ALMOST PERIODIC PROCESSES AND THE EXISTENCE OF ALMOST PERIODIC SOLUTIONS

YOSHIYUKI HINO ^{∗} and SATORU MURAKAMI ^{†}

Department of Mathematics and Informatics, Chiba University 1-33 Yayoicho, Inageku, Chiba 263, Japan

and

Department of Applied Mathematics, Okayama University of Science 1-1 Ridai-cho, Okayama 700, Japan

1. INTRODUCTION

There are many results on the existence of almost periodic solutions for almost periodic systems (cf. [6] and its references). In the methodology, there are three types. One is a separation condition, the others are stability conditions and the existence of some Liapunov function.

To discuss the evolution operator, the concept of processes introduced by Dafermos [1,2] is a usefull tool. In fact, Dafermos [2] has given the existence of almost periodic solutions for almost periodic evolution equations under the separation condition by using general theories for almost periodic processes.

In this paper, we also give existence theorems of almost periodic solutions for some almost periodic evolution equations. Our method is based on the stability property. In Section 2, we shall give some definitions of stabilities that are discussed in this paper. In Section 3, we introduce some equivalent concept, which is called Property (A), to an asymptotically almost periodic integral of almost periodic processes (Theorem 1). Furthermore, we shall show that the existence of an asymptotically almost periodic integral of an almost periodic process implies the existence of an almost periodic integral (Theorem 2) and that uniform asymptotic stability of an almost periodic integral has Property (A) (Theorem 3). In Section 4, we shall give some equivalence relation with respect to the separation condition and some stability property (Theorem 5). In Section 5, we shall give an example as an illustration.

∗Partly supported in part by Grant-in-Aid for Scientific Research (C), No.09640156, Japanese Ministry of Education, Science, Sports and Culture.

†Partly supported in part by Grant-in-Aid for Scientific Research (C), No.09640235, Japanese Ministry of Education, Science, Sports and Culture.

2. PROCESSES AND DEFINITIONS OF STABILITIES

In this section, we shall give the concepts of processes and stability properties for processes. Suppose that X is a separable metric space with metric d and let w : R^{+} × R × X ↦ X , R^{+} := [0, ∞) and R := (−∞, ∞), be a function satisfying the following properties for all t, τ ∈ R^{+}, s ∈ R and x ∈ X:

(p1) w(0, s, x) = x.

(p2) w(t + τ, s, x) = w(t, τ + s, w(τ, s, x)).

(p3) the mapping w : R^{+} × R × X ↦ X is continuous.

We call the mapping w a process on X. Denote by W the set of all processes on X. For τ ∈ R and w ∈ W, we define the translation σ(τ)w of w by

(σ(τ)w)(t, s, x) = w(t, τ + s, x), (t, s, x) ∈ R^{+} × R × X ,

and set γ_{σ}(w) = ⋃_{t∈R} σ(t)w. Clearly γ_{σ}(w) ⊂ W. We denote by H_{σ}(w) all functions v : R^{+} × R × X ↦ X such that for some sequence {τ_{n}} ⊂ R, {σ(τ_{n})w} converges to v pointwise on R^{+} × R × X, that is, lim_{n→∞}(σ(τ_{n})w)(t, s, x) = v(t, s, x) for any (t, s, x) ∈ R^{+} × R × X. The set H_{σ}(w) is considered as a topological space and it is called the hull of w.

Consider a process w on X satisfying

(p4) H_{σ}(w) ⊂ W.

Clearly, H_{σ}(w) is invariant with respect to the tanslation σ(τ), τ ∈ R. Now we suppose that H_{σ}(w) is sequentially compact. Let Ω_{σ}(w) be the ω-limit set of w with respect to the translation semigroup σ(t).

A continuous function μ : R^{+} ↦ X is called an integral on R^{+} of the process w, if w(t, s, μ(s)) = μ(t + s) for all t, s ∈ R^{+} (cf. [3, p.80]). In the following, we suppose that there exists an integral μ on R^{+} of the process w such that the set O^{+}(μ) = {μ(t) : t ∈ R^{+}} is relatively compact in X. For any t ∈ R^{+}, we consider a function π(t) : X × H_{σ}(w) ↦ X × H_{σ}(w) defined by

π(t)(x, v) = (v(t, 0, x), σ(t)v)

for (x, v) ∈ X × H_{σ}(w). π(t) is called the skew product flow of the process w, if the following property holds true:

(p5) π(t)(x, v) is continuous in (t, x, v) ∈ R^{+} × X × H_{σ}(w).

From (p5) we see that π(δ)(μ(s), σ(s)w) = (w(δ, s, μ(s)), σ(s+δ)w) = (μ(s+δ), σ(s+ δ)w) tends to (μ(s), σ(s)w) as δ → 0^{+}, uniformly for s ∈ R^{+}; consequently, the integral μ on R^{+} must be uniformly continuous on R^{+}. From Ascoli-Arzéla’s theorem and the sequential compactness of H_{σ}(w), it follows that for any sequence {τ′_{n}} ⊂ R^{+}, there exist a subsequence {τ_{n}} of {τ′_{n}}, a v ∈ H_{σ}(w) and a function ν : R^{+} ↦ X such that lim_{n→∞} σ(τ_{n})w = v and lim_{n→∞} μ(t + τ_{n}) = ν(t) uniformly on any compact interval in R^{+}. In this case, we write as

(μ^{τ_{n}}, σ(τ_{n})w) → (ν, v) compactly on R^{+},

for simplicity. Denote by H(μ, w) the set of all (ν, v) such that (μ^{τ_{n}}, σ(τ_{n})w) → (ν, v) compactly on R^{+} for some sequence {τ_{n}} ⊂ R^{+}. Clearly, ν is an integral on R^{+} of v for any (ν, v) ∈ H(μ, w). Likewise, for any sequence {τ′_{n}} ⊂ R^{+} with τ′_{n} → ∞ as n → ∞, there exist a subsequence {τ_{n}} of {τ′_{n}}, a v ∈ Ω_{σ}(w) and a function ν : R → X such that lim_{n→∞} σ(τ_{n})w = v and lim_{n→∞} μ(t + τ_{n}) = ν(t) uniformly on any compact interval in R. In this case, we write as

(μ^{τ_{n}}, σ(τ_{n})w) → (ν, v) compactly on R,

for simplicity. Denote by Ω(μ, w) the set of all (ν, v) such that (μ^{τ_{n}}, σ(τ_{n})w) → (ν, v) compactly on R for some sequence {τ_{n}} ⊂ R^{+} with τ_{n} → ∞ as n → ∞. When (ν, v) ∈ Ω(μ, w), we often write ν ∈ Ω_{v}(μ). In this case, ν is an integral on R of v, that is, ν satisfies the relation v(t, s, ν(s)) = ν(t + s) for all t ∈ R^{+} and s ∈ R.

For any x_{0} ∈ X and ε > 0, we set V_{ε}(x_{0}) = {x ∈ X : d(x, x_{0}) < ε}. We shall give the definition of stabilities for the integral μ of the process w.

Definition 1 The integral μ : R^{+} ↦ X of the process w is said to be:

(i) uniformly stable (US) (resp. uniformly stable in Ω_{σ}(w)) if for any ε > 0, there exists a δ := δ(ε) > 0 such that w(t, s, V_{δ}(μ(s))) ⊂ V_{ε}(μ(t + s)) for (t, s) ∈ R^{+} × R^{+} (resp. v(t, s, V_{δ}(ν(s))) ⊂ V_{ε}(ν(t + s)) for (ν, v) ∈ Ω(μ, w) and (t, s) ∈ R^{+} × R^{+});

(ii) uniformly asymptotically stable (UAS) (resp. uniformly asymptotically stable in Ω_{σ}(w)), if it is US (resp. US in Ω_{σ}(w)) and there exists a δ_{0} > 0 with the property that for any ε > 0, there is a t_{0} > 0 such that w(t, s, V_{δ_{0}}(μ(s))) ⊂ V_{ε}(μ(t+s)) for t ≥ t_{0}, s ∈ R^{+} (resp. v(t, s, V_{δ_{0}}(ν(s))) ⊂ V_{ε}(ν(t + s)) for (ν, v) ∈ Ω(μ, w) and t ≥ t_{0}, s ∈ R^{+}).

3. ALMOST PERIODIC INTEGRALS FOR ALMOST PERIODIC PROCESSES

In this section, we shall discuss an existence theorem for an almost periodic integral of almost periodic processes.

A process w : R^{+} × R × X ↦ X is said to be almost periodic if w(t, s, x) is almost periodic in s uniformly with respect to t, x in bounded sets. Let w be an almost periodic process on X. Bochner’s theorem implies that Ω_{σ}(w) = H_{σ}(w) is a minimal set. Also, for any v ∈ H_{σ}(w), there exists a sequence {τ_{n}} ⊂ R^{+} such that w(t, s + τ_{n}, x) → v(t, s, x) as n → ∞, uniformly in s ∈ R and (t, x) in bounded sets of R^{+} × X. Consider a metric ρ on H_{σ}(w) defined by

ρ(u, v) = ∑_{n=0}^{∞} \frac{1}{2^{n}} \frac{ρ_{n}(u, v)}{1 + ρ_{n}(u, v)}

with ρ_{n}(u, v) = sup{|u(t, s, x) − v(t, s, x)| : 0 ≤ t ≤ n, s ∈ R, d(x, x_{0}) ≤ n}, where x_{0} is a fixed element in X. Then v ∈ H_{σ}(w) means that ρ(σ(τ_{n})w, v) → 0 as n → ∞, for some sequence {τ_{n}} ⊂ R^{+}.

Definition 2 An integral μ(t) on R^{+} is said to be asymptotically almost periodic if it is a sum of a continuous almost periodic function ϕ(t) and a continuous function ψ(t) defined on R^{+} which tends to zero as t → ∞, that is

μ(t) = ϕ(t) + ψ(t).

Let μ(t) be an integral on R^{+} such that the set O^{+}(μ) is relatively compact in X. As noted in [5], μ(t) is asymptotically almost periodic if and only if it satisfies the following property:

(L) For any sequence {t′_{n}} such that t′_{n} → ∞ as n → ∞ there exists a subsequence {t_{n}} of {t′_{n}} for which μ(t + t_{n}) converges uniformly on R^{+}.

Now, for an integral μ on R^{+} of the almost periodic process w we consider the following property:

(A) For any ε > 0, there exists a δ(ε) > 0 such that ν(t) ∈ V_{ε}(μ(t + τ)), for all t ≥ 0, whenever (ν, v) ∈ Ω(μ, w), ν(0) ∈ V_{δ(ε)}(μ(τ)), and ρ(σ(τ)w, v) < δ(ε) for some τ ≥ 0.

Theorem 1 Assume that μ(t) is an integral on R^{+} of the almost periodic process w such that the set O^{+}(μ) = {μ(t) : t ∈ R^{+}} is relatively compact in X. Then μ(t) is asymptotically almost periodic if and only if it has Property (A).

Proof Assume that μ(t) has Property (A), and let {t′_{n}} be any sequence such that t′_{n} → ∞ as n → ∞. Then there exist a subsequence {t_{n}} of {t′_{n}} and a (ν, v) ∈ Ω(μ, w) such that (μ^{t_{n}}, σ(t_{n})w) → (ν, v) compactly on R. For any ε > 0, there exists an n_{0}(ε) > 0 such that if n ≥ n_{0}(ε), then ν(0) ∈ V_{δ(ε)}(μ(t_{n})) and ρ(σ(t_{n})w, v) < δ(ε), where δ(ε) is the one for Property (A), which implies

ν(t) ∈ V_{ε}(μ(t + t_{n})) for t ≥ 0.

Thus μ(t) satisfies Property (L) and hence it is asymptotically almost periodic.

Next, suppose that μ(t) is asymptotically almost periodic, but does not have Property (A). Then there exists an ε > 0 and sequences (ν^{n}, v^{n}) ∈ Ω(μ, w), τ_{n} ≥ 0 and t_{n} > 0 such that

ν^{n}(t_{n}) ∈ ∂V_{ε}(μ(t_{n} + τ_{n})),  (1)
ν^{n}(0) ∈ V_{1/n}(μ(τ_{n}))  (2)

and

ρ(v^{n}, σ(τ_{n})w) < \frac{1}{n},  (3)

where ∂V_{ε} is the boundary of V_{ε}. We can assume that for a function ν(t)

d(ν^{n}(t), ν(t)) → 0 uniformly on R as n → ∞,  (4)

because ν^{n} ∈ Ω(μ) and μ(t) is asymptotically almost periodic.

First, we shall show that τ_{n} → ∞ as n → ∞. Suppose not. Then we may assume that for a constant τ ≥ 0, τ_{n} → τ as n → ∞. Since

ρ(σ(τ)w, v^{n}) ≤ ρ(σ(τ)w, σ(τ_{n})w) + ρ(σ(τ_{n})w, v^{n}),

we have by (3)

ρ(σ(τ)w, v^{n}) → 0 as n → ∞.

Here we note that w(t, s+τ, ν(s)) = ν(s+t), because w(t, s+τ, ν(s)) = (σ(τ)w)(t, s, ν(s)) = lim_{n→∞} v^{n}(t, s, ν^{n}(s)) = lim_{n→∞} ν^{n}(t + s) = ν(t + s). Since

d(μ(τ), ν(0)) ≤ d(μ(τ), μ(τ_{n})) + d(μ(τ_{n}), ν^{n}(0)) + d(ν^{n}(0), ν(0))

and

d(μ(τ), μ(τ_{n})) → 0 as n → ∞,

it follows from (2) and (4) that d(μ(τ), ν(0)) = 0. Then μ(t + τ) = w(t, τ, μ(τ)) = w(t, τ, ν(0)) = ν(t) for all t ∈ R^{+}. In particular,

d(μ(t_{n} + τ), ν(t_{n})) = 0 for all n.  (5)

On the other hand, for sufficiently large n we get

d(μ(t_{n} + τ), ν(t_{n})) ≥ d(μ(t_{n} + τ_{n}), ν^{n}(t_{n})) − d(μ(t_{n} + τ_{n}), μ(t_{n} + τ))
−d(ν^{n}(t_{n}), ν(t_{n}))
≥ ε − \frac{ε}{4} − \frac{ε}{4}

by (1), (4) and the uniform continuity of μ(t) on R^{+}. This contradicts (5).

By virtue of the sequential compactness of H_{σ}(w) and the asymptotic almost periodicity of μ(t), we can assume that

(μ^{τ_{n}}, σ(τ_{n})w) → (η, v) compactly on R  (6)

for some (η, v) ∈ Ω(μ, w). Since

d(η(0), ν(0)) ≤ d(η(0), μ(τ_{n})) + d(μ(τ_{n}), ν^{n}(0)) + d(ν^{n}(0), ν(0)),

(2), (4) and (6) imply η(0) = ν(0), and therefore η(t) = v(t, 0, η(0)) = v(t, 0, ν(0)) = lim_{n→∞}(σ(τ_{n})w)(t, 0, ν^{n}(0)) = lim_{n→∞} v^{n}(t, 0, ν^{n}(0)) = lim_{n→∞} ν^{n}(t) = ν(t) by (3), (4) and (6). Hence we have

d(μ(τ_{n} + t), ν^{n}(t)) ≤ d(μ(τ_{n} + t), η(t)) + d(ν(t), ν^{n}(t))
< ε

for all sufficiently large n, which contradicts (1). This completes the proof of Theorem 1.

Theorem 2 If the integral μ(t) on R^{+} of the almost periodic process w is asymptotically almost periodic, then there exists an almost periodic integral of the process w.

Proof Since w(t, s, x) is an almost periodic process, there exists a sequence {t_{n}}, t_{n} → ∞ as n → ∞, such that (σ(t_{n})w)(t, s, x) → w(t, s, x) as n → ∞ uniformly with respect to t, x in bounded sets and s ∈ R. The integral μ(t) has the decomposition μ(t) = ϕ(t) + ψ(t), where ϕ(t) is almost periodic and ψ(t) → 0 as t → ∞. Hence we may assume ϕ(t + t_{n}) → ϕ^{∗}(t) as n → ∞ uniformly on t ∈ R, where we note ϕ^{∗}(t) is almost periodic. Since (ϕ^{∗}, w) ∈ Ω(μ, w), ϕ^{∗}(t) is an almost periodic integral of w.

Lemma 1 Let T > 0. Then for any ε > 0, there exists a δ(ε) > 0 with the property that d(μ(s), ϕ(0)) < δ(ε) and ρ(σ(s)w, v) < δ(ε) imply ϕ(t) ∈ V_{ε}(μ(s + t)) for t ∈ [0, T], whenever s ∈ R^{+} and (ϕ, v) ∈ Ω(μ, w).

Proof Suppose the contrary. Then, for some ε > 0 there exists sequences {s_{n}}, s_{n} ∈ R^{+}, {τ_{n}}, 0 < τ_{n} < T, and (ϕ^{n}, v^{n}) ∈ Ω(μ, w) such that

ρ(σ(s_{n})w, v^{n}) < \frac{1}{n},

ϕ^{n}(0) ∈ V_{1/n}(μ(s_{n})),

ϕ^{n}(t) ∈ V_{ε}(μ(s_{n} + t)) for t ∈ [0, τ_{n})

and

ϕ^{n}(τ_{n}) ∈ ∂V_{ε}(μ(s_{n} + τ_{n})).

Since τ_{n} ∈ [0, T], we can assume that τ_{n} converges to a τ ∈ [0, T] as n → ∞. Since Ω(μ, w) is compact, we may assume (ϕ^{n}, v^{n}) → (ϕ, v) ∈ Ω(μ, w) and (μ^{s_{n}}, σ(s_{n})w) → (η, v) ∈ Ω(μ, w) as n → ∞, respectively. Then ϕ(τ) ∈ ∂V_{ε}(η(τ)). On the other hand, since ϕ(0) = η(0), we get ϕ(t) = η(t) on R^{+} by (p2). This is a contradiction.

Lemma 2 Suppose that w is an almost periodic process on X. If the integral μ(t) on R^{+} is UAS, then it is UAS in Ω_{σ}(w).

Proof Let τ_{k} → ∞ as k → ∞ and (μ^{τ_{k}}, σ(τ_{k})w) → (ν, v) ∈ Ω_{σ}(μ, w) compactly on R. Let any σ ∈ R^{+} be fixed. If k is sufficiently large, we get

ν(σ) ∈ V_{δ(ε/2)/2}(μ(τ_{k} + σ)).

Let y ∈ V_{δ(ε/2)/2}(ν(σ)). Then (σ(τ_{k})w)(t, σ, y) = w(t, τ_{k} + σ, y) ∈ V_{ε/2}(μ(t + τ_{k} + σ)) for t ≥ σ, because y ∈ V_{δ(ε/2)}(μ(τ_{k} + σ)). Since (σ(τ_{k})w)(t, σ, y) → v(t, σ, y) and μ(t + τ_{k} + σ) → ν(t + σ), we get v(t, σ, y) ∈ \overline{V_{ε/2}(ν(t + σ))} for all t ≥ σ, which implies that ν(t) is US.

Now we shall show that ν(t) is UAS. Let y ∈ V_{δ_{0}/2}(ν(σ)) and ν(σ) ∈ V_{δ_{0}/2}(μ(τ_{k} + σ)). Since μ(t) is UAS, we have w(t, τ_{k} + σ, y) ∈ V_{ε/2}(μ(t + τ_{k} + σ)) for t ≥ t_{0}(ε/2) because of y ∈ V_{δ_{0}}(μ(τ_{k} + σ)). Hence v(t, σ, y) ∈ \overline{V_{ε/2}(ν(t + σ))} for t ≥ t_{0}(ε/2).

Theorem 3 Suppose that w is an almost periodic process on X, and let μ(t) be an integral on R^{+} of w such that the set O^{+}(μ) is relatively compact in X. If the integral μ(t) is UAS, then it has Property (A). Consequently, it is asymptotically almost periodic.

Proof Suppose that μ(t) has not Property (A). Then there are sequences {t_{n}}, t_{n} ≥ 0, {r_{n}}, r_{n} > 0, (ϕ^{n}, v^{n}) ∈ Ω(μ, w) and a constant δ_{1}, 0 < δ_{1} < δ_{0}/2, such that

ϕ^{n}(0) ∈ V_{1/n}(μ(t_{n})) and ρ(v^{n}, σ(t_{n})w) < \frac{1}{n}  (7)

and

ϕ^{n}(r_{n}) ∈ ∂V_{δ_{1}}(μ(t_{n} + r_{n})) and ϕ^{n}(t) ∈ V_{δ_{1}}(μ(t + t_{n})) on [0, r_{n}),  (8)

where δ_{0} is the one given for the UAS of μ(t). By Lemma 2, μ(t) is UAS in Ω_{σ}(w). Let δ(·) be the one given for US of μ(t) in Ω_{σ}(w). There exists a sequence {q_{n}}, 0 < q_{n} < r_{n}, such that

ϕ^{n}(q_{n}) ∈ ∂V_{δ(δ_{1}/2)/2}(μ(t_{n} + q_{n}))  (9)

and

ϕ^{n}(t) ∈ \overline{V_{δ_{1}}(μ(t + t_{n}))} \ V_{δ(δ_{1}/2)/2}(μ(t + t_{n})) on [q_{n}, r_{n}],  (10)

for a large n by (7) and (8). Suppose that there exists a subsequence of {q_{n}}, which we shall denote by {q_{n}} again, such that q_{n} converges to some q ∈ R^{+}. It follows from (7) that there exists an n_{0} > 0 such that for any n ≥ n_{0}, q + 1 ≥ q_{n} ≥ 0 and ϕ^{n}(t) ∈ V_{δ(δ_{1}/2)/4}(μ(t_{n}+t)) for t ∈ [0, q+1] by Lemma 1, which contradicts (9). Therefore, we can see that q_{n} → ∞ as n → ∞.

Put p_{n} = r_{n} − q_{n} and suppose that p_{n} → ∞ as n → ∞. Set s_{n} = q_{n} + (p_{n}/2). By (7) and the compactness of Ω(μ, w), we may assume that ((ϕ^{n})^{s_{n}}, σ(s_{n})v^{n}) and (μ^{t_{n}+s_{n}}, σ(t_{n}+ s_{n})w) tend to some (ϕ, v), (η, v) ∈ Ω(μ, w) compactly on R as n → ∞, respectively. For any fixed t > 0, one can take an n_{1} > 0 such that for every n ≥ n_{1}, r_{n} − s_{n} = p_{n}/2 > t, because p_{n} → ∞ as n → ∞. Therefore for n ≥ n_{1}, we have q_{n} < t + s_{n} < r_{n}, and

ϕ^{n}(t + s_{n}) ∉ V_{δ(δ_{1}/2)/2}(μ(t + t_{n} + s_{n}))  (11)

by (10). There exists an n_{2} ≥ n_{1} such that for every n ≥ n_{2}

ϕ^{n}(t + s_{n}) ∈ \overline{V_{δ(δ_{1}/2)/8}(ϕ(t))} and η(t) ∈ \overline{V_{δ(δ_{1}/2)/8}(μ(t + t_{n} + s_{n}))}.  (12)

It follows from (11) and (12) that for every n ≥ n_{2},

d(ϕ(t), η(t)) ≥ d(ϕ^{n}(t + s_{n}), μ(t + t_{n} + s_{n})) − d(μ(t + t_{n} + s_{n}), η(t))
−d(ϕ^{n}(t + s_{n}), ϕ(t))
≥ δ(δ_{1}/2)/4.  (13)

However, since η(0) ∈ V_{δ_{0}/2}(ϕ(0)), the UAS of μ(t) in Ω_{σ}(w) implies d(ϕ(t), η(t)) → 0 as t → ∞, which contradicts (13).

Now we may assume that p_{n} converges to some p ∈ R^{+} as n → ∞, and that 0 ≤ p_{n} < p+1 for all n. Moreover, we may assume that ((ϕ^{n})^{q_{n}}, σ(q_{n})v^{n}) and (μ^{t_{n}+q_{n}}, σ(t_{n} +q_{n})w) tend to some (ψ, u), (ν, u) ∈ Ω(μ, w) as n → ∞, respectively. Since d(ψ(0), ν(0)) = δ(δ_{1}/2)/2 by (9), we have ψ(p) ∈ V_{δ_{1}/2}(ν(p)). However, we have a contradiction by (8), because d(ψ(p), ν(p)) ≥ d(μ(t_{n}+r_{n}), ϕ^{n}(r_{n}))−d(ψ(p), ϕ^{n}(q_{n}+p))−d(ϕ^{n}(q_{n}+p_{n}), ϕ^{n}(q_{n}+ p)) − d(ϕ^{n}(q_{n} + p_{n}), ϕ^{n}(r_{n})) − d(μ(t_{n} + r_{n}), ν(p_{n})) − d(ν(p_{n}), ν(p)) ≥ δ_{1}/2 for all large n. Thus the integral μ(t) must have Property (A).

4. SEPARATION CONDITIONS

In this section, we shall establish an existence theorem of almost periodic integrals under a separation condition.

Definition 3 Ω(μ, w) is said to satisfy a separation condition if for any v ∈ Ω_{σ}(w), Ω_{v}(μ) is a finite set and if ϕ and ψ, ϕ, ψ ∈ Ω_{v}(μ), are distinct integrals of v, then there exists a constant λ(v, ϕ, ψ) > 0 such that

d(ϕ(t), ψ(t)) ≥ λ(v, ϕ, ψ) for all t ∈ R.

To make expressions simple, we shall use the following notations. For a sequence {α_{k}}, we shall denote it by α and β ⊂ α means that β is a subsequence of α. For α = {α_{k}} and β = {β_{k}}, α + β will denote the sequence {α_{k} + β_{k}}. Moreover, L_{α}x will denote lim_{k→∞} x(t + α_{k}), whenever α = {α_{k}} and limit exists for each t.

Lemma 3 Suppose that Ω(μ, w) satisfies the separation condition. Then one can choose a number λ_{0} independent of v ∈ Ω_{σ}(w), ϕ and ψ for which d(ϕ(t), ψ(t)) ≥ λ_{0} for all t ∈ R. The number λ_{0} is called the separation constant for Ω(μ, w).

Proof Obviously, we can assume that the number λ(v, ϕ, ψ) is independent of ϕ and ψ. Let v_{1} and v_{2} are in Ω_{σ}(w). Then there exists a sequence r′ = {r′_{k}} such that

v_{2}(t, s, x) = lim_{k→∞}(σ(r′_{k})v_{1})(t, s, x)

uniformly on R^{+} × S for any bounded set S in R^{+} × X, that is, L_{r′}v_{1} = v_{2} uniformly on R^{+} × S for any bounded set S in R × X. Let ϕ^{1}(t) and ϕ^{2}(t) be integrals in Ω_{v_{1}}(μ). There exist a subsequence r ⊂ r′, (ψ^{1}, v_{2}) ∈ H(ϕ^{1}, v_{1}) and (ψ^{2}, v_{2}) ∈ H(ϕ^{2}, v_{1}) such that L_{r}ϕ^{1} = ψ^{1} and L_{r}ϕ^{2} = ψ^{2} in X compactly on R. Since H(ϕ^{i}, v_{1}) ⊂ Ω(μ, w), i = 1, 2, ψ^{1} and ψ^{2} also are in Ω_{v_{2}}(μ). Let ϕ^{1} and ϕ^{2} be distinct integrals. Then

inf_{t∈R} d(ϕ^{1}(t + r_{k}), ϕ^{2}(t + r_{k})) = inf_{t∈R} d(ϕ^{1}(t), ϕ^{2}(t)) = α_{12} > 0,

and hence

inf_{t∈R} d(ψ^{1}(t), ψ^{2}(t)) = β_{12} ≥ α_{12} > 0,  (14)

which means that ψ^{1} and ψ^{2} are distinct integrals of the process v_{2}(t, s, x). Let p_{1} ≥ 1 and p_{2} ≥ 1 be the numbers of distinct integrals of processes v_{1}(t, s, x) and v_{2}(t, s, x), respectively. Clearly, p_{1} ≤ p_{2}. In the same way, we have p_{2} ≤ p_{1} =: p.

Now, let α = min{α_{ik} : i, k = 1, 2, ⋯, p, i ≠ k} and β = min{β_{jm} : j, m = 1, 2, ⋯, p, j ≠ m}. By (14), we have α ≤ β. In the same way, we have α ≥ β. Therfore α = β, and we may set λ_{0} = α = β.

Theorem 4 Assume that μ(t) is an integral on R^{+} of the almost periodic process w such that the set O^{+}(μ) is relatively compact in X, and suppose that Ω(μ, w) satisfies the separation condition. Then μ(t) has Property (A). Consequently, μ(t) is asymptotically almost periodic.

If for any v ∈ Ω_{σ}(w), Ω_{v}(μ) consists of only one element, then Ω(μ, w) clearly satisfies the separation condition. Thus, the following result (cf. [2]) is an immediate consequence of Theorems 2 and 4.

Corollary 1 If for any v ∈ Ω_{σ}(w), Ω_{v}(μ) consists of only one element, then there exists an almost periodic integral of the almost periodic process w.

Proof of Theorem 4 Suppose that μ(t) has not Property (A). Then there exists an ε > 0 and sequences (ϕ^{k}, v^{k}) ∈ Ω(μ, w), τ_{k} ≥ 0 and t_{k} ≥ 0 such that

d(μ(t_{k} + τ_{k}), ϕ^{k}(t_{k})) = ε(< λ_{0}/2),  (15)

d(μ(τ_{k}), ϕ^{k}(0)) = 1/k  (16)

and

ρ(σ(τ_{k})w, v^{k}) = 1/k,  (17)

where λ_{0} is the separation constant for Ω(μ, w).

First, we shall show that t_{k} + τ_{k} → ∞ as k → ∞. Suppose not. Then there exists a subsequence of {τ_{k}}, which we shall denote by {τ_{k}} again, and a constant τ ≥ 0 and that τ_{k} → τ as k → ∞. Since

ρ(σ(τ)w, v^{k}) ≤ ρ(σ(τ)w, σ(τ_{k})w) + ρ(σ(τ_{k})w, v^{k}),

(17) implies that

ρ(σ(τ)w, v^{k}) → 0 as k → ∞.  (18)

Moreover, we can assume that

(ϕ^{k}, v^{k}) → (ϕ, v) compactly on R

for some (ϕ, v) ∈ Ω(μ, w). It follows from (18) that v = σ(τ)w. Since

d(μ(τ), ϕ(0)) ≤ d(μ(τ), μ(τ_{k})) + d(μ(τ_{k}), ϕ^{k}(0)) + d(ϕ^{k}(0), ϕ(0)) → 0

as k → ∞ by (16), we get μ(τ) = ϕ(0), and hence μ(t + τ) = (σ(τ)w)(t, 0, μ(τ)) = (σ(τ)w)(t, 0, ϕ(0)) = v(t, 0, ϕ(0)) = ϕ(t). However, we have d(μ(t_{k} + τ_{k}), ϕ(t_{k})) ≥ ε/2 for a sufficiently large k by (15), because

d(μ(t_{k} + τ_{k}), ϕ(t_{k})) ≥ d(μ(t_{k} + τ_{k}), ϕ^{k}(t_{k})) − d(ϕ^{k}(t_{k}), ϕ(t_{k})).

This is a contradiction. Thus we must have t_{k} + τ_{k} → ∞ as k → ∞.

Now, set q_{k} = t_{k} + τ_{k} and ν^{k}(t) = ϕ^{k}(t_{k} + t). Then

(μ^{q_{k}}, σ(q_{k})w) ∈ H(μ, w) and (ν^{k}, σ(t_{k})v^{k}) ∈ Ω(μ, w),

respectively. We may assume that (μ^{q_{k}}, σ(q_{k})w) → (μ̄, w̄) compactly on R for some (μ̄, w̄) ∈ Ω(μ, w), because q_{k} → ∞ as k → ∞. Since

ρ(w̄, σ(t_{k})v^{k}) ≤ ρ(w̄, σ(q_{k})w) + ρ(σ(q_{k})w, σ(t_{k})v^{k})
= ρ(w̄, σ(q_{k})w) + ρ(σ(τ_{k})w, v^{k}),

we see that ρ(w̄, σ(t_{k})v^{k}) → 0 as k → ∞ by (17). Hence, we can choose a subsequence {ν^{k_{j}}} of {ν^{k}} and a ν̄ ∈ Ωw̄(μ) such that

(ν^{k_{j}}, σ(t_{k_{j}})v^{k_{j}}) → (ν̄, w̄) compactly on R.

Since

lim_{j→∞}{d(μ(t_{k_{j}} + τ_{k_{j}}), ϕ^{k_{j}}(t_{k_{j}})) − d(ν^{k_{j}}(0), ν̄(0)) − d(μ̄(0), μ(q_{k_{j}}))}

≤ d(μ̄(0), ν̄(0))
≤ lim_{j→∞}{d(μ(t_{k_{j}} + τ_{k_{j}}), ϕ^{k_{j}}(t_{k_{j}})) + d(ν^{k_{j}}(0), ν̄(0)) + d(μ̄(0), μ(q_{k_{j}}))},

it follows from (15) that d(μ̄(0), ν̄(0)) = ε, which contradicts the separation condition of Ω(μ, w).

Theorem 5 Assume that μ(t) is an integral on R^{+} of the almost periodic process w such that the set O^{+}(μ) is relatively compact in X. Then the following statements are equivalent:

(i) Ω(μ, w) satisfies the separation condition;

(ii) there exists a number δ_{0} > 0 with the property that for any ε > 0 there exists a t_{0}(ε) > 0 such that d(ϕ(s), ψ(s)) < δ_{0} implies d(ϕ(t), ψ(t)) < ε for t ≥ s + t_{0}(ε), whenever s ∈ R, v ∈ Ω_{σ}(w) and ϕ, ψ ∈ Ω_{v}(μ).

Consequently, the UAS of the integral μ(t) on R^{+} implies the separation condition on Ω(μ, w).

Proof If we set δ_{0} = λ_{0}, then (i) clearly implies (ii).

We shall show that (ii) implies (i). First of all, we shall verify that any distinct integrals ϕ(t), ψ(t) in Ω_{v}(μ), v ∈ Ω_{σ}(w), satisfy

lim inf_{t→−∞} d(ϕ(t), ψ(t)) ≥ δ_{0}.  (19)

Suppose not. Then for some v ∈ Ω_{σ}(w), there exists two distinct integrals ϕ(t) and ψ(t) in Ω_{v}(μ) which satisfy

lim inf_{t→−∞} d(ϕ(t), ψ(t)) < δ_{0}.  (20)

Since ϕ(t) and ψ(t) are distinct integrals, we have d(ϕ(s), ψ(s)) = ε at some s and for some ε > 0. Then there is a t_{1} such that t_{1} < s − t_{0}(ε/2) and d(ϕ(t_{1}), ψ(t_{1})) < δ_{0} by (20). Then d(ϕ(s), ψ(s)) < ε/2, which contradicts d(ϕ(s), ψ(s)) = ε. Thus we have (19). Since \overline{O^{+}(μ)} is compact, there are a finite number of coverings which consists of m_{0} balls with diameter δ_{0}/4. We shall show that the number of integrals in Ω_{v}(μ) is at most m_{0}. Suppose not. Then there are m_{0} + 1 integrals in Ω_{v}(μ), ϕ^{j}(t), j = 1, 2, ⋯, m_{0} + 1, and a t_{2} such that

d(ϕ^{j}(t_{2}), ϕ^{i}(t_{2})) ≥ δ_{0}/2 for i ≠ j,  (21)

by (19). Since ϕ^{j}(t_{2}), j = 1, 2, ⋯, m_{0} + 1, are in \overline{O^{+}(μ)}, some of these integrals, say ϕ^{i}(t), ϕ^{j}(t)(i ≠ j), are in one ball at time t_{2}, and hence d(ϕ^{j}(t_{2}), ϕ^{i}(t_{2})) < δ_{0}/4, which contradicts (21). Therefore the number of integrals in Ω_{v}(μ) is m ≤ m_{0}. Thus

Ω_{v}(μ) = {ϕ^{1}(t), ϕ^{2}(t), ⋯, ϕ^{m}(t)}  (22)

and

lim inf_{t→−∞} d(ϕ^{j}(t), ϕ^{i}(t)) ≥ δ_{0}, i ≠ j.  (23)

Consider a sequence {τ_{k}} such that τ_{k} → −∞ as k → ∞ and ρ(σ(τ_{k})v, v) → 0 as k → ∞. For each j = 1, 2, ⋯, m, set ϕ^{j,k}(t) = ϕ^{j}(t+τ_{k}). Since (ϕ^{j,k}, σ(τ_{k})v) ∈ H(ϕ^{j}, v), we can assume that

(ϕ^{j,k}, σ(τ_{k})v) → (ψ^{j}, v) compactly on R

for some (ψ^{j}, v) ∈ H(ϕ^{j}, v) ⊂ Ω(μ, w). Then it follows from (23) that

d(ψ^{j}(t), ψ^{i}(t)) ≥ δ_{0}, for all t ∈ R and i ≠ j.  (24)

Since the number of integrals in Ω_{v}(μ) is m, Ω_{v}(μ) consists of ψ^{1}(t), ψ^{2}(t), ⋯, ψ^{m}(t) and we have (24). This shows that Ω(μ, w) satisfies the separation condition.

5. APPLICATION

Consider the equation

\{ \begin{matrix}u_{tt} = & u_{xx} − u_{t} + f(t, x, u), t > 0, 0 < x < 1, \\ u(0, t) = & u(1, t) = 0, t > 0, \end{matrix}  (25)

where f(t, x, u) is continuous in (t, x, u) ∈ R × (0, 1) × R and it is an almost periodic function in t uniformly with respect to x and u which satisfies

|f(t, x, u)| ≤ \frac{1}{6}|u| + J

and

|f(t, x, u_{1}) − f(t, x, u_{2})| ≤ \frac{1}{6}|u_{1} − u_{2}|

for all (t, x, u), (t, x, u_{1}), (t, x, u_{2}) ∈ R × (0, 1) × R and some constant J > 0. We consider a Banach space X given by X = H_{0}^{1}(0, 1) × L^{2}(0, 1) equipped with the norm ∥(u, v)∥ = {∥u_{x}∥_{L^{2}}^{2} + ∥v∥_{L^{2}}^{2}}^{1/2} = {∫_{0}^{1}(u_{x}^{2} + v^{2})dx}^{1/2}. Then (25) can be considered as an abstract equation

\frac{d}{dt}\binom{u}{v} = A\binom{u}{v} + \binom{0}{f(t, x, u)}  (26)

in X, where A is a (unbounded) linear operator in X defined by

A\binom{u}{v} = \binom{v}{u_{xx} − v}

for (u, v) ∈ H^{2}(0, 1) × H_{0}^{1}(0, 1). It is well known that A generates a C_{0} -semigroup of bounded linear operator on X. In the following, we show that each (mild) solution of (26) is bounded in the future, and that it is UAS. Moreover, we show that each solution of (26) has a compact orbit in X. Consequently, one can apply Theorem 3 or Theorem 5 to the process generated by solutions of (26) and its integrals to conclude that (25) has an almost periodic solution which is UAS.

Now, for any solution \binom{u(t)}{v(t)} of (26), we consider a function V (t) defined by

V (t) = ∫_{0}^{1}(u_{x}^{2} + 2λuv + v^{2})dx

with λ = \frac{1}{12}. Since ∥u∥_{L^{2}} ≤ ∥u_{x}∥_{L^{2}} for u ∈ H_{0}^{1}(0, 1), we get the inequality

\frac{11}{12}(∥u_{x}∥_{L^{2}}^{2} + ∥v∥_{L^{2}}^{2}) ≤ V (t) ≤ \frac{13}{12}(∥u_{x}∥_{L^{2}}^{2} + ∥v∥_{L^{2}}^{2}).  (27)

Moreover, we get

\frac{d}{dt}V (t) ≤ 2 ∫_{0}^{1}\{(λ − 1)v^{2} − λu_{x}^{2} − λuv + (λ|u| + |v|)(\frac{|u|}{6} + J)\}dx
≤ 2(λ − 1)∥v∥_{L^{2}}^{2} − 2λ∥u_{x}∥_{L^{2}}^{2} + λ(ε_{1}∥u∥_{L^{2}}^{2} + ε_{1}^{−1}∥v∥_{L^{2}}^{2})

+\frac{λ}{3}∥u∥_{L^{2}}^{2} + λ(ε_{2}∥u∥_{L^{2}}^{2} + ε_{2}^{−1}J^{2}) + \frac{1}{6}(ε_{3}∥u∥_{L^{2}}^{2} + ε_{3}^{−1}∥v∥_{L^{2}}^{2})

+(ε_{4}∥v∥_{L^{2}}^{2} + ε_{4}^{−1}J^{2}) for any positive constants ε_{1}, ε_{2}, ε_{3} and ε_{4}. We set ε_{1} = \frac{1}{10}, ε_{2} = \frac{1}{10}, ε_{3} = \frac{1}{2} and ε_{4} = \frac{1}{3} to get

\frac{d}{dt}V (t) ≤ −\frac{1}{3}∥v∥_{L^{2}}^{2} − \frac{7}{180}∥u_{x}∥_{L^{2}}^{2} + 4J^{2}.  (28)

By (27) and (28), we get

\frac{d}{dt}V (t) ≤ −C_{1}V (t) + C_{2}

for some positive constants C_{1} and C_{2}. Then V (t) ≤ e^{−C_{1}(t−t_{0})}V (t_{0}) + \frac{C_{2}}{C_{1}} for all t ≥ t_{0}, and hence sup_{t≥t_{0}} ∥(u(t), v(t))∥_{X} < ∞. Thus each solution of (26) is bounded in the future.

Next, for any solutions \binom{u(t)}{v(t)} and \binom{ū(t)}{v̄(t)} of (26), we consider a function V̄ (t) defined by

V̄ (t) = ∫_{0}^{1}{(u_{x} − ū_{x})^{2} + 2λ(u − ū)(v − v̄) + (v − v̄)^{2}}dx

with λ = \frac{1}{12}. By almost the same arguments as for the function V (t), we get

\frac{11}{12}\{∥u_{x} − ū_{x}∥_{L^{2}}^{2} + ∥v − v̄∥_{L^{2}}^{2}\} ≤ V̄ (t) ≤ \frac{13}{12}\{∥u_{x} − ū_{x}∥_{L^{2}}^{2} + ∥v − v̄∥_{L^{2}}^{2}\}  (29)

and

\frac{d}{dt}V̄ (t) ≤ −C_{3}V̄ (t)  (30)

for some positive constant C_{3}. Then, by (29) and (30) we get

∥u_{x}(t) − ū_{x}(t)∥_{L^{2}}^{2} + ∥v(t) − v̄(t)∥_{L^{2}}^{2} ≤ \frac{11}{12}V (t)
≤ \frac{11}{12}e^{−C_{3}(t−t_{0})}V (t_{0})
≤ e^{−C_{3}(t−t_{0})}{∥u_{x}(t_{0}) − ū_{x}(t_{0})∥_{L^{2}}^{2}
+∥v(t_{0}) − v̄(t_{0})∥_{L^{2}}^{2}}

for any t ≥ t_{0}, which shows that the solution \binom{u(t)}{v(t)} of (26) is UAS.

Finally, we shall show that each bounded solution (u(t), v(t)) of (26) has a compact orbit. To do this, it sufficies to show that each increasing sequence {τ_{n}} ⊂ R^{+} such that τ_{n} → ∞ as n → ∞ has a subsequence {τ′_{n}} such that {u(τ′_{n}), v(τ′_{n})} is a Cauchy sequence in X. Taking a subsequence if necessary, we may assume that {f(t + τ_{n}, x, u)} converges uniformly in (x, u) in bounded sets and t ∈ R. Therefore, for any γ > 0 there exists an n_{0} > 0 such that if n > m ≥ n_{0}, then

sup_{(t,x,ξ)∈R×(0,1)×O^{+}(u)} |f(t + τ_{n}, x, ξ) − f(t + τ_{m}, x, ξ)|

= sup_{(t,x,ξ)∈R×(0,1)×O^{+}(u)} |f(t + τ_{n} − τ_{m}, x, ξ) − f(t, x, ξ)| < γ.

Fix n and m such that n > m ≥ n_{0}, and set u^{τ}(t) = u(t + τ), v^{τ}(t) = v(t + τ) and f^{τ}(t, x, ξ) = f(t + τ, x, ξ) with τ = τ_{n} − τ_{m}. Clearly, (u^{τ}, v^{τ}) is a solution of (26) with f^{τ} instead of f.

Consider a function U defined by

U(t) = ∫_{0}^{1}{(u_{x} − u_{x}^{τ})^{2} + 2λ(u − u^{τ})(v − v^{τ}) + (v − v^{τ})^{2}}dx

with λ = \frac{1}{12}. Then

\frac{11}{12}\{∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2} + ∥v − v^{τ}∥_{L^{2}}^{2}\} ≤ U(t) ≤ \frac{13}{12}\{∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2} + ∥v − v^{τ}∥_{L^{2}}^{2}\}

and

\frac{d}{dt}U(t) ≤ (2λ − 2)∥v − v^{τ}∥_{L^{2}}^{2} − 2λ∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2}
+2λ ∫_{0}^{1} |u − u^{τ}|{|v − v^{τ}| + |f(t, x, u) − f^{τ}(t, x, u^{τ})|}dx
+2 ∫_{0}^{1} |v − v^{τ}||f(t, x, u) − f^{τ}(t, x, u^{τ})|dx.

Since

|f(t, x, u) − f^{τ}(t, x, u^{τ})| ≤ |f(t, x, u) − f(t, x, u^{τ})| + |f(t, x, u^{τ}) − f^{τ}(t, x, u^{τ})|
≤ \frac{1}{6}|u − u^{τ}| + γ,

we get

\frac{d}{dt}U(t) ≤ −\frac{11}{6}∥v − v^{τ}∥_{L^{2}}^{2} − \frac{1}{6}∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2}

+\frac{1}{12}(\frac{1}{10}∥u − u^{τ}∥_{L^{2}}^{2} + 10∥v − v^{τ}∥_{L^{2}}^{2}) + \frac{1}{36}∥u − u^{τ}∥_{L^{2}}^{2}

+\frac{1}{12}(\frac{1}{2}∥u − u^{τ}∥_{L^{2}}^{2} + 2γ^{2})

+\frac{1}{6}(ε_{5}∥v − v^{τ}∥_{L^{2}}^{2} + \frac{1}{ε_{5}} ∥u − u^{τ}∥_{L^{2}}^{2}) + (ε_{6}∥v − v^{τ}∥_{L^{2}}^{2} + \frac{1}{ε_{6}} γ^{2})

≤ ∥v − v^{τ}∥_{L^{2}}^{2}(−1 + \frac{ε_{5}}{6} + ε_{6})

+∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2}(−\frac{1}{6} + \frac{1}{120} + \frac{1}{36} + \frac{1}{24} + \frac{1}{36ε_{5}} ) + (\frac{1}{6} + \frac{1}{ε_{6}} )γ^{2}.

Putting ε_{5} = 1 and ε_{6} = \frac{1}{3}, we have

\frac{d}{dt}U(t) ≤ −C(∥u_{x} − u_{x}^{τ}∥_{L^{2}}^{2} + ∥v − v_{τ}∥_{L^{2}}^{2}) + 4γ^{2}
≤ −C_{1}U(t) + 4γ^{2},

where C and C_{1} are some constants independent of γ. Then

U(τ_{m}) ≤ e^{−C_{1}τ_{m}}U(0) + \frac{4γ^{2}}{C_{1}}

or

∥u_{x}(τ_{m}) − u_{x}(τ_{n})∥_{L^{2}}^{2} + ∥v(τ_{m}) − v(τ_{n})∥_{L^{2}}^{2} ≤ Ke^{−C_{1}τ_{m}} + \frac{8γ^{2}}{C_{1}} ,

where K = 2U(0). Take n_{1} ≥ n_{0} so that Ke^{−C_{1}τ_{n_{1}}} < \frac{γ^{2}}{C_{1}} . Then

∥u_{x}(τ_{m}) − u_{x}(τ_{n})∥_{L^{2}}^{2} + ∥v(τ_{n}) − v(τ_{m})∥_{L^{2}}^{2} < \frac{9}{C_{1}} γ^{2}

if n ≥ m ≥ n_{1}, which shows that {(u(τ_{n}), v(τ_{n}))} is a Cauchy sequence in X, as required.

REFERENCES

[1] C. M. Dafermos, An invariance principle for compact processes, J. Differential Equations 9 (1971), 239–252.

[2] C. M. Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, Proceedings of a University of Florida International Symposium, Academic Press, New York, 1977, 43–57.

[3] J. K. Hale, Theory of Functional Differential Equations, Applied Math. Sciences 3, Springer-Verlag, New York, 1977.

[4] J. K. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, Rhode Island, 1988.

[5] Y. Hino, S. Murakami and T. Yoshizawa, Stability and existence of almost periodic solutions for abstract functional differential equations with infinite delay, Tohoku Math. J. 49 (1997), 133–147.

[6] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer-verlag, New York, Heidelberg, Berlin, 1975.