staff.ustc.edu.cnstaff.ustc.edu.cn/~wangyi/my papers/CGW10.pdf · 2011. 12. 24. · ASYMPTOTIC...

ASYMPTOTIC BEHAVIOR OF COMPARABLE SKEW-PRODUCTSEMIFLOWS WITH APPLICATIONS

FENG CAO, MATS GYLLENBERG, AND YI WANG

Abstract. The (almost) 1-cover lifting property of omega-limit sets is es-

tablished for nonmonotone skew-product semiflows, which are comparable to

uniformly stable eventually strongly monotone skew-product semiflows. These

results are then applied to study the asymptotic behavior of solutions to the

nonmonotone comparable systems of ODEs, reaction-diffusion systems, differ-

ential systems with time delays and semilinear parabolic equations.

1. Introduction

Monotone dynamical systems have been widely studied because these systems

provide a unified relevant mathematical framework for the qualitative analysis of

many important equations, including second-order parabolic equations and various

classes of systems of ordinary, parabolic and functional differential equations. We

refer to [18, 43] for a comprehensive survey on development of this theory.

The path-breaking work by Hirsch [17] showed that trajectories in strongly

monotone systems have a strong tendency not to be chaotic, i.e., almost all of their

omega-limit sets consist of equilibria. For smooth strongly monotone systems, the

forward orbits are generically convergent to equilibria in the continuous-time case or

to cycles in the discrete-time case (see, e.g. [37, 38, 43]). Recently, non-periodic and

non-autonomous equations have been attracting more attention. A unified frame-

work to study nonautonomous equations is based on the so-called skew-product

semiflows (see [39, 41], etc). In contrast to the autonomous and periodic cases, the

generic convergence cannot hold in strongly monotone skew-product semiflows (See

[41]), even for quasi-periodic or almost periodic cases.

During the past 20 years, many researchers in this field have tried to impose

additional conditions to obtain more useful information of the structure of the

2000 Mathematics Subject Classification. 37B55, 37L15, 34K14, 35B15, 35K57.

Y.W. is partially supported by the Finnish Center of Excellence in Analysis and Dynamics,

and NSF of China No.10971208.

1

2 FENG CAO, MATS GYLLENBERG, AND YI WANG

limit sets of the orbits in monotone systems. One of the popular approaches is to

provide reasonable assumptions to guarantee global convergence of the orbits. Such

assumptions include subhomogeneity ([21, 33, 34, 44, 47, 50, 53]); minimal equilibria

([14, 51]); a first integral with positive gradient ([5, 6, 22, 29, 42, 48]), phase-

translation invariance ([4, 25]). It is worth pointing out that all these additional

assumptions make the systems “orbitally stable” in a certain sense (see [1, 2, 24]) .

However, it is well known that a large amount of important evolution equa-

tions do not generate monotone systems. For example, many population models

with non-cooperative growth functions, and reaction-diffusion systems with non-

quasimonotone reaction terms are among such equations. In order to study proper-

ties of the solutions of such non-monotone evolution equations, an effective approach

is to exhibit and utilize certain comparison techniques. Historically, the motivation

for this approach was to obtain upper (maximal) and lower (minimal) solutions to a

given evolution equation (see the pioneering works in [10, 19] for partial differential

equations, and [26, 31] for ordinary differential equations).

A remarkable comparison technique was derived by Conway and Smoller [9]

for nonmonotone reaction-diffusion systems which admit an invariant rectangle.

This then triggered the successful investigation of global dynamical behavior of

the differential equations with spatial structure arising in mathematical ecology

and population biology (see, e.g., [7, 8, 46], etc). As pointed out in [45, Section

4], it turns out that this comparison technique involves monotone systems in a

very natural way: the original nonmonotone system is comparable with respect to

certain monotone systems.

Motivated by such insight, given a nonmonotone system which is comparable

to some monotone system, one would want to know whether such a nonmonotone

system inherits certain asymptotic behavior from its monotone “partner”. The

answer is generally negative even in autonomous cases unless we impose additional

conditions on the associated monotone systems. To the best of our knowledge,

there are only a few works on the related topics, such as contracting rectangle

techniques in systems of reaction-diffusion equations ([7, 46]) and delayed-equations

([28, 43]), pseudo monotone approaches in functional differential equations ([11, 52])

and sandwich methods in integrodifference equations ([20]), etc. Recently, Jiang

[23] discovered the global convergence of the comparable nonmonotone discrete-

time or continuous-time system provided that all the equilibria of its monotone

partner form into a totally ordered curve in the phase space.

COMPARABLE SKEW-PRODUCT SEMIFLOWS 3

The purpose of our current paper is to study the global dynamics of compara-

ble nonmonotone skew-product semiflows (see Definition 4.1) with respect to some

eventually strongly monotone partner. Since even the strongly monotone skew-

product semiflows can possess very complicated chaotic attractors (see [41]), we

henceforth restrict our attention to the monotone partners which are “uniformly

stable”. According to [24], such monotone skew-product semiflows are globally

tamed-behaved: every precompact trajectory is asymptotic to a copy (also called

1-cover) of the base flow. As a starting point in this paper, for the uniformly sta-

ble eventually strongly monotone skew-product semiflows, we will first investigate

the topological structure of the set of the union of all 1-covers (see Theorem 3.1).

Roughly speaking, this set is a 1-dimensional continuous subbundle on the base,

while each fibre of such bundle is totally ordered and homeomorphic to a closed

interval in R. Moreover, all the fibers share a common “bounded-or-unbounded”

property uniformly for all the base point (see Remark 3.2). Armed with such key

tools, we are able to show the (almost) 1-covering property of omega-limit sets of

comparable nonmonotone skew-product semiflows, whose partner systems are even-

tually strongly monotone and uniformly stable (see Theorem 4.3 in detail). Then

these results are applied to study the asymptotic almost periodicity of solutions to

almost periodic reaction-diffusion systems, ODE systems, differential systems with

time delays and time-recurrent parabolic equations.

This paper is organized as follows. In section 2, we present some basic con-

cepts and preliminary results in the theory of skew-product semiflows and almost

periodic functions which will be important to our proofs. Section 3 is devoted to

the establishment of the topological structure of the union of all 1-covers for even-

tually strongly monotone and uniformly stable skew-product semiflows (Theorem

3.1). In Section 4, we prove the lifting property of the ω-limit sets of compara-

ble nonmonotone skew-product semiflows (Theorem 4.3). In section 5, we apply

our abstract theorems in Sections 3 and 4 to obtain the asymptotic almost peri-

odicity of solutions to comparable systems as non-cooperative ordinary differential

systems (Section 5.1), non-quasimonotone reaction-diffusion systems (Section 5.2)

and time-delayed systems (Section 5.3), and comparable almost periodic parabolic

equations (Section 5.4).


2. Preliminaries

In this section, we summarize some preliminary material to be used in later

sections. First, we summarize some lifting properties of compact dynamical sys-

tems. We then collect definitions and basic facts concerning eventually strongly

monotone skew-product semiflows. Finally, we give a brief review about uniformly

almost periodic functions.

Let Ω be a compact metric space with metric dΩ, and σ : Ω × R → Ω be acontinuous flow on Ω, denoted by (Ω, σ) or (Ω,R). As has become customary,

we denote the value of σ at (ω, t) alternatively by σt(ω) or ω · t. By definition,σ0(ω) = ω and σt+s(ω) = σt(σs(ω)) for all t, s ∈ R and ω ∈ Ω. A subset S ⊂ Ω isinvariant if σt(S) = S for every t ∈ R. A non-empty compact invariant set S ⊂ Ωis called minimal if it contains no non-empty, proper and invariant subset. We

say that the continuous flow (Ω,R) is minimal if Ω itself is a minimal set; distal if

inft∈R

dΩ(σt(ω1), σt(ω2)) > 0 whenever ω1, ω2 ∈ Ω and ω1 6= ω2. Let (Z,R) be anothercontinuous flow. A continuous map p : Z → Ω is called a flow homomorphism ifp(z · t) = p(z) · t for all z ∈ Z and t ∈ R. A flow homomorphism which is ontois called a flow epimorphism and a one-to-one flow epimorphism is referred as a

flow isomorphism. We note that a homomorphism of minimal flows is already an

epimorphism.

Throughout this paper, we always assume that the flow (Ω,R) is minimal and

distal.

We say that a Banach space (V, ‖·‖) is strongly ordered if it contains a closedconvex cone, that is, a non-empty closed subset V+ ⊂ V satisfying V+ + V+ ⊂ V+,αV+ ⊂ V+ for all α ≥ 0, and V+ ∩ (−V+) = {0} with non-empty interior IntV+ 6= ∅(also call that V+ is solid). The cone V+ induces a strong ordering on V via

x1 ≤ x2 if x2 − x1 ∈ V+. We write x1 < x2 if x2 − x1 ∈ V+ \ {0}, and x1 ¿ x2if x2 − x1 ∈ IntV+. Given x1, x2 ∈ V , the set [x1, x2] = {x ∈ V : x1 ≤ x ≤ x2} iscalled a closed order interval in V , and we write (x1, x2) = {x ∈ V : x1 < x < x2}.Let x ∈ V and a subset U ⊂ V . We write x


U is called lower-unbounded (resp. upper-unbounded), if it is not lower-bounded

(resp. upper-bounded); U is called bounded if U is both lower-bounded and upper-

bounded, otherwise it is unbounded. Moreover, U is totally unbounded if it is both

lower-unbounded and upper-unbounded.

A lower bound a0 is said to be the greatest lower bound (g.l.b.), if any other lower

bound a satisfies a ≤ a0. Similarly, we can define the least upper bound (l.u.b.).Let X = [a, b]V with a ¿ b (a, b ∈ V ) or X = V+, or furthermore, X be a closed

order convex subset of V . Our first standing hypothesis is

(H1) Every compact subset in X has both a greatest lower bound and a least

upper bound.

Let R+ = {t ∈ R : t ≥ 0}. We consider a continuous skew-product semiflowΠ : Ω×X × R+ → Ω×X defined by

(2.1) Πt(ω, x) = (ω · t, u(t, ω, x)) , ∀(t, ω, x) ∈ R+ × Ω×X,

satisfying (1) Π0 = Id; (2) the cocycle property : u(t+s, ω, x) = u (s, ω · t, u(t, ω, x)),for each (ω, x) ∈ Ω×X and s, t ∈ R+. Our next standing hypothesis is

(H2) For every (ω, x) ∈ Ω×X, there is a t0 = t0(ω, x) > 0 such that {Π(t, ω, x) :t ≥ t0} is precompact.

A subset A ⊂ Ω×X is positively invariant if Πt(A) ⊂ A for all t ∈ R+; and totallyinvariant if Πt(A) = A for all t ∈ R+. The forward orbit of any (ω, x) ∈ Ω×X isdefined by O+(ω, x) = {Πt(ω, x) : t ≥ 0}, and the omega-limit set of (ω, x) is definedbyO(ω, x) = {(ω̂, x̂) ∈ Ω×X : Πtn(ω, x) → (ω̂, x̂)(n →∞) for some sequence tn →∞}. By (H2), every omega-limit set O(ω, x) is a nonempty, compact and totallyinvariant subset in Ω×X for Πt.

Let P : Ω ×X → Ω be the natural projection. A compact positively invariantset K ⊂ Ω×X is called an almost 1-cover (or almost automorphic extension) of Ωif there exists ω0 ∈ Ω such that P−1(ω0) ∩K consists of a unique element. And,K is a 1-cover of the base flow if P−1(ω) ∩K contains a unique element for everyω ∈ Ω. In this case, we denote the unique element of P−1(ω) ∩K by (ω, c(ω)) andwrite K = {(ω, c(ω)) : ω ∈ Ω}, where c : Ω → X is continuous with

Πt(ω, c(ω)) = (ω · t, c(ω · t)), ∀t ≥ 0,


and hence, K ∩ P−1(ω) = {(ω, c(ω))} for every ω ∈ Ω. For the sake of brevity,we hereafter also write c(·) as a 1-cover of Ω and K ∩ P−1(ω) = (ω, c(ω)) in thecontext without any confusion.

Next, we introduce some definitions concerning compactness and stability of the

skew-product semiflow Πt.

Definition 2.1. (1) (Fiber compactness) Πt is fiber-compact if, there exists a t0 > 0

such that for any ω ∈ Ω and bounded subset B ⊂ X, Πt(ω, B) has compact closurein P−1(ω · t) for every t > t0.

(2) (Uniform stability) A forward orbit O+(ω0, x0) of Πt is said to be uniformly

stable if for every ε > 0 there is a δ = δ(ε) > 0, called the modulus of uniform

stability, such that, for every x ∈ X, if s ≥ 0 and ‖u(s, ω0, x0)− u(s, ω0, x)‖ ≤ δ(ε)then

‖u(t + s, ω0, x0)− u(t + s, ω0, x)‖ < ε for each t ≥ 0.

For skew-product semiflows, we always use the order relation on each fiber

P−1(ω). We write (ω, x1) ≤ω ( 0 such that

Πt(ω, x1) ¿ Πt(ω, x2) whenever (ω, x1) < (ω, x2) and t ≥ t0.

Now we present the third Standing Hypothesis:

(H3) The skew-product semiflow Πt : Ω × X → Ω × X is eventually stronglymonotone, and every forward orbit of Πt is uniformly stable.

The following result is adopted from [24] and will play an important role in our

forthcoming sections.

Lemma 2.3. Assume that (H1)-(H3) hold. Then for any (ω0, x0) ∈ Ω × X, itsomega-limit set O(ω0, x0) is a 1-cover of the base flow (Ω,R).


Proof. See Theorem 4.1 in [24]. ¤

We finish this section with the definitions of almost periodic functions and flows.

A function f ∈ C(R,Rn) is almost periodic if, for any ε > 0, the set T (ε) := {τ :|f(t + τ)− f(t)| < ε, ∀t ∈ R} is relatively dense in R. Let D ⊆ Rm be a subset ofRm. A continuous function f : R×D → Rn; (t, u) 7→ f(t, u), is said to be admissibleif f(t, u) is bounded and uniformly continuous on R × K for any compact subsetK ⊂ D. A function f ∈ C(R×D,Rn)(D ⊂ Rm) is uniformly almost periodic in t,if f is both admissible and almost periodic in t ∈ R.

Let f ∈ C(R ×D,Rn)(D ⊂ Rm) be admissible. Then H(f) = cl{f · τ : τ ∈ R}is called the hull of f , where f · τ(t, ·) = f(t + τ, ·) and the closure is taken underthe compact open topology. Moreover, H(f) is compact and metrizable under the

compact open topology. The time translation g · t of g ∈ H(f) induces a naturalflow on H(f). H(f) is always minimal and distal whenever f is a uniformly almost

periodic function in t (see, e.g., [39] or [41]).

Let f ∈ C(R× Rn,Rn) be uniformly almost periodic, and

(2.2) f(t, x) ∼∑

λ∈Raλ(x)eiλt

be a Fourier series of f (see [49, 41] for the definition and the existence of Fourier

series). Then S = {λ : aλ(x) 6≡ 0} is called the Fourier spectrum of f associated tothe Fourier series (2.2), andM(f) = the smallest additive subgroup of R containingS(f) is called the frequency module of f . Moreover, M(f) is a countable subset ofR. Let f, g ∈ C(R× Rn,Rn) be two uniformly almost periodic functions in t. Themodule containment M(f) ⊂ M(g) if and only if there exists a flow epimorphismfrom H(g) to H(f) (see, [12] or [41, Section 1.3.4]). In particular, M(f) = M(g)if and only if the flow (H(g),R) is isomorphic to the flow (H(f),R).

3. Topological structure of the union of all 1-covers for Πt

Throughout this paper, we always assume that the skew-product semiflow Πt

satisfying (H1)-(H3). Therefore, by Lemma 2.3, the omega-limit set O(ω, x) ofevery (ω, x) ∈ Ω×X is a 1-cover (of Ω) for Πt.

Let C(Π) = {c(·) : c(·) is a 1-cover for Πt}. For a(·), b(·) ∈ C(Π), we writea(·)


[a(·), b(·)]C(Π) = {w(·) ∈ C(Π) : a(·) ≤ w(·) ≤ b(·)}, and the 1-cover open order-interval (a(·), b(·))C(Π) = {w(·) ∈ C(Π) : a(·) ¿ w(·) ¿ b(·)}. Due to the eventu-ally strong monotonicity, one can also write (a(·), b(·))C(Π) = {w(·) ∈ C(Π) : a(·) <w(·) < b(·)}.

Consider the union

A =⋃

K is a 1-cover for Πt

K

of all 1-covers (of Ω) for Πt in Ω×X. Then, A is an invariant subset of Ω×X. Foreach ω ∈ Ω, we write A(ω) = P−1(ω) ∩A. It is also easy to see that

A(ω) = {(ω, u)|u = a(ω) for some a(·) ∈ C(Π)}.

Now we are in a position to describe the topological structure of A, which is our

main result in this section.

Theorem 3.1 (Topological structure of union of 1-covers). Assume that (H1)-(H3)

hold and Πt is fiber-compact. Then, for each ω ∈ Ω, A(ω) is totally ordered andclosed in P−1(ω). Moreover, there is a continuous bijective mapping h : Ω×I → A,where the interval I = {0}, [0, 1], [0,+∞), (−∞, 0] or (−∞,+∞), satisfying

(i) For each α ∈ I, h(·, α) = (·, a(·)) for some a(·) ∈ C(Π);(ii) For each ω ∈ Ω, h(ω, I) = A(ω). In addition, if I is nontrivial (i.e., I 6= {0}),

then h is strictly order-preserving with respect to α ∈ I, i.e.,

h(ω, α1) ¿ h(ω, α2)

for any ω ∈ Ω and any α1, α2 ∈ I with α1 < α2;(iii) I = {0} or [0, 1] corresponds to the case that A is compact, in which A(ω)

is bounded for all ω ∈ Ω;(iv) I = [0,+∞) (resp.(−∞, 0]) corresponds to the case that A(ω) is unbounded

but lower-bounded (resp. upper-bounded), for all ω ∈ Ω;(v) I = (−∞,+∞) corresponds to the case that A(ω) is totally unbounded, for

all ω ∈ Ω.

Remark 3.2. Because of items (iii)-(v) of above Theorem, we say that the fibers

A(ω) share the common “bounded-or-unbounded property” uniformly for all the base

point ω ∈ Ω.

In order to prove Theorem 3.1, we need a series of important lemmas.


Lemma 3.3. Assume that (H1)-(H3) hold. Let p(·), q(·) ∈ C(Π) with p(·) < q(·).Then there exists some r(·) ∈ C(Π) such that p(·) < r(·) < q(·). Moreover, one hasp(·) ¿ r(·) ¿ q(·).

Proof. By the eventually strong monotonicity of Πt, one may assume that p(·) ¿q(·). Suppose that there exists no 1-cover of Ω in (p(·), q(·))C(Π). Then we assertthat the following Order-Interval Dichotomy holds: For each ω ∈ Ω, either

(i) O(ω, x) = p(·), ∀x ∈ (p(ω), q(ω)); or(ii) O(ω, x) = q(·), ∀x ∈ (p(ω), q(ω)).

Before giving the proof of the Order-Interval Dichotomy, we note that such

Order-Interval Dichotomy implies that neither the orbit Πt(ω, q(ω)) nor the orbit

Πt(ω, p(ω)) is uniformly stable, which contradicts our fundamental hypothesis (H3).

Thus, there exists some r(·) ∈ C(Π) such that p(·) < r(·) < q(·). Again by theeventually strong monotonicity of Πt, we obtain that p(·) ¿ r(·) ¿ q(·).

So, it remains to prove the Order-Interval Dichotomy. To end this, suppose on

the contrary that one can find an ω0 ∈ Ω such that neither (i) nor (ii) holds. Denoteby L the open line segment with endpoints (ω0, p(ω0)) and (ω0, q(ω0)). Then, by

the choice of ω0 and the monotonicity of Πt, it is easy to see that

neither p(·) nor q(·) can attract the whole L.(3.1)

Recall that there exists no 1-cover of Ω in (p(·), q(·))C(Π). It then follows fromLemma 2.3 that either

O(ω0, x) ∩ P−1(ω0) = (ω0, p(ω0))

or

O(ω0, x) ∩ P−1(ω0) = (ω0, q(ω0))for every (ω0, x) ∈ L. Combining with (3.1), the monotonicity of Πt implies thatthere exists (ω0, e) ∈ L such that

O(ω0, x) ∩ P−1(ω0) = (ω0, p(ω0))

for (ω0, x) ∈ L with x < e, and

O(ω0, x) ∩ P−1(ω0) = (ω0, q(ω0))

for (ω0, x) ∈ L with x > e. Without loss of generality, one may also assume that

O(ω0, e) ∩ P−1(ω0) = (ω0, p(ω0)).


Let ²0 := ‖q(ω0)− p(ω0)‖ > 0. Since Πt(ω0, e) is uniformly stable, there is a δ0 > 0such that

(3.2) ‖u(t, ω0, e∗)− u(t, ω0, e)‖ < ²02whenever t ≥ 0 and ‖e∗ − e‖ < δ0. Let e∗ > e, (ω0, e∗) ∈ L with ‖e∗ − e‖ < δ0 andchoose tn → +∞ such that ω0 · tn → ω0, it follows from (3.2) that

(3.3) ‖u(tn, ω0, e∗)− u(tn, ω0, e)‖ < ²02 .

Let n →∞ in (3.3), we have

²0 = ‖q(ω0)− p(ω0)‖ ≤ ²02 ,

a contradiction. Thus, we have proved the Order-Interval Dichotomy, which com-

pletes the proof of the lemma. ¤

Lemma 3.4. Assume that (H1)-(H3) hold and Πt is fiber-compact. Let a(·), b(·) ∈C(Π) with a(·) < b(·). Then, for every ω ∈ Ω, there exists a strictly order-preservingcontinuous path

Jω : [0, 1] → A(ω)with endpoints Jω(0) = (ω, a(ω)) and Jω(1) = (ω, b(ω)).

Proof. Similarly as Lemma 3.3, we again assume that a(·) ¿ b(·). Let

Y = {Y ⊂ [a(·), b(·)]C(Π) : Y is a totally ordered subset w.r.t. “≤”}.

Then (Y,⊂) is a partially-ordered set. By Zorn’s lemma, we obtain that Y possessesa maximal element, say Y ∗. Obviously, a(·), b(·) ∈ Y ∗ ⊂ [a(·), b(·)]C(Π).

Now we show that Y ∗ has the following properties:

(i) p(·), q(·) ∈ Y ∗ and p(·) < q(·) implies p(·) ¿ r(·) ¿ q(·) for some r(·) ∈ Y ∗;(ii) Y ∗(ω) := {(ω, w(ω)) : w(·) ∈ Y ∗} is compact for any ω ∈ Ω;(iii) Y ∗(ω) is connected for any ω ∈ Ω.

Before giving the proof of (i)-(iii), we point out that such properties imply the

existence of the path Jω, due to Proposition Y1 in [36, Page 434, Appendix]. More

precisely, if (i)-(iii) are satisfied, then for every ω ∈ Ω, there exists a homeomor-phism

Jω : [0, 1] → Y ∗(ω) ⊂ A(ω),which is order-preserving such that Jω(0) = (ω, a(ω)) and Jω(1) = (ω, b(ω)). So it

remains to prove (i)-(iii). We shall discuss them one by one.


(i) Let p(·), q(·) ∈ Y ∗ with p(·) ¿ q(·). It then follows from Lemma 3.3 thatthere exists some c(·) ∈ (p(·), q(·))C(Π). Suppose that (p(·), q(·))C(Π) ∩ Y ∗ = ∅.Then c(·) is order-related to any element in Y ∗, because Y ∗ is totally ordered. Asa consequence, the set Y ′ = Y ∗ ∪ {c(·)} is a totally ordered set in [a(·), b(·)]C(Π),which contradicts the maximality of Y ∗. Hence, (p(·), q(·))C(Π) ∩ Y ∗ 6= ∅, that is,one can find a r(·) ∈ Y ∗ such that p(·) ¿ r(·) ¿ q(·).

(ii) Note that Πt is fiber-compact and Y ∗(ω) ⊂ Πt0(ω ·−t0, [a(ω ·−t0), b(ω ·−t0)]).Then Y ∗(ω) has compact closure for every ω ∈ Ω. So in order to prove Y ∗(ω) iscompact, it suffices to show that

(3.4) Y ∗(ω) is closed for every ω ∈ Ω.

Indeed, given any ω ∈ Ω and any sequence {(ω, xm(ω))}∞m=1 ⊂ Y ∗(ω) such thatxm(ω) → c as m → ∞, it is easy to see that c ∈ [a(ω), b(ω)]. Since Πt(ω, c) isuniformly stable, for every ² > 0 there is a positive integer M = M(²) ∈ N suchthat

(3.5) ‖u(t, ω, xm(ω))− u(t, ω, c)‖ < ², ∀t ≥ 0, m ≥ M(²).

Let c∗(·) ∈ C(Π) be the ω-limit set of (ω, c), and choose a sequence tn → +∞ suchthat ω · tn → ω as n →∞. Recall that xm(·) ∈ C(Π) for all m = 1, 2, · · · . It thenfollows from (3.5) that

‖xm(ω)− c∗(ω)‖ ≤ ², for all m ≥ M(²).

Letting m →∞, it yields that

‖c− c∗(ω)‖ ≤ ².

Note that ² > 0 is arbitrarily chosen. So, one obtains that (ω, c) = (ω, c∗(ω)) ∈A(ω). Recall that Y ∗ is totally ordered. Then for any y(·) ∈ Y ∗, there exists asubsequence of {xm(·)}∞m=1, still denoted by {xm(·)}∞m=1, such that either xm(·) ≤y(·) for all m ∈ N, or xm(·) ≥ y(·) for all m ∈ N. As a consequence, one obtainsthat y(·) and c∗(·) is related by “≤”. By arbitrariness of y(·) ∈ Y ∗, c∗(·) is relatedto any element in Y ∗. Suppose that c∗(·) /∈ Y ∗. Then Y ′ = Y ∗ ∪ {c∗(·)} is alsototally ordered. This contradicts the maximality of Y ∗. Accordingly we conclude

that c∗(·) ∈ Y ∗, and hence (ω, c) = (ω, c∗(ω)) ∈ Y ∗(ω), which completes the proofof (3.4).


(iii) Suppose that Y ∗(ω) is not connected. Then one can find a nonempty proper

subset S ( Y ∗(ω), which is both open and closed in Y ∗(ω). Choose some (ω, b) ∈Y ∗(ω) \ S. Since Y ∗(ω) is totally ordered with respect to “¿”, we define the setS− = {(ω, y) ∈ S : (ω, y) ¿ (ω, b)} and S+ = {(ω, y) ∈ S : (ω, y) À (ω, b)}. ThenS = S− ∪ S+, and hence, one may assume without loss of generality that S− 6= ∅.Recall that Y ∗(ω) is compact. Then a(ω) , l.u.b.{S−} exists. Moreover a(ω) ∈ S−,because S is closed and (ω, b) /∈ S . On the other hand, note that a(ω) ∈ S and Sis also open in Y ∗(ω). Then there exists (ω, c) ∈ S− such that a(ω) ¿ (ω, c), whichcontracts the definition of a(ω). Thus, we have proved that Y ∗(ω) is connected.

This completes the proof. ¤

Lemma 3.5. Assume that a(·), b(·) ∈ C(Π) with a(·) ¿ b(·). Given ω0 ∈ Ω, letJ : [0, 1] → A(ω0) be a strictly order-preserving continuous path in A(ω0), withendpoints J(0) = (ω0, a(ω0)) and J(1) = (ω0, b(ω0)). Then, for any ω ∈ Ω andx ∈ [a(ω), b(ω)], there exists some τ = τ(ω, x) ∈ [0, 1] such that

O(ω, x) ∩ P−1(ω0) = J(τ).

Proof. For any ω ∈ Ω and x ∈ [a(ω), b(ω)], it follows from Lemma 2.3 thatO(ω, x) = c(·) for some c(·) ∈ C(Π). Let α and β, respectively, be the largestand the smallest numbers in [0, 1] ⊂ R satisfying

(3.6) J(α) ≤ (ω0, c(ω0)) ≤ J(β).

Thus, it suffices to prove that α = β. Suppose on the contrary that α < β. Then,

either c(ω0) 6= J(α) or c(ω0) 6= J(β). Note that Πt is eventually strongly monotone.Then by (3.6), one obtains that either J(α) ¿ (ω0, c(ω0)) or (ω0, c(ω0)) ¿ J(β).But this contradicts the definition of α or β. Thus we conclude that α = β =

τ(ω, x) ∈ [0, 1]. ¤

Now we are ready to prove Theorem 3.1.

Proof of Theorem 3.1. First we claim that any two distinct elements of C(Π) are

related with respect to “¿”. Indeed, take u(·), v(·) ∈ C(Π). For each ω ∈ Ω, wedefine

x(ω) = g.l.b.{u(ω), v(ω)}

and

y(ω) = l.u.b.{u(ω), v(ω)}.


Now fix an ω0 ∈ Ω, let p(·) = O(ω0, x(ω0)) and q(·) = O(ω0, y(ω0)), with p(·), q(·) ∈C(Π), respectively. Using the eventually strong monotonicity of Πt, one obtains

p(·) ≤ u(·) and p(·) ≤ v(·). Similarly, u(·) ≤ q(·) and v(·) ≤ q(·). Accordingly,

u(·), v(·) ∈ [p(·), q(·)]C(Π)with p(·) ¿ q(·). By virtue of Lemma 3.4, there exists a strictly order-preservingcontinuous path Jω0 : [0, 1] → A(ω0) with endpoints (ω0, p(ω0)) and (ω0, q(ω0)).By virtue of Lemma 3.5, we have

(ω0, u(ω0)) = O(ω0, u(ω0)) ∩ P−1(ω0) = Jω0(τ1)

and

(ω0, v(ω0)) = O(ω0, v(ω0)) ∩ P−1(ω0) = Jω0(τ2),for some τ1, τ2 ∈ [0, 1]. So (ω0, u(ω0)) and (ω0, v(ω0)) are order-related, and henceu(·) and v(·) are related w.r.t. “¿”. Thus we have proved the claim.

Based on this claim, A(ω) is totally-ordered for every ω ∈ Ω. By repeating thearguments for proving (3.4) in Lemma 3.4, we further deduce that A(ω) is closed,

and hence, A(ω) is locally compact because Πt is fiber-compact. Thus, one can

obtain that A(ω) is connected by repeating the proof in (iii) of Lemma 3.4. It then

follows from Propositions Y1-Y2 in [36, Page 434, Appendix] that for every ω ∈ Ω,A(ω) is either a singleton or coincides with the image of a strictly order-preserving

continuous path

(3.7) Jω : Iω → A(ω),

where the interval Iω = [0, 1], [0,+∞), (−∞, 0] or (−∞,+∞). Here Iω = [0, 1]corresponds to the case that A(ω) is bounded; Iω = [0,+∞) (resp. (−∞, 0]) cor-responds to the case that A(ω) is unbounded but lower-bounded (resp. upper-

bounded); while Iω = (−∞,+∞) corresponds to the case that A(ω) is totallyunbounded.

Now fix an ω0 ∈ Ω and let the interval I = Iω0 . Define the mapping

(3.8) h : Ω× I → A; (ω, α) 7→ O(Jω0(α)) ∩ P−1(ω),

where Jω0 comes from (3.7) with ω replaced by ω0. It is easy to see that h is well-

defined on Ω× I. In the following, we will show that h satisfies all the statementsin Theorem 3.1:

We first note that h is surjective from Ω× I onto A. Indeed, for any (ω, x) ∈ A,it follows from Lemma 2.3 that there exists some b(·) ∈ C(Π) such that O(ω, x) ∩


P−1(ω∗) = (ω∗, b(ω∗)), for all ω∗ ∈ Ω. In particular, x = b(ω). Note alsothat (ω0, b(ω0)) ∈ A(ω0). Then, by (3.7), one can find an α̂ ∈ I such thatJω0(α̂) = (ω0, b(ω0)). Choose a sequence sn → +∞ such that ω0 · sn → ω.Then Πsn(Jω0(α̂)) = Πsn(ω0, b(ω0)) = (ω0 · sn, b(ω0 · sn)) → (ω, b(ω)) = (ω, x),as n → +∞. Thus, (ω, x) ∈ O(Jω0(α̂))∩P−1(ω), which implies that h is surjective.

Moreover, h is also injective. In fact, let (ωi, αi) ∈ Ω×I, i = 1, 2, with h(ω1, α1) =h(ω2, α2). By Lemma 2.3, O(Jω0(αi)) is a 1-cover of Ω, denoted by ci(·) ∈ C(Π), fori = 1, 2. Moreover, (ω0, ci(ω0)) = Jω0(αi) since Jω0(αi) ∈ A(ω0), for i = 1, 2. Notealso that (ω1, c1(ω1)) = h(ω1, α1) = h(ω2, α2) = (ω2, c2(ω2)). Then ω1 = ω2 , ω∗

and c1(ω∗) = c2(ω∗). Since Ω is minimal, there is a sequence tn → ∞ such thatω∗ · tn → ω0. Then Πtn(ω∗, ci(ω∗)) = (ω∗ · tn, ci(ω∗ · tn)) → (ω0, ci(ω0)) = Jω0(αi)as n → ∞, for i = 1, 2. Recall that c1(ω∗) = c2(ω∗), then we obtain Jω0(α1) =Jω0(α2), which implies that α1 = α2. Thus, h is also injective.

In order to prove that h is continuous, we choose any sequence {(ωk, αk)}∞k=1 ⊂Ω × I with (ωk, αk) → (ω∞, α∞) as k → ∞. Again by Lemma 2.3, for eachk = 1, 2, · · · ,∞, there exists an ak(·) ∈ C(Π) such that

(3.9) h(ω, αk) = O(Jω0(αk)) ∩ P−1(ω) = (ω, ak(ω)), ∀ω ∈ Ω.

In particular, (ω0, ak(ω0)) = Jω0(αk), for k = 1, 2, · · · ,∞. Recall that αk → α∞ ask →∞, then one has ak(ω0) → a∞(ω0) as k →∞. It then follows from assumption(H3) that, for any ε > 0, there exists a positive integer K ∈ N such that

‖(ω0 · t, ak(ω0 · t))− (ω0 · t, a∞(ω0 · t))‖ = ‖Πt(ω0, ak(ω0))−Πt(ω0, a∞(ω0))‖ < ε/3,

for all k ≥ K and t ≥ 0. This implies that, if k ≥ K then

(3.10) ‖ak(ω)− a∞(ω)‖ < ε/3,

uniformly for all ω ∈ Ω. Moreover, for such ε and K (choose K larger if necessary),it is easy to see that

(3.11) ‖ωk − ω∞‖ < ε/3 and ‖a∞(ωk)− a∞(ω∞)‖ < ε/3,

for all k ≥ K. By virtue of (3.9), (3.10) and (3.11), we have

‖h(ωk, αk)− h(ω∞, α∞)‖ = ‖(ωk, ak(ωk))− (ω∞, a∞(ω∞))‖≤ ‖ωk − ω∞‖+ ‖ak(ωk)− a∞(ωk)‖+ ‖a∞(ωk)− a∞(ω∞)‖< ε/3 + ε/3 + ε/3 = ε,

for all k ≥ K. We have proved that h is continuous.


Now we will show that h satisfies (i)-(v).

Statement (i) is obvious because O(Jω0(α)) is a 1-cover of Ω for each α ∈ I.(ii): Since h is surjective, h(ω, I) = A(ω) for each ω ∈ Ω. Let I be nontrivial

and assume α1, α2 ∈ I with α1 < α2. By the monotonicity of Jω0(·) and Πt, onehas h(ω, α1) ≤ h(ω, α2). Note also that h is injective. We obtain that h(ω, α1) <h(ω, α2), and hence, h(ω, α1) ¿ h(ω, α2) by the eventually strong monotonicity ofΠt.

According to the statement right after (3.7), in order to prove (iii)-(v), it suffices

to prove that

(3.12) Iω ≡ I, for all ω ∈ Ω.

To end this, suppose that there is an ω∗ ∈ Ω such that Iω∗ 6= I(= Iω0). We nowassert that A(ω0) is upper-bounded (resp. lower-bounded) if and only if A(ω∗) is

upper-bounded (resp. lower-bounded). (We only prove the “upper-bounded” case.

The other case is similar). Indeed, if A(ω0) is upper-bounded, then (ω0, w) =

l.u.b.A(ω0) exists because A(ω0) is totally-ordered. Note also that A(ω0) is closed.

Then (ω0, w) ∈ A(ω0). Let z ∈ X be such that (ω∗, z) = O(ω0, w) ∩ P−1(ω∗).Then (ω∗, z) ∈ A(ω∗). Moreover, A(ω∗) is upper-bounded by (ω∗, z) (Otherwise,one can find (ω∗, z′) ∈ A(ω∗) such that (ω∗, z) < (ω∗, z′). By the eventually strongmonotonicity, (ω0, w) < O(ω∗, z′)∩P−1(ω0) ∈ A(ω0), contradicting to the definitionof w). Noticing that ω0 and ω∗ are symmetric, we have proved the assertion.

Based on this assertion, it is easy to see that A(ω0) is bounded (or unbounded

but lower-bounded or unbounded but upper-bounded, or totally unbounded) if and

only if A(ω∗) is of the same type. This implies that Iω0 = Iω∗ , a contradiction.

Consequently, Iω ≡ Iω0 = I for all ω ∈ Ω. Therefore, the statements in (iii)-(v)have been obtained. In particular, by the continuity of h on Ω× I, A is compact ifI = {0} or [0, 1]. Thus, we have completed the proof. ¤

4. Abstract results for comparable systems

Definition 4.1 (Comparable skew-product semiflow). A skew-product semiflow

Γt(ω, x) = (ω · t, v(t, ω, x)) on Ω × X is called lower-comparable (resp. upper-comparable) with respect to Πt, if Γt(ω, x) ≥ Πt(ω, y) (resp. Γt(ω, x) ≤ Πt(ω, y))whenever (ω, x), (ω, y) ∈ Ω×X with (ω, x) ≥ (ω, y) (resp. (ω, x) ≤ (ω, y)).

In this section, we will establish the 1-cover property for the ω-limit sets of the

comparable skew-product semiflows Γt.


Lemma 4.2. Assume that (H1)-(H3) hold. Assume also a skew-product semiflow

Γt is lower-comparable (resp. upper-comparable) with respect to Πt. Let K ⊂Ω ×X be an ω-limit set of Γt. Then there exist a 1-cover a∗(·) ∈ C(Π) such that(ω, a∗(ω)) ≤ K ∩ P−1(ω) (resp. (ω, a∗(ω)) ≥ K ∩ P−1(ω)), for all ω ∈ Ω.

Proof. We only prove the case that Γt is lower-comparable w.r.t. Πt. The other

case is similar.

Since K is an ω-limit set of Γt, one can write K = OΓ(ω0, x0) for some (ω0, x0) ∈Ω×X. For such (ω0, x0), it follows from Lemma 2.3 that there exists some a∗(·) ∈C(Π) such that OΠ(ω0, x0)∩P−1(ω) = (ω, a∗(ω)), for all ω ∈ Ω. Then we claim that(ω, a∗(ω)) ≤ K∩P−1(ω) for all ω ∈ Ω. As a matter of fact, for any (ω, y) ∈ K, thereexists a sequence {tn} → ∞ such that Γtn(ω0, x0) → (ω, y) as n →∞. By taking asubsequence, if necessary, one may assume that Πtn(ω0, x0) → (ω, z) ∈ OΠ(ω0, x0).It then follows from the 1-cover property of OΠ(ω0, x0) that a∗(ω) = z. Notealso that Γt is lower-comparable w.r.t. Πt. Then one has a∗(ω) = z ≤ y. Since(ω, y) ∈ K is arbitrarily chosen, we obtain that (ω, a∗(ω)) ≤ K ∩ P−1(ω) for allω ∈ Ω, which completes the proof. ¤

Theorem 4.3. Assume that a skew-product semiflow Γt is lower-comparable (resp.

upper-comparable) with respect to Πt. Let K ⊂ Ω×X be the ω-limit set of a forwardorbit O+(ω0, x0) of Γt. Then we have the following:

(i) If there exists an ω∗ ∈ Ω such that K ∩ P−1(ω∗) ≤ (ω∗, b) (resp. K ∩P−1(ω∗) ≥ (ω∗, b)) for some (ω∗, b) ∈ A(ω∗), then K contains a unique minimalset M . Moreover, M is an almost 1-cover of Ω w.r.t. Γt.

(ii) For any ω ∈ Ω, if there exists some (ω, b) ∈ A(ω) such that K ∩ P−1(ω) ≤(ω, b) (resp. K ∩ P−1(ω) ≥ (ω, b)), then K is a 1-cover of Ω w.r.t. Γt.

(iii) If I = [0,+∞) or (−∞,+∞) (resp. I = (−∞, 0] or (−∞,+∞)) in Theorem3.1 for Πt, then K is a 1-cover of Ω w.r.t. Γt.

Proof. We only prove the case that Γt is lower-comparable w.r.t. Πt. The other

case is similar.

(i) By virtue of Lemma 4.2, there exists a 1-cover a∗(·) ∈ C(Π) such that(ω, a∗(ω)) ≤ K∩P−1(ω) for all ω ∈ Ω. Hence, we define a nonempty set Y ⊂ C(Π),for which

(4.1) Y = {y(·) ∈ C(Π) : (ω, y(ω)) ≤ K ∩ P−1(ω), for all ω ∈ Ω}.


Recall that Πt is monotone and A(ω) is totally ordered for all ω ∈ Ω (by Theorem3.1). Then C(Π) is also totally-ordered with respect to ¿. Accordingly, l.u.b.Yexists. Moreover, it is also easy to check that l.u.b.Y ∈ Y.

Denote p(·) = l.u.b.Y. Now we claim that there exists an ω̃ ∈ Ω such that K ∩P−1(ω̃) = {(ω̃, p(ω̃))}. Suppose on the contrary that, for any ω ∈ Ω, there is some(ω, xω) ∈ K such that (ω, p(ω)) < (ω, xω). Then, by the strong monotonicity of Πtand the comparability of Γt w.r.t. Πt, one can assume without loss of generality

that (ω, p(ω)) ¿ (ω, xω) for all ω ∈ Ω. In particular, for the ω∗ ∈ Ω given in ourassumption, one has

(4.2) (ω∗, p(ω∗)) ¿ (ω∗, xω∗) ≤ (ω∗, b(ω∗)).

For such p(·) and b(·), it follows from Lemma 3.4 that there exists a strictly order-preserving continuous path Jω∗ : [0, 1] → A(ω∗) with endpoints Jω∗(0) = (ω∗, p(ω∗))and Jω∗(1) = (ω∗, b(ω∗)). As a consequence, one can find some q(·) ∈ C(Π) suchthat (ω∗, p(ω∗)) ¿ (ω∗, q(ω∗)) ¿ (ω∗, xω∗). Since K is the ω-limit set of (ω0, x0)with respect to Γt, there exists some sequence tn → +∞ such that

Γtn(ω0, x0) → (ω∗, xω∗) ∈ K as n →∞.

By choosing a subsequence, if necessary, we also obtain that

Πtn(ω0, q(ω0)) → (ω∗, q(ω∗)) as n →∞.

Accordingly, there exists N À 1 such that

ΠtN (ω0, q(ω0)) ¿ ΓtN (ω0, x0).

Again, from the eventually strong monotonicity of Πt and the comparability of Γt

w.r.t. Πt, it follows that

(4.3) Πt+tN (ω0, q(ω0)) ¿ ΠtΓtN (ω0, x0) ≤ Γt+tN (ω0, x0), ∀t ≥ 0.

For any (ω, x) ∈ K, there exists sn → +∞ such that

Γsn(ω0, x0) → (ω, x) as n →∞.

Given such sequence {sn}, let t = sn − tN in (4.3) for all n sufficiently large. Thisimplies that

Πsn(ω0, q(ω0)) ¿ Γsn(ω0, x0)


for all n sufficiently large. By Letting n → ∞, it yields that (ω, q(ω)) ≤ (ω, x).Since (ω, x) ∈ K is arbitrary, we obtain that (ω, q(ω)) ≤ K ∩P−1(ω) for all ω ∈ Ω,contradicting the definition of p(·). Thus we have proved the claim.

Using this claim, we deduce that there is a unique minimal set M ⊂ K. Other-wise, suppose on the contrary that K contains two distinct minimal sets M1 and M2,

then it follows from the minimality of Ω that Mi ∩P−1(ω) are nonempty, for everyω ∈ Ω and i = 1, 2. In particular, ∅ 6= Mi ∩ P−1(ω̃) ⊂ K ∩ P−1(ω̃) = {(ω̃, p(ω̃))},for i = 1, 2. Hence M1 ∩ P−1(ω̃) = M2 ∩ P−1(ω̃) = {(ω̃, p(ω̃))}. This implies thatM1 = M2, a contradiction. Thus, K only contains a unique minimal set M , with

M ∩ P−1(ω̃) = {(ω̃, p(ω̃))}. By [41, Definition 1.2.11 and Corollary 1.2.15], M isan almost automorphic extension of Ω, i.e., M is an almost 1-cover of Ω. Thus we

have proved the first statement.

(ii) Assume that for any ω ∈ Ω, there exists some (ω, b) ∈ A(ω) such thatK ∩ P−1(ω) ≤ (ω, b). Then we assert that K is a 1-cover of Ω w.r.t. Γ, satisfyingK∩P−1(ω) = (ω, p(ω)) for ω ∈ Ω. Here p(·) = l.u.b.Y is defined in (4.1). Otherwise,there exist an ω1 ∈ Ω and some (ω1, c) ∈ K ∩ P−1(ω1) such that (ω1, p(ω1)) <(ω1, c). Let ω2 = ω1 ·t1 for some t1 > t0 > 0. By the eventually strong monotonicityof Πt and the comparability of Γt w.r.t. Πt, one has (ω2, p(ω2)) ¿ Γt1(ω1, c) ∈ K ∩P−1(ω2). According to our assumption, for such ω2, we can find (ω2, b(ω2)) ∈ A(ω2)such that

(ω2, p(ω2)) ¿ Γt1(ω1, c) ≤ (ω2, b(ω2)),which is exactly (4.2), with ω∗ replaced by ω2, respectively. Hence, by repeating

the same arguments following (4.2) in the proof of (i), one obtains a contradiction

to the definition of p(·). So, we have proved the assertion, which completes ourproof of (ii).

(iii) Finally, assume that I = [0,+∞) or (−∞,+∞) in Theorem 3.1 for Πt. Then,by Theorem 3.1(iv)-(v), A(ω) is upper-unbounded for any ω ∈ Ω. Note also thatK is bounded. Consequently, for every ω ∈ Ω, there exists some (ω, b(ω)) ∈ A(ω)such that K ∩P−1(ω) ≤ (ω, b(ω)). Thus, (iii) is a direct corollary of statement (ii).We have completed our proof. ¤

5. Asymptotic almost periodicity in comparable systems

It is well known that there are many differential equations generate monotone

skew-product semiflows satisfying (H1)-(H3) (cf. [23, 24, 35, 41] and references

therein). As a consequence, our results have wide applications to various types of


comparable nonmonotone systems of differential equations. We do not intend to

present them all, but give four typical examples to illustrate how our main theorems

are applied to study the asymptotic almost periodicity of solutions to nonmonotone

comparable almost periodic ODEs, reaction-diffusion systems, delayed differential

systems and semilinear parabolic equations.

5.1. Almost periodic comparable ODE systems. Let Rn+ = {(x1, ..., xn) ∈Rn : xi ≥ 0,∀1 ≤ i ≤ n}. Consider the n-dimensional system of ordinary differentialequations

(5.1) ẋ = f(t, x),

where f satisfies the following hypotheses:

(A1) f ∈ C1(R× Rn+,Rn) is C1-admissible and uniformly almost periodic in t;(A2) fi(t, 0) = 0 (1 ≤ i ≤ n);(A3) (Cooperativity and Strong Irreducibility) ∂fi∂xj (t, x) ≥ 0 for all 1 ≤ i 6= j ≤ n.

Moreover, there is a δ > 0 such that if two nonempty subsets I, J of {1, 2, · · · , n}form a partition of {1, 2, · · · , n}, then for any (t, x) ∈ R × Rn+, there exist i ∈ I,j ∈ J such that | ∂fi

∂xj(t, x)| ≥ δ > 0;

(A4) There is a C1-function L : Rn+ → R such that gradL(x) À 0 at each x ∈ Rn+,and 〈gradL(x), f(t, x)〉 = 0 for all (t, x) ∈ R× Rn+.

Hypothesis (A4) says that L(x) has positive gradient at each x ∈ Rn+ and is a firstintegral of system (5.1), i.e., L(x) is a constant along every solution of (5.1). There

are many models possessing such invariant L, such as gross-substitute systems in

economics (cf. [32, 40]) and master equation in Markov processes ([13, 27]), etc.

Let H(f) be the hull of f . Then it is easy to check that each g ∈ H(f) satisfies(A1)-(A4) as well. Note that the time translation g ·t of g ∈ H(f) induces a naturalalmost periodic minimal flow on H(f). It then follows that system (5.1) induces

the following (local) skew-product flow:

Πt : H(f)× Rn+ → H(f)× Rn+; (g, x) 7→ (g · t, φ(t, x; g)), t ∈ R,

where φ(t, x; g) is the solution of

(5.1g) ẋ = g(t, x)

with φ(0, x; g) = x. By virtue of (A3), Πt is eventually strongly monotone (see [41,

Lemma 3.4.5]). Assume also that


(A5) Every solution of (5.1g), g ∈ H(f), is bounded.

Then (A4) implies that every forward orbit of Πt is uniformly stable (see [42, Lemma

3.1]). Thus, Hypotheses (H1)-(H3) are satisfied for Πt.

Now we consider the following nonmonotone system

(5.2) ẋ = F (t, x),

where F satisfies (A1) and its frequency module M(F ) = M(f). Moreover, weassume that either

(5.2-UC) Fi(t, x) ≥ 0 if xi = 0, and F (t, x) ≤ f(t, x) for all (t, x) ∈ R× Rn+;

or

(5.2-LC) F (t, x) ≥ f(t, x) for all (t, x) ∈ R× Rn+.

Then system (5.2) generates a (local) skew-product flow on H(F )× Rn+:

Γt : H(F )× Rn+ → H(F )× Rn+; (G, x) 7→ (G · t, ψ(t, x;G)), t ∈ R,

where ψ(t, x;G) is the solution of

ẋ = G(t, x)

with G ∈ H(F ) and ψ(0, x;G) = x.

Theorem 5.1. Let (A1)-(A5) hold. Assume also either (5.2-UC) or (5.2-LC) is

satisfied. Then every bounded solution of the comparable system (5.2) will be as-

ymptotic to an almost periodic solution.

Proof. Note that M(F ) = M(f), as we mentioned in the end of Section 2, the flow(H(F ), ·) is isomorphic to the flow (H(f), ·), and hence one may write Ω := H(f) ∼=H(F ). By virtue of Kamke Theorem (cf. [26, 31, 23]), we further obtain that Γt is

upper-comparable with respect to Πt (when (5.2-UC) holds); or lower-comparable

with respect to Πt (when (5.2-LC) holds).

Let K be the ω-limit set of a bounded solution of the comparable system (5.2).

(i) Assume that (5.2-UC) holds, i.e., Γt is upper-comparable with respect to Πt.

Note that K ⊂ Ω×Rn+, and 0(·) ∈ C(Π) by (A2). Then K∩P−1(ω) ≥ (ω, 0) ∈ A(ω)for any ω ∈ Ω. By Theorem 4.3(ii), we obtain that K is a 1-cover of Ω w.r.t. Γt.

(ii) Assume that (5.2-LC) holds, i.e., Γt is lower-comparable with respect to Πt.

We will utilize Theorem 4.3(iii) to deduce the 1-cover property of K. Accordingly,

it suffices to show that I = [0,+∞) in Theorem 3.1. By virtue of Theorem 3.1(iv),


this corresponds to that A(ω) is lower-bounded but upper-unbounded, for all ω ∈ Ω.By (A2), it is easy to see that A(ω) is lower-bounded. So, we only need to show

that A(ω) is upper-unbounded for all ω ∈ Ω. Suppose on the contrary that A(ω0)is upper-bounded for some ω0 ∈ Ω. Then (ω0, w) := l.u.b.A(ω0) exists becauseA(ω0) is totally-ordered (see Theorem 3.1). Note also that A(ω0) is closed. Then

(ω0, w) ∈ A(ω0). Now choose z ∈ Rn+ such that w ¿ z. Then

Πt(ω0, w) ¿ Πt(ω0, z), for all t ≥ 0.

Recall that (ω0, w) ∈ A(ω0), we choose a sequence tn →∞ such that ω0 · tn → ω0and φ(tn, w;ω0) → w as n →∞. For such {tn}, choose a subsequence, if necessary,such that φ(tn, z;ω0) → z∗ as n →∞. By Lemma 2.3 and monotonicity of Πt, onehas (ω0, z∗) ∈ A(ω0) and w ≤ z∗. On the other hand, it follows from (A4) thatL(w) = L(φ(t, w;ω0)) < L(φ(t, z;ω0)) = L(z∗) for all t ≥ 0. Then w < z∗, whichcontradicts to the definition of w. Thus, we have proved that I = [0,+∞) holds inour case. By Theorem 4.3(iii), we obtain that K is a 1-cover of Ω w.r.t. Γt. ¤

5.2. Almost periodic comparable reaction-diffusion systems. Consider the

almost periodic reaction-diffusion system with Neumann boundary condition:

(5.3)

∂ui∂t

= di(t)∆ui + Fi(t, x, u1, · · · , un), x ∈ Ω, t > 0,∂ui∂ν

(t, x) = 0, x ∈ ∂Ω, t > 0,

ui(0, x) = u0,i(x), x ∈ Ω, 1 ≤ i ≤ n,

where Ω is a bounded domain in RN with smooth boundary. Of course, ∆ is the

Laplacian operator on RN .

Let d = (d1(·), · · · , dn(·)) ∈ C(R,Rn) be an almost periodic vector-valued func-tion bounded below by a positive real vector. The nonlinearity F = (F1, · · · , Fn) :R × Ω̄ × Rn → Rn is a C1-admissible (with D = Ω̄ × Rn ⊂ RN+n) and uniformlyalmost periodic in t, vector-valued function. Let u = (u1, · · · , un), X = C(Ω̄,Rn)and the standard cone X+ = C(Ω̄,Rn+), together with the requirement that F

points into X+ along the boundary of X+:

(B1) Fi(t, x, u) ≥ 0, for any u ∈ X+ with ui = 0, and x ∈ Ω̄, t ∈ R+.

Here we do not assume F has any monotonicity properties.

Let Y = H(d, F ) be the hull of the function (d, F ). By the standard theory

of reaction-diffusion systems (see, e.g., [18], Chapter 6), it follows that for every


u0 ∈ X+ and y = (µ,G) ∈ Y , the system

(5.4)

∂ui∂t

= µi(t)∆ui + Gi(t, x, u), x ∈ Ω, t > 0,∂ui∂ν

(t, x) = 0, x ∈ ∂Ω, t > 0,

u(0, x) = u0(x), x ∈ Ω, 1 ≤ i ≤ nadmits a (locally) unique regular solution u(t, x, u0, y) in X+. This solution also

continuously depends on y ∈ Y and u0 ∈ X+ (see, e.g., [16, Sec.3.4]). Therefore,(5.4) defines a (local) skew-product semiflow Γ on X+ × Y with

Γt(u0, y) = (u(t, ·, u0, y), y · t), ∀ (u0, y) ∈ X+ × Y, t ≥ 0.

Now we assume that there exists a vector-valued function f : R × Rn+ → Rnsatisfying (A1)-(A5) in Section 5.1, with its frequency module M(f) = M(F ),such that either

(B2+) F (t, x, u) ≤ f(t, u) for all (t, x, u) ∈ R× Ω̄× Rn+;

or

(B2−) F (t, x, u) ≥ f(t, u) for all (t, x, u) ∈ R× Ω̄× Rn+.

We consider the hull H(d, f). Since M(f) = M(F ), one has Y = H(d, F ) ∼=H(d, f). Then it is easy to see that, for any (µ,G) ∈ H(d, F ), there exists a(µ, g) ∈ H(d, f) such that either

(i) G(t, x, u) ≤ g(t, u) for all (t, x, u) ∈ R× Ω̄× Rn+ (when (B2+) holds); or(ii) G(t, x, u) ≥ g(t, u) for all (t, x, u) ∈ R× Ω̄× Rn+ (when (B2−) holds).For such (µ, g) ∈ H(d, f), we introduce the following new reaction-diffusion

system:

(5.5)

∂vi∂t

= µi(t)∆vi + gi(t, u), x ∈ Ω, t > 0,∂vi∂ν

(t, x) = 0, x ∈ ∂Ω, t > 0,

v(0, x) = v0(x) ∈ X+, x ∈ Ω, 1 ≤ i ≤ n.Then system (5.5) induces the following global skew-product semiflow:

Πt : X+×H(d, f) → X+×H(d, f); (v0, (µ, g)) 7→ (v(t, ·, v0;µ, g), (µ, g)·t), t ∈ R+,

where v(t, ·, v0;µ, g) is the unique regular global solution of (5.5) in X+. Accordingto assumption (A3) and [24, Sec.6.2], one can obtain that Πt is fibre-compact and

satisfies standard hypotheses (H1)-(H3).


Lemma 5.2. (i) If (B2+) holds, then Γt is upper-comparable with respect to Πt;

(ii) If (B2−) holds, then Γt is lower-comparable with respect to Πt.

Proof. We only prove (i). The proof of (ii) is similar. To end this, we follow the

comparison arguments in [9]. Let u0, v0 ∈ X+ with u0 ≤ v0, and define the functionw by

w(t, x) = v(t, x, v0;µ, g)− u(t, x, u0;µ,G).

Then w satisfies

(5.6)

∂wi∂t

= µi(t)∆wi + Qi(t, x, w), x ∈ Ω, t > 0,∂wi∂ν

(t, x) = 0, x ∈ ∂Ω, t > 0,

w(0, x) = w0(x) ≥ 0, x ∈ Ω, 1 ≤ i ≤ n,

where

Qi(t, x, w) = gi(t, w + u(t, x, u0;µ,G))−Gi(t, x, u(t, x, u0;µ,G)),

for each 1 ≤ i ≤ n. We claim that the rectangle Rn+ is invariant for Q = (Q1, · · · , Qn).From [18, Proposition 6.2], it follows that we only need to show that for any i,

Qi(t, x, w) ≥ 0 if w ≥ 0 with wi = 0. Indeed, recall that g is cooperative. Then

gi(t, w + u) = gi(t, w1 + u1, · · · , ui, · · · , wn + un)≥ gi(t, u1, · · · , ui, · · · , un) = gi(t, u) ≥ Gi(t, x, u),

for any (t, x, u) ∈ R+ × Ω̄ × Rn+. As a consequence, it follows that Qi(t, x, w) ≥ 0if w ≥ 0 with wi = 0. Thus we complete the proof. ¤

In the following, we write (B2±) as either (B2+) or (B2−) holds.

Theorem 5.3. Let (B1) and (B2±) hold for system (5.3). Then every L∞-bounded

solution of (5.3) is asymptotic to an almost periodic and spatially homogeneous

solution.

Proof. Let u(t, ·, u0, d, F ) be a L∞-bounded solution of (5.3) in X+. Then, followingfrom the work in [16] and the standard a priori estimates for parabolic equations,

we know that the solution u becomes a globally defined classical solution on X+;

moreover, one obtains that {u(t, ·, u0, d, F ) : t ≥ τ} is precompact in X+. Thus theω-limit set of (u0, (d, F )) = (u0, y) ∈ X+ × Y , with respect to Γt, is a nonemptyand compact set in X+. We denote such ω-limit set by K.


(i) If (B2+) holds, then it follows from Lemma 5.2 that Γt is upper-comparable

with respect to Πt. Note that K ⊂ X+ × Y , and 0(·) ∈ C(Π) by (A2). ThenK ∩ P−1(y) ≥ (0, y) ∈ A(y) for any y ∈ Y . By Theorem 4.3(ii), we obtain that Kis a 1-cover of Ω w.r.t. Γt.

(ii) if (B2−) holds, then Γt is lower-comparable with respect to Πt. Again, we

will utilize Theorem 4.3(iii) to deduce the 1-cover property of K. Accordingly, it

suffices to show that I = [0,+∞) in Theorem 3.1. By virtue of Theorem 3.1(iv), thiscorresponds to that A(y) is lower-bounded but upper-unbounded, for all y ∈ Y . By[24, Theorem 6.3], we note that A ⊂ Rn+ × Y . Then one can repeat the argumentsin Case (5.2-LC) in Section 5.1 to obtain that I = [0,+∞) holds in our case. Itthen follows from Theorem 4.3(iii) that K is a 1-cover of Ω w.r.t. Γt. ¤

5.3. Almost periodic comparable delayed differential systems. Consider

an almost periodic n-compartmental system with pipes describing by the following

differential systems with time delays (see e.g., [6, 24, 51]):

(5.7)

dui(t)dt

=n∑

j=1

fij(t− τij , uj(t− τij))−n∑

j=1

fji(t, ui(t)), t > 0, 1 ≤ i ≤ n,

u(s) = ϕ(s), s ∈ [−τ, 0],

where τij ≥ 0, τ = max1≤i,j≤n{τij} and ϕ ∈ X , C([−τ, 0],Rn). Obviously, X isa strongly ordered Banach space with solid cone X+ = C([−τ, 0],Rn+). We assumethat

(C1) Each fij ∈ C(R2,R) is C1-admissible and uniformly almost periodic in t;(C2) fij(t, 0) ≡ 0, for all t ∈ R and 1 ≤ i, j ≤ n;(C3) There exists a δ > 0 such that ∂fij∂u (t, u) ≥ δ > 0, ∀(t, u) ∈ R2, 1 ≤ i, j ≤ n.

Let f = (fij)1≤i,j≤n and let H(f) be the hull of f . For each (g, ϕ) ∈ H(f)×X,let u(t, g, ϕ) be the unique solution of (5.7g) (i.e., (5.7) with f replaced by g), with

the initial function ϕ ∈ X. Then system (5.7) generates a skew-product semiflow:

Πt : H(f)×X → H(f)×X; (g, ϕ) 7→ (g · t, ut(g, ϕ)), t ≥ 0,

where

[ut(g, ϕ)](s) = u(t + s, g, ϕ), ∀s ∈ [−τ, 0].


Now we consider the nonmonotone system:

(5.8)

dui(t)dt

=n∑

j=1

Fij(t− τij , uj(t− τij))−n∑

j=1

fji(t, ui(t)), t > 0, 1 ≤ i ≤ n,

u(s) = ϕ(s), s ∈ [−τ, 0],for which F = (Fij)1≤i,j≤n satisfies (C1) and its frequency M(F ) = M(f) (hence(H(F, f), ·) is flow isomorphic to the flow (H(f), ·)). We also assume that either

(5.8-UC) Fij(t, u) ≤ fij(t, u) for all (t, u) ∈ R2 and 1 ≤ i, j ≤ n;

or

(5.8-LC) Fij(t, u) ≥ fij(t, u) for all (t, u) ∈ R2 and 1 ≤ i, j ≤ n.

Denote Ω := H(F, f) ∼= H(f). For each (ψ, ω) ∈ X × Ω, we write v(t, ψ, ω) theunique solution of (5.8ω) (i.e., (5.8) with (F, f) replaced by ω), with the initial

function ψ ∈ X. System (5.8) generates a (local) skew-product semiflow on X ×Ω:

Γt : Ω×X → Ω×X; (ω, ψ) 7→ (ω · t, vt(ψ, ω)), t ≥ 0,

where [vt(ω, ψ)](s) = v(t + s, ω, ψ),∀s ∈ [−τ, 0].

Theorem 5.4. Let (C1)-(C3) hold. Assume also that (5.8-LC) or (5.8-UC) is satis-

fied. Then every bounded solution of the comparable system (5.8) will be asymptotic

to an almost periodic solution.

Proof. It follows from (C3) that Πt is eventually strongly monotone (see [6, 51]).

Recall that system (5.7) possesses a first integral J : H(f)×X → R defined by (see[24, Lemma 6.2]):

J(g, ϕ) :=n∑

i=1

ϕi(0) +n∑

i,j=1

∫ 0−τij

gij(s, ϕj(s))ds.

J is also strictly order-preserving in the sense that

(5.9) J(g, ϕ) < J(g, ψ), for all ϕ < ψ in X and g ∈ H(f).

Then by [24, Lemmas 6.3 and 6.4], (H2) holds and Πt is uniformly stable. Thus

Πt satisfies hypotheses (H1)-(H3). Furthermore, it follows from [15, Theorem 4.1]

that Πt is fiber-compact.

For such Πt, we will show that I = (−∞ + ∞) in Theorem 3.1. By virtue ofTheorem 3.1(v), this corresponds to that A(g) is both lower-unbounded and upper-

unbounded, for all g ∈ H(f). Otherwise, suppose without loss of generality that


A(g0) is lower-bounded for some g0 ∈ H(f). Then (g0, ϕ) := g.l.b.A(g0) ∈ A(g0),because A(g0) is totally-ordered and closed (see Theorem 3.1). Now choose ψ ∈ Xsuch that ψ ¿ ϕ. Then

Πt(g0, ψ) ¿ Πt(g0, ϕ), for all t > 0 sufficiently large.

Choose a sequence tn → ∞ such that g0 · tn → g0 and utn(g0, ϕ) → ϕ as n →∞. One may also assume that utn(g0, ψ) → ψ∗ as n → ∞. By Lemma 2.3 andmonotonicity of Πt, one has (g0, ψ∗) ∈ A(g0) and ψ∗ ≤ ϕ. On the other hand, itfollows from (5.9) that J(g0, ϕ) = J(Πt(g0, ϕ)) > J(Πt(g0, ψ)) = J(g0, ψ∗) for all

t ≥ 0. Then ψ∗ < ϕ, which contradicts to the definition of ϕ. Thus, we have provedthat I = (−∞,+∞) holds in our case.

By virtue of the comparison Theorem (see [43, Corollary 5.3.4] or [23]), Γt is

upper-comparable with respect to Πt (when (5.8-UC) holds); or lower-comparable

with respect to Πt (when (5.8-LC) holds).

Recall that the interval I = (−∞,+∞). It then follows from Theorem 4.3(iii)that every bounded solution of the comparable system (5.8) will be asymptotic to

an almost periodic solution. This completes our proof. ¤

5.4. Almost periodic comparable semilinear parabolic equations. Consider

the following initial-boundary value problem for semilinear parabolic equations

(5.10)

∂u

∂t=

N∑

i,j=1

aij(x)uxixj + f(t, x, u), x ∈ Ω, t > 0,

Bu =0, x ∈ ∂Ω, t > 0,u(0, x) = u0(x), x ∈ Ω.

Here Ω is a bounded domain in RN (N ≥ 1) with the boundary ∂Ω of class C2+θfor some θ ∈ (0, 1). The operator Lu := ∑ aijuxixj satisfies uniform ellipticitycondition

(D1)∑

aij(x)ξiξj ≥ c1|ξ|2 (x ∈ Ω̄, ξ = (ξ1, · · · , ξN ) ∈ RN ) for some positiveconstant c1 > 0.

The nonlinearity f : R × Ω̄ × R → R is assumed to be a admissible (withD = Ω̄×R ⊂ RN+1) and uniformly almost periodic in t, real-valued function withcertain regularity condition

(D2) aij ∈ Cθ(Ω̄), f is continuous on R × Ω̄ × R together with its derivativeswith respect to u. Moreover, f is locally Hölder continuous in (t, x).


We consider a time-independent regular linear boundary operator B on ∂Ω of Neu-mann (Bu = ∂u∂ν ) or Robin (Bu = ∂u∂ν +αu) type. Here ν is the unit outward normalvector field on ∂Ω. We also assume that

(D3) f(t, x, 0) = 0 and there is an M > 0 such that f(t, x, u) ≤ −δ0, for any(t, x) ∈ R× Ω̄ and u ≥ M .

(D4) f(t, x, su) ≥ sf(t, x, u) for all s ∈ [0, 1] and (t, x, u) ∈ R× Ω̄× R+.

Let X = C(Ω̄,R). Then X is a strongly ordered Banach space with solid cone

X+ = C(Ω̄,R+). Let also Y = H(f) be the hull of the nonlinearity f . For any

g ∈ Y , the function g is uniformly almost periodic in t and satisfies all the aboveassumptions (D1)-(D4) with the same M . As a consequence, (5.10) gives rise to a

family of equations associated to each g ∈ Y :

(5.10g)

∂u

∂t=

N∑

i,j=1

aij(x)uxixj + g(t, x, u), x ∈ Ω, t > 0,

Bu =0, x ∈ ∂Ω, t > 0,u(0, x) = u0(x), x ∈ Ω.

For every u0 ∈ X+, equation (5.10g) admits a (locally) unique regular solutionu(t, x, u0, g) in X+. This solution also continuously depends on g ∈ Y and u0 ∈ X+(see e.g. [16, Sec.3.4]). Therefore, (5.10g) defines a (local) skew-product semiflow

Π on X+ × Y with

Πt(u0, g) = (u(t, ·, u0, g), g · t), ∀ (u0, g) ∈ X+ × Y, t ≥ 0.

It follows from the strong parabolic maximum principle that Πt is eventually

strongly monotone (see, e.g. [18, Theorem 6.15]). Moreover, Πt is fiber-compact

(see [18, Proposition 6.13]). (D3) implies that there is an L∞-bound on {u(t, ·, u0, g) :t ≥ 0} uniformly for u0 ∈ X+ and g ∈ Y . Once an L∞-bound is established, onecan obtain a C1-bound on {u(t, ·, u0, g) : t ≥ τ} for any τ > 0 (see [3, Theorem 2.4]).Therefore, following from the work in [16] and the standard a priori estimates for

parabolic equations, we know that the solution u becomes a globally defined clas-

sical solution on X+; moreover, one gets that {u(t, ·, u0, g) : t ≥ τ} is precompactfor any u0 ∈ X+ and g ∈ Y .

Meanwhile, one can also deduce from (D3) that there exist minimal sets K1 =

{(0, g) : g ∈ Y } and K2 ⊂ [0,M ]X × Y such that

(i) K1 ≤ K2, which means that, (0, g) ≤ K2 ∩ P−1(g) for any g ∈ Y .


(ii) For each (u0, g) ∈ X+ × Y with (u0, g) ≥ K2 ∩ P−1(g),

limt→∞

d(Πt(u0, g),K2) = 0.

For the following linearized almost periodic parabolic equation

(5.11)

∂v

∂t=

N∑

i,j=1

aij(x)vxixj + f(t, x, 0)v, x ∈ Ω, t > 0,

Bv =0, x ∈ ∂Ω, t > 0.

According to [30, 42], there exists a unique principal spectrum point λ(f(·, ·, 0))associated with (5.11). By [41, Proposition 2.4.1], one has λK1 = λ(f(·, ·, 0)). HereλKi is the upper Lyapunov exponent on Ki, i = 1, 2 (see [41, Definition 2.4.3]).

Hereafter, we assume that

(D5) λK1 = λ(f(·, ·, 0)) ≤ 0.

Now we consider the following IBVP for comparable parabolic equations of the

following form

(5.12)

∂u

∂t=

N∑

i,j=1

aij(x)uxixj + F (t, x, u), x ∈ Ω, t > 0,

Bu =0, x ∈ ∂Ω, t > 0,u(0, x) = u0(x), x ∈ Ω,

for which (D1)-(D2) hold for aij ,F with the frequency M(F ) = M(f) (hence(H(F ), ·) is flow isomorphic to the flow (H(f), ·)). Furthermore, we assume that

(5.12-UC) 0 ≤ F (t, x, u) ≤ f(t, x, u), ∀(t, x, u) ∈ R× Ω̄× R+.

Denote Y := H(F ) ∼= H(f). For each (v0, y) ∈ X+ × Y , we write v(t, v0, y)the unique classical solution of (5.12y) (i.e., (5.12) with F replaced by y), with the

initial function v0 ∈ X+. System (5.12) generates a (local) skew-product semiflowon X+ × Y :

Γt : X+ × Y → X+ × Y ; (v0, y) 7→ (v(t, v0, y), y · t), t ≥ 0.

Theorem 5.5. Let (D1)-(D5) and (5.12-UC) hold. Then every solution of the

comparable system (5.12) will be asymptotic to an almost periodic solution.

Proof. We first show that (D4)-(D5) implies that Πt is uniformly stable. As a

matter of fact, by virtue of (D4), Πt is subhomogeneous (or called concave) in the


sense that

u(t, ·, su0, g) ≥ su(t, ·, u0, g), ∀s ∈ [0, 1] and (u0, g) ∈ X+ × Y.

As a consequence, when λK1 = λ(f(·, ·, 0)) < 0, it follows from [24, Theorem 6.1(1)]that K1 = K2 and limt→∞ d(Πt(u0, g),K1) = 0, for all (u0, g) ∈ X+×Y . So, in thiscase, Πt is globally uniformly asymptotically stable. When λK1 = λ(f(·, ·, 0)) = 0,it then follows from [33, Theorems 6.7 and 7.2] that λK2 = 0, K2 is a 1-cover of Y

and the order-interval [K1,K2] is foliated with 1-covers of Y , i.e., µK1 + (1−µ)K2is a 1-cover, for all µ ∈ [0, 1]. Thus, Πt is uniformly stable on X+. Thus we haveshown that Πt is uniformly stable.

It follows from the Maximum Principle that Γt is upper-comparable with respect

to Πt. Let K be the ω-limit set of any solution of the comparable system (5.10).

Then, by (5.12-UC) and Lemma 4.2, one has K1 = {(0, y) : y ∈ Y } ≤ K ≤ K2. Itthen follows from Theorem 4.3(ii) that K is a 1-cover of Y . In other words, we have

proved that every solution of the comparable system (5.10), satisfying (5.12-UC),

will be asymptotic to an almost periodic solution. ¤

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Department of Mathematics, Nanjing University of Aeronautics and Astronautics,

Nanjing, Jiangsu 210016, P. R. China

E-mail address: [email protected]

Department of Mathematics and Statistics, University of Helsinki, FIN-00014, Helsinki,

Finland

E-mail address: [email protected]

aDepartment of Mathematics and Statistics, University of Helsinki, P.O. Box 68,

FIN-00014, Finland, bDepartment of Mathematics, University of Science and Technol-

ogy of China, Hefei, Anhui 230026, P. R. China

E-mail address: [email protected], [email protected]

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