{k} as j → +∞ , which implies u_{j} → u uniformly on [−kT, kT].

Therefore

(φ′_{k}(u_{j}) − φ′_{k}(u))(u_{j} − u) → 0,
∥u_{j} − u∥_{L_{2kT}^{2}} → 0

and

∫_{−kT}^{kT} (V_{u}(t, u_{j}(t)) − V_{u}(t, u(t)), u_{j}(t) − u(t)) dt → 0

as j → +∞.

By an easy computation, we can see that

(φ′_{k}(u_{j}) − φ′_{k}(u))(u_{j} − u) = ∥u̇_{j} − u̇∥_{L_{2kT}^{2}}^{2} − ∫_{−kT}^{kT} (V_{u}(t, u_{j}(t)) − V_{u}(t, u(t)), u_{j}(t) − u(t)) dt.

Hence we have ∥u̇_{j} − u̇∥_{L_{2kT}^{2}}^{2} → 0, and so u_{j} → u in E_{k}. Step 2. There are constants ρ > 0, α > 0 independent of k, such that φ_{k}|_{S_{ρ}} ≥ α, where S_{ρ} = {u ∈ E_{k}|∥u∥_{E_{k}} = ρ}.

Choose ρ = \frac{ρ_{0}}{C_{1}} , then for u ∈ S_{ρ} we have ∥u∥_{L_{2kT}^{∞}} ≤ ρ_{0}. Therefore, |u| ≤ ρ_{0} for all t ∈ [−kT, kT], and then by (A_{5}), W (t, u) ≤ 0. Together with (A_{2}), we obtain

φ_{k}(u) = ∫_{−kT}^{kT} [\frac{1}{2}|u̇(t)|^{2} + K(t, u(t)) − W (t, u(t))] dt + ∫_{−kT}^{kT} (f_{k}(t), u(t)) dt
≥ \frac{1}{2} ∫_{−kT}^{kT} |u̇(t)|^{2} dt + \underline{b} ∫_{−kT}^{kT} |u(t)|^{2} dt − ∥f_{k}∥_{L_{2kT}^{2}} ∥u(t)∥_{E_{k}}
≥ min\{\frac{1}{2}, \underline{b}\}∥u(t)∥_{E_{k}}^{2} − C_{0}∥u∥_{E_{k}}
= min\{\frac{1}{2}, \underline{b}\}ρ^{2} − C_{0}ρ
= min\{\frac{1}{2}, \underline{b}\}\begin{matrix}ρ_{0} & 2 \\ 2C_{1} & 2 \end{matrix} := α.

Step 3. For the ρ defined as above, there exists e_{k} ∈ E_{k} such that ∥e_{k}∥_{E_{k}} > ρ, φ_{k}(e_{k}) ≤ 0.

By (A_{5}),

\frac{W (t, u)}{|u|^{2}} ≥ \frac{2π^{2}}{T^{2}}

for all |u| > R and t ∈ [−T, T].

Let δ = max_{{t∈[−T,T],|u|≤R}} |W (t, u)|, we obtain

W (t, u) ≥ \frac{2π^{2}}{T^{2}} (|u|^{2} − R^{2}) − δ  (2.5)

for all u ∈ ℝ^{n}, t ∈ ℝ.

Set

ē_{k}(t) = \{ \begin{matrix}s sin(ωt)e, & t ∈ [−T, T] \\ 0, & t ∈ [−kT, kT]\[−T, T] \end{matrix}  (2.6)

where ω = \frac{π}{T} , e = (1, 0, ⋯, 0). Let e_{k} be the 2kT-periodic extension of ē_{k}, then e_{k} ∈ E_{k}, and ∥e_{k}∥_{E_{k}} → ∞ as s → ∞. We can assume s is large enough such that ∥e_{k}∥_{E_{k}} ≥ ρ. Combining (A_{2}) and (2.5), we obtain

φ_{k}(e_{k}(t)) = ∫_{−kT}^{kT} [\frac{1}{2}|ė_{k}(t)|^{2} + K(t, e_{k}(t)) − W (t, e_{k}(t)) + (f_{k}(t), e_{k}(t))] dt
≤ ∫_{−T}^{T} \frac{1}{2}|ė_{k}(t)|^{2}dt + b̄ ∫_{−T}^{T} |e_{k}(t)|^{2}dt − \frac{2π^{2}}{T^{2}} ∫_{−T}^{T} |e_{k}(t)|^{2}dt

+(\frac{2π^{2}R^{2}}{T^{2}} + δ)2T + C_{0}( ∫_{−T}^{T} |e_{k}(t)|^{2} dt)^{\frac{1}{2}}

≤ \frac{1}{2}s^{2}ω^{2} ∫_{−T}^{T} | cos(ωt)|^{2} dt + (b̄s^{2} − \frac{2π^{2}s^{2}}{T^{2}} ) ∫_{−T}^{T} | sin(ωt)|^{2} dt

+C_{0}s( ∫_{−T}^{T} | sin(ωt)|^{2} dt)^{\frac{1}{2}} + 2T (δ + \frac{2π^{2}R^{2}}{T^{2}} )

= (\frac{1}{2}ω^{2} + b̄ − \frac{2π^{2}}{T^{2}} )s^{2}T + C_{0}sT^{\frac{1}{2}} + 2T (δ + \frac{2π^{2}R^{2}}{T^{2}} )
≤ −\frac{π^{2}}{T} s^{2} + C_{0}sT^{\frac{1}{2}} + 2T (δ + \frac{2π^{2}R^{2}}{T^{2}} ) → −∞

as s → ∞. So for all k ∈ ℕ, we can choose an s large enough such that e_{k} defined as above satisfies ∥e_{k}∥_{E_{k}} > ρ and φ_{k}(e_{k}) ≤ 0.

Therefore, by the Mountain Pass Theorem, φ_{k} possesses a critical value c_{k} defined by

c_{k} = inf_{g∈Γ_{k}} max_{s∈[0,1]} φ_{k}(g(s))

satisfying c_{k} ≥ α, where

Γ_{k} = {g ∈ C([0, 1], E_{k}) : g(0) = 0, g(1) = e_{k}}.

By Lemma 2.1 we know for each k ∈ ℕ, there exists u_{k} ∈ E_{k} such that

φ_{k}(u_{k}) = c_{k}, φ′_{k}(u_{k}) = 0.

Consequently u_{k} is a classical 2kT -periodic solution of (1.2). Moreover, since c_{k} ≥ α > 0, u_{k} is a nontrivial solution.

In the following, we will show that there exists a subsequence of {u_{k}} which almost uniformly converges to a C^{1} function. We denote C_{loc}^{p}(ℝ, ℝ^{n})(p ∈ ℕ∪{0}), the space of C^{p} functions on ℝ with values in ℝ^{n} under the topology of almost uniformly convergence of functions and all derivatives up to the order p. We have

Lemma 2.2 Let {u_{k}}_{k∈ℕ} be the sequence given as above. Then it possesses a subsequence also denoted by {u_{k}} and a C^{1} function u_{0} : ℝ → ℝ^{n} such that u_{k} → u_{0} in C_{loc}^{1}(ℝ, ℝ^{n}), as k → +∞.

Proof We will prove this lemma by the Arzela-Ascoli Theorem. We first show that the sequence {c_{k}}_{k∈ℕ} and {∥u_{k}∥_{E_{k}}}_{k∈ℕ} are bounded.

For each k ∈ ℕ, let g_{k} : [0, 1] → E_{k} be a curve given by g_{k}(s) = se_{k} where e_{k} is defined as above. Then g_{k} ∈ Γ_{k} and φ_{k}(g_{k}(s)) = φ_{1}(g_{1}(s)) for all k ∈ ℕ and s ∈ [0, 1].

Hence

c_{k} = inf_{g∈Γ_{k}} max_{s∈[0,1]} φ_{k}(g_{k}(s)) ≤ max_{s∈[0,1]} φ_{k}(g_{k}(s)) = max_{s∈[0,1]} φ_{1}(g_{1}(s)) := M_{0}

independently of k ∈ ℕ.

Therefore

φ_{k}(u_{k}) = c_{k} ≤ M_{0}

for all k ∈ ℕ.

As φ′_{k}(u_{k}) = 0, we have

(1 + ∥u_{k}∥_{E_{k}})∥φ′_{k}(u_{k})∥ = 0 ≤ M_{0}

for all k ∈ ℕ.

Along the proof of Step 1. in Lemma 2.1, it is easy to prove that{∥u_{k}∥_{E_{k}}} is bounded uniformly in k ∈ ℕ, which means there exists a constant M_{1} > 0 independent of k such that

∥u_{k}∥_{E_{k}} ≤ M_{1}.

In order to show that {u_{k}}_{k∈ℕ} and {u̇_{k}}_{k∈ℕ} are equicontinuous, we first prove {u_{k}}_{k∈ℕ}, {u̇_{k}}_{k∈ℕ} and {ü_{k}}_{k∈ℕ} are uniformly bounded in L_{2kT}^{∞}.

By Proposition 1,

∥u_{k}∥_{L_{2kT}^{∞}} ≤ C_{1}M_{1} := M_{2}  (2.7)

for all k ∈ ℕ.

By (1.2) and the definition of f_{k}, it is clear that

|ü_{k}(t)| ≤ |f_{k}(t)| + |V_{u}(t, u_{k}(t))| = |f(t)| + |V_{u}(t, u_{k}(t))|

for t ∈ [−kT, kT].

Together with (2.7), (A_{1}) and (A_{6}), we obtain there exists M_{3} > 0 independent of k such that

∥ü_{k}∥_{L_{2kT}^{∞}} ≤ M_{3}.  (2.8)

Since for each k ∈ ℕ and t ∈ ℝ there exists ξ ∈ [t − 1, t] such that

u_{k}(t) − u_{k}(t − 1) = u̇_{k}(ξ),

we can see

|u̇_{k}(t)| = |∫_{ξ}^{t} ü_{k}(s) ds + u̇_{k}(ξ)|
≤ ∫_{t−1}^{t} |ü_{k}(s)| ds + |u_{k}(t) − u_{k}(t − 1)|
≤ M_{3} + 2M_{2} := M_{4}.

Thus we have

∥u̇_{k}∥_{L_{2kT}^{∞}} ≤ M_{4}.

Therefore

|u_{k}(t) − u_{k}(t′)| = |∫_{t′}^{t} u̇_{k}(s) ds| ≤ ∫_{t′}^{t} |u̇_{k}(s)| ds ≤ M_{4}|t − t′|.

That is {u_{k}}_{k∈ℕ} are equicontinuous.

Analogously, {u̇_{k}}_{k∈ℕ} are also equicontinuous.

By the Arzela-Ascoli Theorem, there is a subsequence of {u_{k}}, still denoted by {u_{k}}, which converges to a C^{1} function u_{0} in C_{loc}^{1}(ℝ, ℝ^{n}).

Now we are coming to the point of proving Theorem 1. We need the following results from [3].

Proposition 2 Let u : ℝ → ℝ^{n} be a continuous mapping. If a weak derivative u̇ : ℝ → ℝ^{n} is continuous at t_{0}, then u is differential at t_{0} and

lim_{t→t_{0}} \frac{u(t) − u(t_{0})}{t − t_{0}} = u̇(t_{0}).

Proposition 3 Let u : ℝ → ℝ^{n} be a continuous mapping such that u̇ ∈ L_{loc}^{2}(ℝ, ℝ^{n}) (the space of functions on ℝ with values in ℝ^{n} locally square integrable). For every t ∈ ℝ the following inequality holds:

|u(t)| ≤ \sqrt{2} (∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} (|u(s)|^{2} + |u̇(s)|^{2}) ds)^{\frac{1}{2}}  (2.9)

Proof of Theorem 1 We prove u_{0} is exactly our desired homoclinic solution of (1.1). Arguing just as Lemma 2.9 in [3], for each k ∈ ℕ and t ∈ ℝ, u_{k} satisfies

ü_{k}(t) = f_{k}(t) − V_{u}(t, u_{k}(t)).  (2.10)

Since u_{k} → u_{0} and f_{k} → f almost uniformly on ℝ, we obtain

ü_{k} → f(t) − V_{u}(t, u_{0}(t)).

For any finite interval [a, b], there is k_{0} ∈ ℕ such that for all k ≥ k_{0} and t ∈ [a, b], (2.10) becomes

ü_{k}(t) = f(t) − V_{u}(t, u_{k}(t)).

So ü_{k}(t) is continuous on [a, b] for each k ≥ k_{0}. By Proposition 2, ü_{k}(t) is a derivative of u̇_{k}(t) in (a, b) for each k ≥ k_{0}.

Combining ü_{k} → f(t) − V_{u}(t, u_{0}(t)) and u̇_{k} → u̇_{0} almost uniformly on ℝ, we obtain

ü_{0}(t) = f(t) − V_{u}(t, u_{0}(t))

in (a, b) and then in ℝ. So u_{0} satisfies (1.1).

We now prove u_{0}(t) → 0 as t → ±∞. Obviously, for each i ∈ ℕ there is k_{i} ∈ ℕ such that for all k ≥ k_{i},

∫_{−iT}^{iT} (|u_{k}(t)|^{2} + |u̇_{k}(t)|^{2}) dt ≤ ∥u_{k}∥_{E_{k}}^{2} ≤ M_{1}^{2}.

Letting k → +∞, we obtain

∫_{−iT}^{iT} (|u_{0}(t)|^{2} + |u̇_{0}(t)|^{2}) dt ≤ M_{1}^{2}.

As i → ∞, we have

∫_{−∞}^{+∞} (|u_{0}(t)|^{2} + |u̇_{0}(t)|^{2}) dt ≤ M_{1}^{2}.

Hence we get

∫_{|t|≥ρ} (|u_{0}(t)|^{2} + |u̇_{0}(t)|^{2}) dt → 0  (2.11)

as ρ → +∞.

By Proposition 3 and (2.11), for all t > ρ + \frac{1}{2} we have

|u_{0}(t)| ≤ \sqrt{2} (∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} (|u_{0}(s)|^{2} + |u̇_{0}(s)|^{2}) ds)^{\frac{1}{2}}
≤ \sqrt{2} (∫_{ρ}^{+∞} (|u_{0}(s)|^{2} + |u̇_{0}(s)|^{2}) ds)^{\frac{1}{2}}
≤ \sqrt{2} (∫_{|t|≥ρ} (|u_{0}(s)|^{2} + |u̇_{0}(s)|^{2}) ds)^{\frac{1}{2}} .

Therefore, as ρ → +∞, t → +∞, we get u_{0}(t) → 0. Using the same method, we can obtain u_{0}(t) → 0 as t → −∞.

In the following we prove that u̇_{0}(t) → 0 as t → ±∞. Since u_{0}(s) → 0 as s → ±∞, and ∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} |f(s)|^{2} ds → 0 as t → ±∞, by (A_{1}), we have

∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} |ü_{0}(s)|^{2} ds = ∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} (|V_{u}(s, u_{0}(s))|^{2} + |f(s)|^{2}) ds
−2 ∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} (V_{u}(s, u_{0}(s)), f(s)) ds → 0  (2.12)

as t → ±∞.

From Proposition 3, we get

|u̇_{0}(t)|^{2} ≤ 2 ∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} (|u_{0}(s)|^{2} + |u̇_{0}(s)|^{2}) ds + 2 ∫_{t− \frac{1}{2}}^{t+ \frac{1}{2}} |ü_{0}(s)|^{2} ds.

Together with (2.11) and (2.12), we obtain u̇(t) → 0 as t → ±∞. Since f is nonzero, we know u_{0} ≢ 0. So u_{0} is a homoclinic solution of (1.1).

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(Received November 8, 2009)