Ouhinou, Variation of constants formula and almost-periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applica
Trang 1solutions for partial functional differential equations
with infinite delay Khalil Ezzinbia, Samir Fatajoua, Gaston Mandata N’gu´er´ekatab,∗
a Universit´e Cadi Ayyad, Facult´e des Sciences Semlalia, D´epartement de Math´ematiques, B.P 2390 Marrakech, Morocco
b Department of Mathematics, Morgan State University, 1700 East Cold Spring Lane, Baltimore, MD 21251, USA
Received 23 May 2007; accepted 29 June 2007
Abstract
In this paper, we study the existence of C(n)-almost-periodic solutions for partial functional differential equations with infinite
delay We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition We use the reduction principle developed recently in [M Adimy, K Ezzinbi, A Ouhinou, Variation of constants formula and almost-periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications
317 (2006) 668–689] to prove the existence of a C(n)-almost- periodic solution when there is at least one bounded solution in R+
We give an application to the Lotka–Volterra model with diffusion
c 2007 Elsevier Ltd All rights reserved
MSC: 34C27; 34K14; 35R10
Keywords: Hille–Yosida condition; Infinite delay; C0-semigroup; Integral solution; Fading memory space; Reduction principle; C (n)
-almost-periodic solution; Exponential dichotomy
1 Introduction
The aim of this work is to study the existence of C(n)-almost-periodic solutions for the following partial functional
differential equation with infinite delay
d
where A : D(A) → X is a not necessarily densely defined linear operator on a Banach space X, for every t ∈ R, the history function xt ∈B is defined by
xt(θ) = x(t + θ) for θ ∈ (−∞, 0],
∗ Corresponding author Tel.: +1 443 885 3964; fax: +1 443 885 8216.
E-mail addresses: ezzinbi@ucam.ac.ma (K Ezzinbi), gnguerek@morgan.edu (G.M N’gu´er´ekata).
0362-546X/$ - see front matter c 2007 Elsevier Ltd All rights reserved.
doi:10.1016/j.na.2007.06.041
Trang 2where B is a normed linear space of functions mapping(−∞, 0] to X and satisfying some fundamental axioms given
by Hale and Kato in [7] L is a bounded linear operator from B to X and f is an almost-periodic X -valued function
on R They assume that the undelayed part A satisfies the Hille–Yosida condition
(H0) there exist M0≥1,ω0∈ R such that(ω0, +∞) ⊂ ρ(A) and
|(λI − A)−n| ≤ M0
(λ − ω0)n for n ∈ N and λ > ω0, whereρ(A) is the resolvent set of A Recall in [10], Massera proposed a new approach to show the existence of periodic solutions of ordinary differential equations in finite dimensional spaces For some kind of ordinary differential equations, the author proved the existence of periodic solutions under a minimal condition, namely, the existence of
a bounded solution on R+ is enough to get periodic solutions Many authors used Massera’s approach to prove the existence of periodic, almost periodic or C(n)-almost-periodic solutions in the context of differential equations.
In [2], the authors proved that the existence of a bounded solution on R+implies the existence of an almost-periodic solution Firstly, they established a new variation of constants formula for Eq.(1.1) Secondly, they used the spectral decomposition of the phase space to get a (new) reduction principle of Eq.(1.1)to a finite dimensional space when B
is a uniform fading memory space More precisely, they established a relationship between the bounded solutions of
Eq.(1.1)on R+with the bounded solutions on R for an ordinary differential equation in a finite dimensional space
In this work, we propose to use the reduction principle established in [2] and the Massera’s approach to show the existence of C(n)-almost-periodic solutions of Eq.(1.1).
C(n)-almost-periodic functions are functions such that the i th derivative are almost periodic for i = 1, , n; they have many applications in dynamical systems For more details, we refer to [1] and references therein In [4], the authors discussed some properties of C(n)-almost-periodic functions taking values in Banach spaces.
In [3], the authors proved the existence of a C(n)-almost-periodic solution for the following nonautonomous
differential equation
d
where A(t) generates an exponentially stable family in a Banach space, they showed that when θ is C(n)-almost
periodic, then the only bounded solution of Eq.(1.2)is also Cn-almost periodic In [9], the authors proved the existence
of C(n)-almost-periodic solutions for some ordinary differential equations by using the exponential dichotomy
approach
In this work, we first discuss the existence of C(n)-almost- periodic solution for the following ordinary differential
equation
d
where G is a constant n × n-matrix and e : R → Rnis C(n)-almost periodic We then prove the Massera-type theorem
for the existence of C(n)-almost-periodic solution, more precisely, we prove that the existence of a bounded solution
in R+implies the existence of a C(n)-almost-periodic solution Moreover, we show that every bounded solution in R
is C(n+1)-almost periodic We use the reduction principle developed recently in [2] to obtain a Massera-type theorem
for the existence of C(n)-almost-periodic solution of Eq.(1.1).
This work is organized as follows, in Section2, we recall some results on the existence of solutions of Eq.(1.1)and
we give the variation of constants formula that will be used in this work In Section3, we give the reduction principle
of Eq.(1.1)to a finite dimensional ordinary differential equation Section4is devoted to state some results on C(n)
-almost-periodic functions In Section5, we use the reduction principle to show that the existence of a bounded solution
on R+implies the existence of an C(n)-almost-periodic solution of Eq.(1.1) In Section6, we prove the existence and
uniqueness of an C(n)-almost-periodic solution of Eq.(1.1)where the solution semigroup of Eq.(1.1)with f = 0
has an exponential dichotomy Finally, for illustration, we propose to study the existence of a C(n)-almost-periodic
solution for a Lotka–Volterra model with diffusion
Trang 32 Integral solutions and variation of constants formula
We use the (classical) axiomatic approach of Hale and Kato [7] for the phase space B We assume that(B, k·k)
is a normed space of functions mapping(−∞, 0] into a Banach space X and satisfying the following fundamental axioms:
(A) there exist a positive constant N , a locally bounded function M(·) on [0, +∞) and a continuous function K (·) on [0, +∞), such that if x : (−∞, a] → X is continuous on [σ, a] with x ∈ B, for some σ < a, then for all t ∈ [σ, a], (i) xt ∈B,
(ii) t → xt is continuous with respect to k·k on [σ, a],
(iii) N |x(t)| ≤ kxtk ≤ K(t − σ ) sup
σ ≤s≤t|x(s)| + M(t − σ) kxσk
(B) B is a Banach space
The following lemma is well known
Lemma 2.1 ([2, p 140]) Assume that(H0) holds Let A0be the part of the operator A in D(A), which is defined by
D(A0) = {x ∈ D(A) : Ax ∈ D(A)}
A0x = Ax
Then A0generates a C0-semigroup(T0(t))t ≥0on D(A)
To Eq.(1.1), we associate the following Cauchy problem
(d
dtx(t) = Ax(t) + L(xt) + f (t) for t ≥ σ,
The following results are taken from [2]
Definition 2.2 ([2]) Letφ ∈ B A function u : R → X is called an integral solution of Eq.(2.1)on R if the following conditions hold:
(i) u is continuous on [σ, ∞),
(ii) uσ =φ,
(iii) Rt
σu(s)ds ∈ D(A) for t ≥ σ ,
(iv) u(t) = φ(0) + A Rt
σu(s)ds + Rt
σ L(us)ds + Rt
σ f(s)ds for t ≥ σ For simplicity, integral solutions will be called solutions in this work
Theorem 2.3 ([2]) Assume that(H0), (A) and (B) hold Then for all φ ∈ B such that φ(0) ∈ D(A), Eq.(2.1)has a unique solution u = u(·, φ, L, f ) on R which is given by
u(t) =
T0(t − σ )φ(0) + lim
λ→+∞
Z t
σ T0(t − s)λR(λ, A) [L(us) + f (s)] ds, for t ≥ σ, φ(t) for t ≤ σ
Let BA= {φ ∈ B : φ(0) ∈ D(A)} be the phase space corresponding to Eq.(2.1) For t ≥ 0, we define the operator
U(t) for φ ∈ BA, by
U(t)φ = ut(·, φ, L, 0),
where u(·, φ, L, 0) is the solution of Eq.(2.1)with f = 0 andσ = 0
Theorem 2.4 ([2]) Assume that(H0), (A) and (B) hold Then (U(t))t ≥0is a C0-semigroup onBA That is
(i) U(0) = I d,
(ii) U(t + s) = U(t)U(s) for t, s ≥ 0,
(iii) for allφ ∈ B , t 7→ U(t)φ is continuous from [0, ∞) to B
Trang 4Moreover,(U(t))t ≥0satisfies, for t ≥0, φ ∈ BA, the translation property
(U(t)φ) (θ) =
(U(t + θ)φ) (0), for t + θ ≥ 0 φ(t + θ), for t + θ ≤ 0
Due to the relationship between the semigroup and its generator, it is fundamental to compute the infinitesimal generator of(U(t))t ≥0; to this end, we assume furthermore that
(D1) If (φn)nis a sequence in B such thatφn→0 in B as n → +∞, then for allθ ≤ 0, φn(θ) → 0 in X as n → +∞ (D2) B ⊂ C((−∞, 0]; X), where C((−∞, 0]; X) is the space of continuous functions from (−∞, 0] into X (D3) There exists λ0∈ R such that for all λ ∈ C with Re λ > λ0and x ∈ X , we have eλ·x ∈B and
Re λ>λ0,x∈X
x 6=0
eλ·x
|x | < ∞, where
(eλ·x)(θ) = eλθx forθ ∈ (−∞, 0] and x ∈ X
The aim of the following result is the computation of the infinitesimal generator of(U(t))t ≥0
Lemma 2.5 ([2]) Assume that(H0), (A), (B), (D1) and (D2) hold Then the infinitesimal generator AU of(U(t))t ≥0
is given by
D(AU) = {φ ∈ C1((−∞, 0]; X) ∩ BA :φ0∈BA,φ(0) ∈ D(A) and φ0(0) = Aφ(0) + L(φ)},
AUφ = φ0
Recently in [2], a variation of constants formula for Eq.(2.1)has been established In order to recall this formula,
we need to give some preliminary results Firstly, we consider the space X := BA⊕ hX0i, where
hX0i = {X0x : x ∈ X }
and X0xis the discontinuous function defined by
(X0x) (θ) =0 forx forθ ∈ (−∞, 0)θ = 0.
The space X endowed with the norm
kφ + X0x k = kφk + |x|
is a Banach space
According to Axiom (D3), we define for each complex number λ such that Re λ > λ0, the linear operator 1(λ) : D(A) → X by
1(λ) = λI − A − L(eλ.I),
where L(eλ.I) is a bounded linear operator on X, which is defined by
L(eλ.I)(x) = L(eλ.x) for x ∈ X
Theorem 2.6 ([2]) Assume that(H0), (A), (B), (D1)–(D3) hold Then the extensionAfU of the operator AU defined
on X by
D(AfU) = {φ ∈ BA:φ0∈BAandφ(0) ∈ D(A)},
f
AUφ = φ0+X0(Aφ(0) + Lφ − φ0(0)),
satisfies the Hille–Yosida condition on X More precisely, there existsω ∈ R such that (ω, ∞) ⊂ ρ AfU and for
λ > ω, φ ∈ BA, x ∈ X, n ∈ N∗, one has
λ −AU−n
(φ + X0x) = (λ − AU)−nφ + (λ − AU)n−1(eλ.1(λ)−1x)
Trang 5Theorem 2.7 ([2]) Assume that(H0), (A), (B), (D1)–(D3) hold Let φ ∈ BA Then the corresponding solution u of
Eq.(2.1)is given by the following variation of constants formula
ut =U(t − σ)φ + lim
n→+∞
Z t
where fBn=n(nI −AfU)−1for n large enough
3 Reduction principle in uniform fading memory spaces
In this work, we assume that B satisfies Axioms (A), (B), (D1)–(D3) Let C00 be the space of all X -valued continuous functions on(−∞, 0] with compact support We suppose the following axiom: (C) if a uniformly bounded sequence(ϕn)nin C00converges to a functionϕ compactly in (−∞, 0], then ϕ is in B and kϕn−ϕk → 0 as n → ∞ Let(S0(t))t ≥0be the strongly continuous semigroup defined on the subspace
B0= {φ ∈ B : φ(0) = 0}
by
(S0(t)φ) (θ) =φ(t + θ) for t + θ ≤ 00 for t +θ ≥ 0.
Definition 3.1 Assume that the space B satisfies Axioms(A)–(C) B is said to be a fading memory space if for all
φ ∈ B0,
S0(t)φ −→
t →∞0 in B
Moreover, B is said to be a uniform fading memory space, if
kS0(t)k −→
t →∞0
The following results give some properties of fading memory spaces
Lemma 3.2 ([8, p 190]) The following statements hold
(i) If B is a fading memory space, then the functions K(·) and M(·) in Axiom (A) can be chosen to be constants (ii) If B is a uniform fading memory space, then the functions K(·) and M(·) can be chosen such that K (·) is constant and M(t) → 0 as t → ∞
Proposition 3.3 ([8]) If B is a fading memory space, then the space BC((−∞, 0]; X) of all bounded and continuous
X -valued functions on(−∞, 0], endowed with the uniform norm topology, is continuously embedding in B
ByProposition 3.3, one can observe that if B is a fading memory space then(D3) is satisfied with λ0≥0
In order to study the qualitative behavior of the semigroup(U(t))t ≥0, we suppose the following assumption that (H1) T0(t) is compact on D(A), for each t > 0
Let V be a bounded subset of a Banach space Y The Kuratowskii measure of noncompactnessα(V ) of V is defined by
α(V ) = infd > 0 such that there exists a finite number of sets V1, , Vnwith
diam(Vi) ≤ d such that V ⊆ ∪n
i =1Vi
Moreover, for a bounded linear operator P on Y , we define |P|αby
|P|α =inf{k> 0 : α(P (V )) ≤ kα(V ) for any bounded set V of Y }
Trang 6For the semigroup(U(t))t ≥0, we define the essential growth boundωess(U) by
ωess(U) = lim
t →∞
1
t log |U(t)|α
We have the following fundamental result
Theorem 3.4 ([5]) Assume that(H0), (H1) hold and B is a uniform fading memory space Then
ωess(U) < 0
Definition 3.5 Let C be a densely defined operator on Y The essential spectrum of C denoted byσess(C) is the set of
λ ∈ σ(C) such that one of the following conditions holds:
(i) Im(λI − C) is not closed,
(ii) the generalized eigenspace Mλ(C) = Sk≥1Ker(λI − C)kis of infinite dimension,
(iii) λ is a limit point of σ (C) \ {λ}
The essential radius of any bounded operator T is defined by
ress(T ) = sup{|λ| : λ ∈ σess(C)}
Theorem 3.6 ([2]) Assume that(H0), (H1) hold and B is a uniform fading memory space Then σ+(AU) = {λ ∈
σ (AU) : Re(λ) ≥ 0} is a finite set of the eigenvalues of AU which is not in the essential spectrum More precisely,
λ ∈ σ+(AU) if and only if there exists x ∈ D(A)\{0} which solves the following characteristic equation
1(λ)x = λx − Ax − L(eλ·x) = 0
We have the following spectral decomposition result
Theorem 3.7 ([2]) Assume that(H0), (H1) hold and B is a uniform fading memory space Then the phase space BA
is decomposed as
BA=S ⊕ V,
whereS, V are two closed subspaces of BAwhich are invariant under the semigroup(U(t))t ≥0 Let US(t) be the restriction of U(t) on S Then there exist positive constants N and µ such that
US(t)φ ≤Ne−µtkφk for φ ∈ S
On the other hand,V is a finite dimensional space Then the restriction UV(t) of U(t) on V becomes a group Let ΠSand ΠVdenote the projections on S and V respectively and d = dim V Take a basis {φ1, , φd} in V Then there exist d-elements {ψ1, , ψd} in the dual space B∗Aof BAsuch that hψi, φji =δi j, where
δi j = 1 if i = j
0 if i 6= j,
and hψi, φi = 0, for φ ∈ S and i = 1, , d, with h·, ·i being the canonical pairing between the dual space and the original space Denote by Φ =(φ1, , φd) and by Ψ the transpose of (ψ1, , ψd) One has
hΨ, Φi = IR d,
where IRd is the identity d × d-matrix For eachφ ∈ BA, ΠVφ is computed as
ΠVφ = ΦhΨ, φi,
=
d
X
hψi, φiφi
Trang 7Let u be the solution of Eq.(2.1)andζ(t) = (ζ1(t), , ζd(t)) be the components of ΠVut in the basis Φ Then
ΠVut =Φζ(t) and ζ(t) = hΨ, uti
Since UV(t)t ≥0is a group on the finite dimensional space V, then there exists a d × d-matrix G such that
UV(t)φ = ΦeGthΨ, φi for t ∈ R and φ ∈ V
This means that
UV(t)Φ = ΦeGt
for t ∈ R
Let n0∈ N such that n0> ω We define, for n ∈ N such that n ≥ n0and i ∈ {1, , d}, the functional x∗i
n by
hx∗in, xi = hψi,fBn(X0x)i for all x ∈ X
ByTheorem 2.6, we have fBn(X0x) = nen 1−1(n)x, for n ≥ n0 Then, we choose n0large enough such that
fBn(X0x) ≤ M0|x | for all x ∈ X and n ≥ n0
This implies that xn∗i is a bounded linear operator on X with xn∗i ≤ M0|ψi| Define the d-columns vector xn∗as an element of L(X, Rd) given by the transpose of x∗1, , x∗d Then, for all n ≥ n0and x ∈ X , we have
hx∗n, xi = hΨ,fBn(X0x)i and sup
n≥n 0
xn∗ ≤M0 sup
i =1 , ,d
|ψi|< ∞
We have the following important result
Theorem 3.8 ([2]) Assume that (H0), (H1) hold and B is a uniform fading memory space Then the sequence
xn∗
n≥n 0converges weakly inL(X, Rd), in the sense that there exists x∗∈L(X, Rd) such that
hx∗n, xi −→
n→∞hx∗, xi for x ∈ X
Corollary 3.9 ([2]) Assume that(H0), (H1) hold and B is a uniform fading memory space Then for any continuous function h : [σ, T ] → X, we have for all t ∈ [σ, T ]
lim
n→∞
Z t
σ U
V(t − s)ΠV
fBn(X0h(s)) ds = ΦZ t
σ e (t−s)Ghx∗, h(s)ids
In the next theorem, we state a finite dimensional reduction principle of Eq.(1.1)
Theorem 3.10 ([2]) Assume that(H0), (H1) hold and B is a uniform fading memory space Let u be a solution of
Eq.(1.1)on R Then ζ (t) = hΨ , utifor t ∈ R, is a solution of the following ordinary differential equation
˙
Conversely, if f is bounded andζ is a solution of(3.1), then the function
Φζ(t) + lim
n→+∞
Z t
−∞
US(t − s)ΠS
is a solution of Eq.(1.1)on R
4 C(n)-almost-periodic functions
We recall some properties about C(n)-almost-periodic functions Let BC(R, X) be the space of all bounded and continuous functions from R to X, equipped with the uniform norm topology Let h ∈ BC(R, X) and τ ∈ R; we define the function hτ by
hτ(s) = h(τ + s) for s ∈ R
Trang 8Let C(n)(R, X) be the space of all continuous functions which have a continuous nth derivative on R and Cn
b(R, X)
be the subspace of C(n)(R, X) of functions satisfying
sup
t ∈R
i =n
X
i =0
f(i)(t) <∞,
f(i)denotes the i th derivative of f Then Cn
b(R, X) is a Banach space equipped with the following norm
| f |n=sup
t ∈R
i =n
X
i =0
f(i)(t)
Definition 4.1 ([6]) A bounded continuous function h : R → X is said to be almost periodic if
{hτ :τ ∈ R} is relatively compact in BC(R, X)
Definition 4.2 ([3]) Letε > 0 and f ∈ Cn
b(R, X) A number τ ∈ R is said to be a |.|n−ε almost periodic of the function f if
| fτ − f |n< ε
The set of all |.|n−ε almost periodic of the function f is denoted by E(n)(ε, f )
Definition 4.3 ([3]) A function f ∈ Cbn(R, X) is said to be a C(n)-almost-periodic function if for everyε > 0, the set E(n)(ε, f ) is relatively dense in R
Definition 4.4 A P(n)(R, X) denotes the space of the all C(n)-almost-periodic functions R → X.
Since it is well known that for any almost-periodic functions f and g andε > 0, there exists a relatively dense set
of their commonε almost periods, we get the following result
Proposition 4.5 f ∈ A P(n)(R, X) if and only if f(i)∈ A P(R, X) for i = 1, , n
Proposition 4.6 ([4]) A P(n)(R, X) provided with the norm |.|n is a Banach space
The following example of a C(n)-almost-periodic function has been given in [4].
Example Let g(t) = sin(αt) + sin(βt), where αβ 6∈ Q Then the function f (t) = eg(t) is a C(n)-almost-periodic
function for any n ≥ 1
In [4], one can find example of a function which is C(n)-almost periodic but not C(n+1)-almost periodic.
The following theorem provides the sufficient and necessary condition for the existence of C(n)-almost-periodic
solutions of Eq.(1.3)
Theorem 4.7 Assume that e is a C(n)-almost-periodic function If Eq.(1.3)has a bounded solution on R+, then it has
an C(n+1)-almost-periodic solution Moreover, every bounded solution of Eq.(1.3)on R is C(n+1)-almost periodic.
Proof We assume that Eq.(1.3)has a bounded solutionχ in R+ Since e is an almost-periodic solution, then there exists a sequence(tn)n, tn→ ∞such that
e(t + tn) → e(t), uniformly in t ∈ R
Using the diagonal extraction process, we show that the sequence χ(t + tn) has a subsequence which converges compactly to the solution of Eq.(1.3)which is bounded and defined in R For m = 0, it has been proved in [6], that if
eis an almost-periodic function then every bounded solution of equation in R is almost periodic Let x be a bounded solution of Eq.(1.3), then x is almost periodic G x is also almost periodic and
x0(t) = Gx(t) + e(t) for t ∈ R,
Trang 9it follows that x0 is almost periodic Since the function f is C(n)-almost periodic and for i = 1, , n, we have the formula,
x(i)(t) = Gx(i−1)(t) + e(i−1)(t) for t ∈ R
Consequently, we deduce that x(i) is almost periodic for i = 1, , n and byProposition 4.5, we deduce that x is
C(n)-almost periodic Moreover
x(n+1)(t) = Gx(n)(t) + e(n)(t) for t ∈ R,
which implies that x(n+1)is almost periodic and x is C(n+1)-almost periodic.
5 C(n)-almost-periodic solutions
In the following, we assume that
(H2) f is a C(n)-almost-periodic function.
Theorem 5.1 Assume that(H0), (H1), (H2) hold and B is a uniform fading memory space If there is at least φ ∈ B such that Eq.(2.1)has a bounded solution on R+, then Eq.(1.1)has a C(n)-almost-periodic solution Moreover every
bounded solution on the whole line is a C(n)-almost-periodic solution.
Proof Let u be a bounded solution of Eq.(1.1)on R+ ByTheorem 3.10, the function z(t) = hΨ, uti, for t ≥ 0, is a solution of the ordinary differential equation(3.1)and z is bounded on R+ Moreover, the function
%(t) = hx∗, f (t)i for t ∈ R,
is C(n)-almost periodic from R to Rd ByTheorem 4.7, we get that the reduced system (3.1)has a C(n)
-almost-periodic solutionezand Φez(.) is a C(n)-almost-periodic function on R FromTheorem 3.10, we know that the function
u(t) = v(t)(0), where
v(t) = Φez(t) + lim
n→+∞
Z t
−∞
US(t − s)ΠS
e
BnX0f(s) ds for t ∈ R,
is a solution of Eq.(1.1)on R We claim that v is C(n)-almost periodic In fact, let y be defined by
y(t) = lim
n→+∞
Z t
−∞
US(t − s)ΠS
e
BnX0f(s) ds for t ∈ R
Then y ∈ Cbn(R, X) Since f is C(n)-almost periodic Letε > 0 and τ be a |.|n−ε almost periodic of the function f that is
| fτ− f |n< ε
It follows that
y(t + τ) − y(t) = lim
n→+∞
0
US(s)ΠS
e
BnX0( f (t + τ − s) − f (t − s)) ds and we get for some positive constantγ that
|yτ −y|n≤γ ε | fτ− f |n,
which implies that y is C(n)-almost periodic.
Letv be a bounded solution on the whole line then v is given by the
v(t) = Φz(t) + lim
n→+∞
Z t
−∞
US(t − s)ΠS
e
BnX0f(s) ds for t ∈ R, where
z(t) = hΨ, u i for t ∈ R
Trang 10is a solution of the reduced system(3.1), which is C(n)-almost periodic byTheorem 4.7, and arguing as above, one
can prove that the function
n→+∞
Z t
−∞
US(t − s)ΠS
e
BnX0f(s) ds for t ∈ R,
is also C(n)-almost periodic.
6 Exponential dichotomy
Definition 6.1 The semigroup(U(t))t ≥0is said to have an exponential dichotomy if
σ(AU) ∩ iR = ∅
Sinceωess(U) < 0, then we get the following result on the spectral decomposition of the phase space BA Theorem 6.2 ([2]) Assume that(H0), (H1) hold and B is a uniform fading memory space If the semigroup (U(t))t ≥0
has an exponential dichotomy, then the spaceBAis decomposed as a direct sumBA =S ⊕ U of two U(t) invariant closed subspacesS and U such that the restricted semigroup on U is a group and there exist positive constants N0 andε0such that
|U(t)ϕ| ≤ N0e−ε 0 t|ϕ| for t ≥ 0 and ϕ ∈ S
|U(t)ϕ| ≤ N0eε 0 t|ϕ| for t ≤ 0 and ϕ ∈ U
As a consequence of the exponential dichotomy, we get the following result on the uniqueness of the bounded solution of Eq.(1.1)
Theorem 6.3 Assume that(H0), (H1) hold and B is a uniform fading memory space If the semigroup (U(t))t ≥0has
an exponential dichotomy, then for any bounded continuous function f on R, Eq.(1.1)has a unique bounded solution
on R Moreover, this solution is C(n)-almost periodic if f is Cn-almost periodic
Proof Since the semigroup(U(t))t ≥0has an exponential dichotomy, then Eq.(1.1)has one and only one bounded solution on R which is given for t ∈ R by the following formula
lim
n→+∞
Z t
−∞
US(t − s)ΠS
e
BnX0f(s) ds + lim
n→+∞
Z t +∞
UU(t − s)ΠU
e
BnX0f(s) ds(0)
ByTheorem 3.10, we conclude that this solution is C(n)-almost periodic when f is C(n)-almost periodic.
7 Application
To illustrate the previous results, we consider the following Lotka–Volterra model with diffusion
∂
∂tv(t, ξ) =
∂2
∂ξ2v(t, ξ) +Z 0
−∞
η(θ)v(t + θ, ξ)dθ + ρ(t)F(ξ) for t ∈ R and 0 ≤ ξ ≤ π, v(t, 0) = v(t, π) = 0 for t ∈ R
(7.1)
whereη is a positive function on (−∞, 0] , ρ : R → R is Cn-almost periodic, for example, we consider
g(t) = sin(αt) + sin(βt) for t ∈ R
whereα
β 6∈ Q Then the function ρ(t) = eg(t) F : [0, π] → R is a continuous function Let X = C([0, π] ; R) be the space of all continuous functions from [0, π] to R endowed with the uniform norm topology Consider the operator
A : D(A) ⊂ X → X defined by
D(A) = {z ∈ C2([0, π] ; R) : z(0) = z (π) = 0},
Az = z00
... periodic.The following theorem provides the sufficient and necessary condition for the existence of C< small>(n)-almost-periodic
solutions of Eq.(1.3)
Theorem. ..
Using the diagonal extraction process, we show that the sequence χ(t + tn) has a subsequence which converges compactly to the solution of Eq.(1.3)which is bounded and defined in R For. .. t|ϕ| for t ≤ and ϕ ∈ U
As a consequence of the exponential dichotomy, we get the following result on the uniqueness of the bounded solution of Eq.(1.1)
Theorem 6.3 Assume that(H0),