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Nonlinear Analysis 71 (2009) 3705–3714
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Nonlinear Analysis
journal homepage: www.elsevier.com/locate/na
Weighted pseudo-almost periodic solutions of a class of abstractdifferential equationsI
Liuwei Zhang ∗, Yuantong XuDepartment of Mathematics, Sun Yat-sen University, Guangzhou 510275, PR China
a r t i c l e i n f o
Article history:Received 22 May 2008Accepted 6 February 2009
Keywords:Weighted pseudo-almost periodicityHille–Yosida conditionIntegral solutionSpectral analysisPartial functional differential equations
a b s t r a c t
For abstract linear functional differential equations with a weighted pseudo-almostperiodic forcing term, we prove that the existence of a bounded solution onR+ implies theexistence of a weighted pseudo-almost periodic solution. Our results extend the classicaltheorem due to Massera on the existence of periodic solutions for linear periodic ordinarydifferential equations. To illustrate the results, we consider the Lotka–Volterra model withdiffusion.
Crown Copyright© 2009 Published by Elsevier Ltd. All rights reserved.
1. Introduction
In this paper, we consider the existence of weighted pseudo-almost periodic solutions to the following partial functionaldifferential equation
ddtx(t) = Ax(t)+ Lxt + f (t), t ∈ R, (1.1)
where A : D(A)→ X is not necessary densely defined linear operator on a Banach space X, and satisfies the Hille–Yosidacondition: there existM ≥ 0, ω ∈ R such that (ω,∞) ⊂ ρ(A) and
|((λ− A)−n)| ≤M
(λ− ω)n, for n ∈ N, λ > ω,
where ρ(A) is the resolvent set of A and C = C([−r, 0],X) is the space of continuous functions from [−r, 0] to X endowedwith the uniform norm topology. L is a bounded linear operator from C to X and f is a weighted pseudo-almost periodicfunction from R to X. The function xt ∈ C is defined by
xt(θ) = x(t + θ), for θ ∈ [−r, 0].
We employ the variation of constants formula obtained in [1] and new fundamental results about the spectral analysis ofthe solutions to establish a new principle reduction obtained in [2], We prove that the existence of a bounded solution onR+ implies the existence of a weighted pseudo-almost periodic solution of Eq. (1.1).The existence and uniqueness of almost periodic type solutions of differential equations are always among the most
attractive topics due to their significance and application in areas such as physics, control theory, biology, and so on. In [3–5],
I Supported by the National Natural Science Foundation of China (No. 10471155).∗ Corresponding author.E-mail addresses: [email protected] (L. Zhang), [email protected], [email protected] (Y. Xu).
0362-546X/$ – see front matter Crown Copyright© 2009 Published by Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2009.02.032
3706 L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714
Zhang firstly introduces the notion of pseudo-almost periodicity as a natural generalization of the classical almost periodicityin the sense of Bohr [6,7] and a implement of the asymptotic almost periodicity due to Fréchet [8,9]. After that, in [10],Diagana introduces new classes of functions called weighted pseudo-almost periodic functions which implement Zhang’spseudo-almost periodic functions. For more details, we refer to [10,11]. In [11], Diagana considered the following abstractdifferential equation
ddtu(t) = Au(t)+ f (t, u(t)), t ∈ R, (1.2)
where A is the infinitesimal generator of an exponentially stable c0-semigroup on a Banach space X and f : R × X → Xis a weighted pseudo-almost function in the fist variable and uniformly in the second variable. Under some assumptions,the author showed that Eq. (1.2) has an unique cl(ρ)-pseudo-almost periodic solution. In [12], by means of the variation ofconstants formula and exponential dichotomy, we proved that the following functional differential equations
ddtx(t) = Lxt + f (t), t ∈ R, (1.3)
ddtx(t) = L(t)xt + f (t), t ∈ R, (1.4)
have one and only oneweighted pseudo-almost periodic solutionwhen f is weighted pseudo-almost periodic and the aboveequations have a bounded solution respectively. In [13], Diagana studied the integral equation
u(t) = f (u(h1(t)))+∫∞
tQ (s, u(s), u(h2(s)))C(t − s)ds+ g(t), t ∈ R, (1.5)
where f , g , h1, h2, C : R → R are continuous functions with hi(R) = R for i = 1, 2, and Q : R × R × R → R is jointlycontinuous. Under some suitable assumptions, the existence and uniqueness of a weighted pseudo-almost periodic solutionto Eq. (1.5) is attained.This paper is organized as follows, in Section 2, we recall the definition and basic properties of weighted pseudo-almost
periodic functions and prove the equivalence of the bounded solution on R+ and the weighted pseudo-almost periodicityof the solution to the linear differential equations with constant coefficients. In Section 3, we give the variation of constantsformula thatwill be used in this paper and give the reduction principle of Eq. (1.1) to a finite dimensional ordinary differentialequation. In Section 4, we use the reduction principle to show that the existence of a bounded solution on R+ impliesthe existence of a weighted pseudo-almost periodic solution of Eq. (1.1). In hyperbolic case, we prove that Eq. (1.1) has aunique bounded solution which is weighted pseudo-almost periodic. To illustrate our results, in the last section, we studythe Lotka–Volterra equation with diffusion.
2. Weighted pseudo-almost periodic functions
In this section, we recall weighted pseudo-almost periodic functions and their basic properties. Let (X, ‖ · ‖) be a Banachspace and BC(R,X) denote the collection of all X-valued bounded continuous functions. The space BC(R,X) equipped withthe sup norm defined by
‖u‖ = supt∈R‖u(t)‖
is a Banach space. Furthermore, C(R,X) denotes the class of continuous functions from R in X.
Definition 2.1. A function f ∈ C(R,X) is called (Bohr) almost periodic if for each ε > 0 there exists l(ε) > 0 such thatevery interval of length l(ε) contains a number τ with the property that
‖f (t + τ)− f (t)‖ < ε, for all t ∈ R.
The number τ above is called an ε-translation number of f , and the collection of all such functions will be denoted as AP(X).
Let U denote the collection of functions (weights) ρ : R→ (0,∞), which are locally integrable over R such that ρ > 0almost everywhere. If ρ ∈ U and for T > 0, then set
µ(T , ρ) :=∫ T
−Tρ(x)dx.
Thus the space of weights U∞ is defined by
U∞ := {ρ ∈ U : limT→∞
µ(T , ρ) = ∞}.
In particular, ρ(x) = 1 for each x ∈ R is in U∞. In addition, define UB by
UB := {ρ ∈ U∞ : ρ is bounded with infx∈Rρ(x) > 0}.
L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714 3707
Obviously, UB ⊂ U∞ ⊂ U, with strict inclusions.Let ρ ∈ U∞. Set
PAP0(X) :={f ∈ BC(R,X) : lim
T→∞
12T
∫ T
−T‖f (σ )‖dσ = 0
}.
and
PAP0(X, ρ) :={f ∈ BC(R,X) : lim
T→∞
1µ(T , ρ)
∫ T
−T‖f (σ )‖ρ(σ)dσ = 0
}.
Definition 2.2 ([3]). A function f ∈ BC(R,X) is called pseudo-almost periodic if it can be expressed as f = g + ϕ, whereg ∈ AP(X) and ϕ ∈ PAP0(X). The collection of such functions will be denoted by PAP(X).
Definition 2.3. Let ρ ∈ U∞. A function f ∈ BC(R,X) is called weighted pseudo-almost periodic (or ρ-pseudo-almostperiodic) if it can be expressed as f = g + ϕ, where g ∈ AP(X) and ϕ ∈ PAP0(X, ρ). The collection of such functions will bedenoted by PAP(X, ρ).
Remark 2.4. (i) Note that PAP0(X) = PAP0(X, ρ), when ρ(x) = 1 for all x ∈ R, so the weighted pseudo-almost periodicfunctions implement Zhang’s pseudo-almost periodic functions.
(ii) It is not hard to see that PAP0(X, ρ) is a closed subspace of (BC(R,X), ‖ · ‖).(iii) From [11], for ρ ∈ U∞, the decomposition of a ρ-pseudo-almost periodic function f = g + ϕ, where g ∈ AP(X) and
ϕ ∈ PAP0(X, ρ), is unique.
Definition 2.5. Let ρ1, ρ2 ∈ U∞. One says that ρ1 is equivalent to ρ2, denoting this as ρ1 ≺ ρ2, ifρ1ρ2∈ UB.
For ρ1, ρ2, ρ3 ∈ U∞. It is clear that ρ1 ≺ ρ1 (reflexivity); if ρ1 ≺ ρ2, then ρ2 ≺ ρ1 (symmetry); and if ρ1 ≺ ρ2 andρ2 ≺ ρ3, then ρ1 ≺ ρ3 (transitivity). So, ≺ is a binary equivalence relation on U∞. Thus the equivalence class of a givenweighted ρ ∈ U∞ will then be denoted by
cl(ρ) = {$ ∈ U∞ : ρ ∈ $ }.
It is then clear that U∞ =⋃ρ∈U∞ cl(ρ).
Let ρ ∈ U∞. If ρ1, ρ2 ∈ cl(ρ), then PAP(X, ρ1) = PAP(X, ρ2). In particular, if ρ ∈ UB, then PAP(X, ρ) = PAP(X, cl(1)) =PAP(X).For more details about weighted pseudo-almost periodic functions, one can refer to [10,11].Now we consider the systems of the form
dYdt= BY + F , (2.1)
where B = (bij) is an n× nmatrix and F ∈ PAP(Rn, ρ), where ρ ∈ U∞.
Theorem 2.6. Eq. (2.1) has a bounded solution on R+ if and only if it has a unique ρ-pseudo-almost periodic solution on R.
Proof. Note that the ρ-pseudo-almost periodic functions are bounded, we only need to prove the necessity. The existenceof the bounded solution on R+ implies matrix B has no eigenvalues with null real part, so there exists a bounded solutionof Eq. (2.1) on the whole R.Firstly, we consider the scale differential equation
dydt= λy+ p(t), (2.2)
where λ = µ+ iν ∈ Cwith µ 6= 0 and p(t) ∈ BC(R,R).Claim I: If p ∈ AP(R) (or PAP0(R, ρ)), then Eq. (2.2) has a bounded solution y(x) ∈ AP(R) (or PAP0(R, ρ)).In fact, the unique bounded solution of Eq. (2.2) is
y(t) =
−
∫∞
te−λ(t−s)p(s)ds, µ > 0,∫ t
−∞
eλ(t−s)p(s)ds, µ < 0.
We only consider the case µ > 0, one can proceed analogously when µ < 0.
3708 L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714
If p ∈ AP(R), for all t ∈ R, then
‖y(t + τ)− y(t)‖ ≤1µsups∈R|p(s+ τ)− p(s)|, for all τ ∈ R.
This indicates that every εµ-transition of p is an εµ-transition of y, thus y ∈ AP(R).If P ∈ PAP0(R, ρ), then
1µ(T , ρ)
∫ T
−T‖y(t)‖ρ(t)dt =
1µ(T , ρ)
∫ T
−T
∥∥∥∥− ∫ ∞te−λ(t−s)p(s)ds
∥∥∥∥ ρ(t)dt≤
1µ(T , ρ)
∫ T
−Tρ(t)dt
∫∞
t‖p(s)‖eµ(t−s)ds
=1
µ(T , ρ)
∫ T
−Tρ(t)dt
∫ T
t‖p(s)‖eµ(t−s)ds+
1µ(T , ρ)
∫ T
−Tρ(t)dt
×
∫∞
T‖p(s)‖eµ(t−s)ds. (2.3)
Set
J1 :=1
µ(T , ρ)
∫ T
−Tρ(t)dt
∫ T
t‖p(s)‖eµ(t−s)ds
≤1
µ(T , ρ)
∫ T
−T‖p(t)‖ρ(t)dt
∫ T
teµ(t−s)ds
=1
µ(T , ρ)
∫ T
−T‖p(t)‖ρ(t)
1µ(1− eµ(t−T ))dt. (2.4)
Note that 1µ(1− eµ(t−T )) is bounded since t ≤ T and µ > 0. Then
limT→∞
1µ(T , ρ)
∫ T
−T‖p(t)‖ρ(t)dt = 0
implies J1 → 0 as T →∞.Set
J2 :=1
µ(T , ρ)
∫ T
−Tρ(t)dt
∫∞
T‖p(s)‖eµ(t−s)ds
≤1
µ(T , ρ)
∫ T
−Tρ(t)dt
∫∞
T‖p(s)‖ds. (2.5)
Then J2 → 0 as T →∞. Note that
1µ(T , ρ)
∫ T
−T‖y(t)‖ρ(t)dt = J1 + J2,
thus y(t) ∈ PAP0(R, ρ). We proved Claim I.Since the decomposition of aρ-pseudo-almost periodic function is unique, then the bounded solution y(t) is in PAP(R, ρ),
provided p ∈ PAP(R, ρ).
Claim II: If F ∈ PAP(Rn, ρ), the bounded solution of Eq. (2.1) is ρ-pseudo-almost periodic. In fact, without loss of generality,we can suppose that the matrix B is triangular, i.e.,
dy1dt= λ1y1 + b12y2 + · · · + b1nyn + f1
dy2dt= λ2y2 + · · · + b2nyn + f2
...dyndt= λnyn + fn.
(2.6)
L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714 3709
Note that the last equation of (2.6) has the form of Eq. (2.2). Therefore, by Claim I, there exists a unique solution yn ∈PAP(R, ρ) for the last equation, that is,
yn(t) =
−
∫∞
te−λn(t−s)fn(s)ds, Re λn > 0,∫ t
−∞
eλn(t−s)fn(s)ds, Re λn < 0.
Substituting this yn in the second last equation of Eq. (2.6),we get for yn−1 an equation of the formEq. (2.2). Since Re λn−1 6= 0,it follows that we have a unique bounded solution of the second last equation. Applying successively the Claim I, one canprove the Claim II. The proof of Theorem 2.6 is complete. �
Remark 2.7. The unique weighted solution Y of Eq. (2.1) is in PAP(Rn, cl(ρ)).
Theorem 2.6 plays an important role in the proof of Theorem 4.1 in Section 4.
3. Variation of constants formula and spectral analysis
In this section, we give the variation of constants formula that will be used in the paper and establish fundamental resultsabout the spectral decomposition of solutions to Eq. (1.1). Throughout this paper, we assume that(H1): Operator A satisfies the Hille–Yosida condition.
Definition 3.1. We say that a continuous function u from [−r,∞) into X is an integral solution of Eq. (1.1), if the followingconditions hold:
(i)∫ t0 x(s)ds ∈ D(A), for t ≥ 0;
(ii) x(t) = ϕ(0)+ A∫ t0 x(s)ds+
∫ t0 (Lxs + f (s))ds, for t ≥ 0;
(iii) x0 = ϕ.
Let us consider the part A0 of the operator A inD(A)which defined by{D(A0) = {x ∈ D(A) : Ax ∈ D(A)},A0x = Ax, for x ∈ D(A0).
From [14], we know that operator A0 defined above generates a strongly continuous semigroup (T0(t))t≥0 onD(A).For the existence of the integral solutions, one has the following result.
Theorem 3.2 ([15]). Assume that (H1) holds, then for all ϕ ∈ C such that ϕ(0) ∈ D(A), Eq. (1.1) has a unique integral solutionx on [−r,∞). Moreover, x is given by
x(t) = T0(t)ϕ(0)+ limλ→∞
∫ t
0T0(t − s)Bλ(Lxs + f (s)), for t ≥ 0,
where Bλ = λR(λ− A)−1, for λ > ω.For simplicity, we call integral solutions just solutions.The phase space C0 of Eq. (1.1) is defined by
C0 = {ϕ ∈ C : ϕ(0) ∈ D(A)}.
For each t ≥ 0, define the linear operator U on C0 by
Uϕ = vt(·, ϕ),
where v(·, ϕ) is the solution of the following homogeneous equation{ ddtv(t) = Av(t)+ Lvt , for t ≥ 0,
v0 = ϕ ∈ C0.
In [15], we know that (U(t))t≥0 is a strongly continuous semigroup of linear operators on C0:
(i) for all t ≥ 0, U(t) is a bounded linear operator on C0;(ii) U(0) = I;(iii) U(t + s) = U(t)U(s), for all t, s ≥ 0;(iv) for all ϕ ∈ C0, U(t)ϕ is a continuous function of t ≥ 0 with values in C0.
3710 L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714
Moreover,(v) (U(t))t≥0 satisfies, for t ≥ 0 and θ ∈ [−r, 0], the following translation property:
U(t)ϕ(θ) ={(U(t + θ))(0), if t + θ ≥ 0,ϕ(t + θ), if t + θ ≤ 0.
Theorem 3.3 ([1]). Let Au be defined on C0 by{D(Au) = {ϕ ∈ C1([−r, 0];X) : ϕ(0) ∈ D(A), ϕ′(0) ∈ D(A) and ϕ′(0) = Aϕ(0)+ L(ϕ)},Auϕ = ϕ′, for ϕ ∈ D(Au).
Then, Au is the infinitesimal generator of the semigroup (U(t))t≥0 on C0.
Let 〈X0〉 be the space defined by
〈X0〉 = {X0c : c ∈ X}.
where the function X0c is defined by
(X0c)(θ) ={0, if θ ∈ [−r, 0),c, if θ = 0.
The space C0 ⊕ 〈X0〉 is equipped with the norm |ϕ + X0c| = |ϕ|C + |c| for (ϕ, c) ∈ C0 × X, is a Banach space. Consider theextension Au of the operator Au defined on C0 ⊕ 〈X0〉 by{
D(Au) = {ϕ ∈ C1([−r, 0];X) : ϕ(0) ∈ D(A), ϕ′(0) ∈ D(A)},Auϕ = ϕ′ + X0(Aϕ(0)+ Lϕ − ϕ′(0)).
Under the assumption that (H1) holds, from [1], we know that Au satisfies the Hille–Yosida condition on C0⊕〈X0〉, i.e., thereexist M ≥ 0, ω ∈ R such that (ω,∞) ⊂ ρ(Au) and
|(λ− Au)−n| ≤M
(λ− ω)n, for n ∈ N, λ > ω.
Moreover, the part of Au onD(Au) = C0 is exactly the operator Au.
Theorem 3.4 ([1]). Assume that (H1) holds. Then, for t ≥ 0,
xt = U(t)ϕ + limλ→∞
∫ t
0U(t − s)(Bλ(X0)f (s))ds, for t ≥ 0,
where Bλ = λ(λ− Au)−1, for λ > ω.
In the following, we assume that(H2): the operator T0(t) is compact onD(A), for every t > 0,then U(t) defined above is compact for t > r . Therefore, we have that the spectrum σ(Au) is the point spectrum and
σ(Au)+ {λ ∈ C : ker∆(λ) 6= {0}}, where the linear operator∆(λ) : D(A)→ X is defined by
∆(λ) = λI − A− L(eλ·I)
and eλ·I : X→ C is defined by
(eλ·x)(θ) = eλθx, x ∈ X, θ ∈ [−r, 0].
From [1], we know that C0 can be decomposed as follows:
C0 = S⊕
V,
where S is U-invariant and there are positive constants α and N such that
|U(t)ϕ|C ≤ Ne−αt |ϕ|C , for each t ≥ 0 and ϕ ∈ S.
V is a finite dimensional space and the restriction of U to V is a group.LetUS(t) andUV(t) denote the restriction ofU(t) respectively on S andV which correspond to the above decomposition.Let d = dimV with a vector basisΦ = {φ1, . . . , φd}. Then, there exist d-elements {ψ1, . . . , ψd} in C∗0 , the dual space of
C0, such that{〈ψi, φj〉 = δij,〈ψi, φ〉 = 0, for all φ ∈ S and i = 1, . . . , d,
where 〈·, ·〉 denotes the duality pairing between C∗0 and C0 and δij is 1 for i = j and 0 otherwise.
L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714 3711
Let Ψ = col{ψ1, . . . , ψd}, 〈Ψ ,Φ〉 is a (d× d)-matrix, where the (i, j)-component is 〈ψi, φj〉. Denote byΠS andΠV theprojections respectively on S and V . For every ϕ ∈ C0, we have
ΠVϕ = Φ〈Ψ , ϕ〉.
Since (UV(t))t≥0 is a group on V , then there exists a (d× d)-matrix B such that
UV(t)Φ = ΦetB, for t ∈ R.Moreover, σ(B) = {λ ∈ σ(Au) : Re (λ) ≥ 0}.For n, n0 ∈ N such that n ≥ n0 ≥ ω and i = 1, . . . , d. Define the linear operator x∗i,n by
x∗i,n(a) = 〈ψi, BnX0a〉, for a ∈ X .
Since Bn ≤ nn−ω M , for each n ≥ n0, then x
∗
i,n is a bounded linear operator from X to R such that
|x∗i,n| ≤n
n− ωM|ψi|, for each n ≥ n0.
Define the d-column vector x∗ = col(x∗1,n, . . . , x∗
d,n). Then there is the following theorem.
Theorem 3.5 ([2]). There exists x∗ ∈ L(X,Rd), such that (x∗n)n≥n0 converges weakly to x∗ in the sense that
〈x∗n, x〉 → 〈x∗, x〉, as n→∞, for all x ∈ X.
Thus, for all continuous function h : R→ X, we have
limn→∞
∫ t
σ
UV(t − s)ΠV(BnX0h(s))ds = Φ∫ t
σ
e(t−s)B〈x∗, h(s)〉ds, for all t, σ ∈ R.
Theorem 3.6 ([2]). Assume that (H1) and (H2) hold. Let u be a solution of Eq. (1.1) on R. Then, z(t) = 〈Ψ , ut〉 is a solution ofthe ordinary differential equation
dz(t)dt= Bz(t)+ 〈x∗, f (t)〉, t ∈ R. (3.1)
Conversely, if f is a bounded function on R and z is a solution of Eq. (3.1) on R, then the function u given by
u(t) =(Φz(t)+ lim
n→∞
∫ t
−∞
US(t − s)ΠS(BnX0f (s))ds)(0), for t ∈ R,
is a solution of Eq. (1.1) on R.
4. Main results
Theorem 4.1. Assume that (H1), (H2) hold and f is a ρ-pseudo-almost periodic function. If Eq. (1.1) has a bounded solution onR+, then it has a unique ρ-pseudo-almost periodic solution.
Proof. Let u be a bounded solution of Eq. (1.1) on R+. By Theorem 3.6, the function z(t) = 〈Ψ , ut〉, for t ≥ 0, is a solutionof the ordinary differential equation (2.1) and z is bounded on R+.Moreover, the function
h(t) = 〈x∗, f (t)〉, for t ∈ R
is a ρ-pseudo-almost periodic from R to Rd. By Theorem 2.6, we get that the reduced system (3.1) has a ρ-pseudo-almost periodic solution z and Φ z(·) is ρ-pseudo-almost periodic on R. From Theorem 3.6, we know that the functionu(t) = v(t)(0), where
v(t) = Φ z(t)+ limn→∞
∫ t
−∞
US(t − s)ΠS(BnX0f (s))ds, for t ∈ R,
is a solution of Eq. (1.1) on R. In the sequence, we will show that v is a ρ-pseudo-almost periodic function. In fact, since f isρ-pseudo-almost periodic, then assume that f = g + ϕ, where g is its almost periodic component and ϕ satisfies
limT→∞
1µ(t, ρ)
∫ T
−T‖ϕ(t)‖ρ(t)dt = 0.
It suffices to show that
p(t) =: limn→∞
∫ t
−∞
US(t − s)ΠS(BnX0g(s))ds
3712 L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714
is almost periodic and
q(t) =: limn→∞
∫ t
−∞
US(t − s)ΠS(BnX0ϕ(s))ds
satisfies
limT→∞
1µ(t, ρ)
∫ T
−T‖q(t)‖ρ(t)dt = 0.
Let (αm)m≥0 be any real sequence. Then, there exists a subsequence (αmk)k≥0 such that g(t + αmk) converges uniformlyon R to some function g1. By the Lebesgue’s dominated convergence theorem, we get that the sequence of the functions
p(t + αmk) = limn→∞
∫ t+αmk
−∞
US(t + αmk − s)ΠS(BnX0g(s))ds
= limn→∞
∫ t
−∞
US(t − s)ΠS(BnX0g(αmk + s))ds (4.1)
converges uniformly for t ∈ R to the following functions
limn→∞
∫ t
−∞
US(t − s)ΠS(BnX0g1(s))ds.
This holds for any sequence (αm)m≥0, which implies that p(t) is almost periodic.On the other hand, we get a positive constantM such that
limT→∞
1µ(t, ρ)
∫ T
−T‖q(t)‖ρ(t)dt = lim
T→∞
1µ(t, ρ)
∫ T
−T
∥∥∥∥ limn→∞∫ t
−∞
US(t − s)ΠS(BnX0ϕ(s))ds∥∥∥∥ ρ(t)dt
= limT→∞
1µ(t, ρ)
∫ T
−T
∥∥∥∥ limn→∞∫∞
0US(t − s)ΠS(BnX0ϕ(t − s))ds
∥∥∥∥ ρ(t)dt= limT→∞
1µ(t, ρ)
∫ T
−TM‖ϕ(t)‖ρ(t)dt
= 0, (4.2)
that is, q(t) ∈ PAP0(Rn, ρ). This completes the proof of Theorem 4.1. �
Remark 4.2. In Theorem 4.1, the unique weighted pseudo-almost solution is in PAP(Rn, cl(ρ)). Note that if ρ ∈ UB, thenPAP(X, ρ) = PAP(X, cl(1)) = PAP(X), Theorem 4.1 implies Eq. (1.1) has a pseudo-almost periodic solution on R if and onlyif it has a bounded solution on R+ and ρ ∈ UB.
Definition 4.3. We say that the semigroup (U(t))t≥0 is hyperbolic if
σ(Au)⋂iR = ∅.
From the compactness of the semigroup (U(t))t≥0 and from [16], we get the following result.
Theorem 4.4. Assume that (H2) holds. If the semigroup (U(t))t≥0 is hyperbolic, then the space C0 is decomposed as a direct sumC0 = S
⊕U of two U(t) invariant closed subspaces S andU such that the restricted semigroup onU is a group and there exist
positive constants N0 and ε0 such that
|U(t)ϕ| ≤ N0e−ε0t |ϕ|, ∀ t ≥ 0, ϕ ∈ S,
|U(t)ϕ| ≤ N0eε0t |ϕ|, ∀ t ≤ 0, ϕ ∈ U.
As a consequence of the hyperbolicity we get the uniqueness of the bounded solution of Eq. (1.1).
Theorem 4.5. Assume that (H2) holds and the semigroup (U(t))t≥0 is hyperbolic. If f is bounded on R, then Eq. (1.1) has uniquebounded solution on R which is ρ-pseudo almost periodic if f is ρ-pseudo almost periodic.Proof. By the hyperbolicity of (U(t))t≥0, then Eq. (1.1) has one and only one bounded solution on R which is given by thefollowing formula(
limn→+∞
∫ t
−∞
US(t − s)ΠS(Bn(X0(f (s))))ds− limn→+∞
∫+∞
tUU(t − s)ΠU(Bn(X0(f (s))))ds
)(0).
By Theorem3.6 and the proof of Theorem4.1, we conclude that this solution isρ-pseudo almost periodicwhen f is ρ-pseudoalmost periodic. The proof is complete. �
L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714 3713
5. Application
To illustrate the previous results, we consider the following Lotka–Volterra model with diffusion ∂
∂tv(t, ξ) =
∂2
∂ξ 2v(t, s)+
∫ 0
−rη(θ)v(t + θ, s)dθ + F(t, s), t ∈ R, t ≥ 0, s ∈ [0, π],
v(t, 0) = v(t, π) = 0, t ≥ 0,(5.1)
where η is a positive function on (−∞, 0] and F : R × [0, π] → R is a continuous and ρ-pseudo almost periodic int uniformly with respect to x ∈ [0, π]. Let X = C([0, π],R) be the space of all continuous functions from [0, π] to Rendowed with the uniform norm topology. Consider the operator A : D(A) ⊂ X defined by{
D(A) = {z ∈ C2([0, π],R) : z(0) = z(π) = 0},Az = z ′′.
From [17], we know that A satisfies the Hille–Yosida condition on X.Obviously,D(A) = {ψ ∈ C([0, π],R) : ψ(0) = ψ(π) = 0}.Define(L(φ))(x) =
∫ 0
−rη(θ)φ(θ)(x)d(θ), x ∈ [0, π], φ ∈ C ,
f (t)(x) = F(t, x), t ∈ R, x ∈ [0, π].
One can see that L is a bounded linear operator from C to X and f : R → X is ρ-pseudo almost periodic. Putx(t)(s) = v(t, s) for t ∈ R+ and s ∈ [0, π]. Then Eq. (5.1) takes the abstract form
dx(t)dt= Ax(t)+ Lxt + f (t). (5.2)
The part A0 of A in D(A) is given by{D(A0) = {z ∈ C2([0, π],R) : z(0) = z(π) = z ′′(0) = z ′′(π) = 0},A0z = z ′′.
From [16], A0 generates a compact c0-semigroup onD(A).
Theorem 5.1. Assumes 0 <∫ 0−r η(θ)dθ < 1. Then Eq. (5.1) has a unique ρ-pseudo almost periodic solution.
Proof. By Definition 4.3 and Theorem 4.5, it suffices to show that σ(Au)⋂iR = ∅. In fact, let λ ∈ σ(Au), Re λ > 0, then
there exists x ∈ D(A), x 6= 0 such that(λ− A−
∫ 0
−rη(θ)eλθdθ
)x = 0.
The spectrum σ(A) is reduced to the point spectrum σp(A) and σp(A) = {−n2 : n ∈ N∗}. So, λ ∈ σ(Au)with Re λ > 0 if andonly if
λ−
∫ 0
−rη(θ)eλθdθ = −n2, n ≥ 1.
Then
Re (λ) =∫ 0
−rη(θ)eRe λθ cos(Im(λ))dθ − n2 (5.3)
≤
∫ 0
−rη(θ)dθ − n2. (5.4)
Since 0 <∫ 0−∞
η(θ)dθ < 1, then there is a contradiction with Re (λ) ≥ 0. Consequently, semigroup (U(t))t≥0 is hyperbolic.The proof is complete. �
3714 L. Zhang, Y. Xu / Nonlinear Analysis 71 (2009) 3705–3714
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