Directory UMM :Journals:Journal_of_mathematics:EJQTDE:
Electronic Journal of Qualitative Theory of Differential Equations
2010, No. 22, 1-26; http://www.math.u-szeged.hu/ejqtde/
EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS TO
SOME CLASSES OF NONAUTONOMOUS HIGHER-ORDER
DIFFERENTIAL EQUATIONS
TOKA DIAGANA
Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential
equations with Stepanov almost automorphic forcing terms. A few illustrative
examples are discussed at the very end of the paper.
1. Introduction
The main motivation of this paper comes from the work of Andres, Bersani, and
Radov´
a [8], in which the existence (and uniqueness) of almost periodic solutions
was established for the class of n-order autonomous differential equations
n
X
(1.1)
ak u(n−k) (t) = f (u) + p(t), t ∈ R,
u(n) (t) +
k=1
where f, p : R 7→ R are (Stepanov) almost periodic, f is Lipschitz, and ak ∈ R
for k = 1, ..., n are given real constants such that the real part of each root of the
characteristic polynomial associated with the (linear) differential operator on the
left-hand side of Eq. (1.1), that is,
Q(λ) := λn +
n
X
ak λn−k
k=1
is at least nonzero.
The method utilized in [8] makes extensive use of a very complicated representation formula for solutions to Eq. (1.1). For details on that representation formula,
we refer the reader to [9] and [10] and the references therein.
Let H be a Hilbert space. In this paper, we study a more general equation than
Eq. (1.1). Namely, using similar techniques as in [14, 27], we study and obtain
some reasonable sufficient conditions, which do guarantee the existence of almost
automorphic solutions to the class of nonautonomous n-order differential equations
(1.2)
u(n) (t) +
n−1
X
ak (t)u(k) (t) + a0 (t)Au(t) = f (t, u), t ∈ R,
k=1
where A : D(A) ⊂ H 7→ H is a (possibly unbounded) self-adjoint linear operator on
H whose spectrum consists of isolated eigenvalues
0 < λ1 < λ2 < ... < λl → ∞ as l → ∞
1991 Mathematics Subject Classification. 43A60; 34B05; 34C27; 42A75; 47D06; 35L90.
Key words and phrases. exponential dichotomy; Acquistapace and Terreni conditions; evolution families; almost automorphic; Stepanov almost automorphic, nonautonomous higher-order
differential equation.
EJQTDE, 2010 No. 22, p. 1
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the
corresponding eigenspace, the functions ak : R 7→ R (k = 0, 1, ..., n − 1) are almost
automorphic with
inf a0 (t) = γ0 > 0,
t∈R
and the function f : R×H 7→ H is Stepanov almost automorphic in the first variable
uniformly in the second variable.
Consider the time-dependent polynomial defined by
Qlt (ρ) := ρn +
n−1
X
ak (t)ρk + λl a0 (t)
k=1
and denote its roots by
ρlk (t) = µlk (t) + iνkl (t),
k = 1, 2, ..., n, l ≥ 1, and t ∈ R.
In the rest of this paper, we suppose that there exists δ0 > 0 such that
#
"
l
l
l
(1.3)
sup
max µ1 (t), µ2 (t), ..., µn (t) ≤ −δ0 < 0.
l≥1,t∈R
To deal with Eq. (1.2), the main idea consists of rewriting it as a nonautonomous
first-order differential equation on Xn = H × H × H.... × H (n-times) involving the
family of n × n-operator matrices {A(t)}t∈R .
Indeed, assuming that u is differentiable n times and setting
u
u′
u′′
z :=
u(3)
.
,
u(n−1)
then Eq. (1.2) can be rewritten in the Hilbert space Xn in the following form
(1.4)
z ′ (t) = A(t)z(t) + F (t, z(t)), t ∈ R,
where A(t) is the family of n × n-operator matrices defined by
0
IH
0
0 .... 0
0
0
IH .... 0
(1.5)
A(t) =
.
.
. IH
.
.
.
.
.
.
.
.
.
.
−a0 (t)A −a1 (t)IH .
.
.
. −an−1 (t)IH
whose domains D(A(t)) are constant in t ∈ R and are precisely given by
D = D(A) × H × H... × H := D(A) × Xn−1
EJQTDE, 2010 No. 22, p. 2
for all t ∈ R.
Moreover, the semilinear term F appearing in
some α ∈ (0, 1) by
0
0
0
F (t, z) := 0
.
.
,
f (t, u)
the real interpolation space between Xn and D(A(t)) given by Xnα =
, with
Hα := (H, D(A))α,∞ .
Xnα is
n−1
where
Hα × X
Eq. (1.4) is defined on R × Xnα for
Under some reasonable assumptions, it will be shown that the linear operator
matrices A(t) satisfy the well-known Acquistapace-Terreni conditions [3], which do
guarantee the existence of an evolution family U (t, s) associated with it. Moreover,
it will be shown that U (t, s) is exponentially stable under those assumptions.
The existence of almost automorphic solutions to higher-order differential equations is important due to their (possible) applications. For instance when n = 2,
we have thermoelastic plate equations [14, 27] or telegraph equation [31] or SineGordon equations [26]. Let us also mention that when n = 2, some contributions
on the maximal regularity, bounded, almost periodic, asymptotically almost periodic solutions to abstract second-order differential and partial differential equations
have recently been made, among them are [11], [12], [44], [45], [46], and [47]. In
[8], the existence of almost periodic solutions to higher-order differential equations
with constant coefficients in the form Eq. (1.1) was obtained in particular in the
case when the forcing term is almost periodic. However, to the best of our knowledge, the existence of almost automorphic solutions to higher-order nonautonomous
equations in the form Eq. (1.2) in the case when the forcing term is Stepanov almost
automorphic is an untreated original question, which in fact constitutes the main
motivation of the present paper.
The paper is organized as follows: Section 2 is devoted to preliminaries facts
needed in the sequel. In particular, facts related to the existence of evolution
families as well as preliminary results on intermediate spaces will be stated there.
In addition, basic definitions and classical results on (Stepanov) almost automorphic
functions are also given. In Sections 3 and 4, we prove the main result. In Section
5, we provide the reader with an example to illustrate our main result.
2. Preliminaries
Let H be a Hilbert space equipped with the norm k·k and the inner product h·, ·i.
In this paper, A : D(A) ⊂ H 7→ H stands for a self-adjoint (possibly unbounded)
linear operator on H whose spectrum consists of isolated eigenvalues
0 < λ1 < λ2 < ... < λl → ∞ as l → ∞
EJQTDE, 2010 No. 22, p. 3
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the
corresponding eigenspace.
Let {ekj } be a (complete) orthonormal sequence of eigenvectors associated with
the eigenvalues {λj }j≥1 .
)
(
∞
2
X
2
λj
Ej u
< ∞ , we have
Clearly, for each u ∈ D(A) := u ∈ H :
j=1
Au =
∞
X
λj
j=1
where Ej u =
γj
X
γj
X
hu, ekj iekj =
∞
X
λj Ej u
j=1
k=1
hu, ekj iekj .
k=1
Note that {Ej }j≥1 is a sequence of orthogonal projections on H. Moreover, each
u ∈ H can written as follows:
∞
X
Ej u.
u=
j=1
It should also be mentioned that the operator −A is the infinitesimal generator of
an analytic semigroup {T (t)}t≥0 , which is explicitly expressed in terms of those
orthogonal projections Ej by, for all u ∈ H,
T (t)u =
∞
X
e−λj t Ej u.
j=1
In addition, the fractional powers Ar (r ≥ 0) of A exist and are given by
)
(
∞
2
X
2r
r
λj
Ej u
< ∞
D(A ) = u ∈ H :
j=1
and
Ar u =
∞
X
r
λ2r
j Ej u, ∀u ∈ D(A ).
j=1
Let (X,
·
) be a Banach space. If L is a linear operator on the Banach space
X, then:
• D(L) stands for its domain;
• ρ(L) stands for its resolvent;
• σ(L) stands for its spectrum;
• N (L) stands for its null-space or kernel; and
• R(L) stands for its range.
If L
: D
= D(L) ⊂ X 7→
X
is a closed linear operator, one denotes its graph norm
by
·
. Clearly, (D,
·
) is a Banach space. Moreover, one sets
D
D
R(λ, L) := (λI − L)−1
for all λ ∈ ρ(A).
EJQTDE, 2010 No. 22, p. 4
We set Q = I − P for a projection P . If Y, Z are Banach spaces, then the
space B(Y, Z) denotes the collection of all bounded linear operators from Y into Z
equipped with its natural topology. This is simply denoted by B(Y) when Y = Z.
2.1. Evolution Families. Hypothesis (H.1). The family of closed linear operators A(t) for t ∈ R on X with domain D(A(t)) (possibly not densely defined) satisfy
the so-called
Acquistapace-Terreni conditions, that is, there exist constants ω ∈ R,
θ ∈ π2 , π , K, L ≥ 0 and µ, ν ∈ (0, 1] with µ + ν > 1 such that
n o
K
(2.1)
Sθ ∪ 0 ⊂ ρ A(t) − ω ∋ λ,
R λ, A(t) − ω
≤
1 + λ
and
h
i
µ −ν
(2.2)
A(t) − ω R λ, A(t) − ω R ω, A(t) − R ω, A(s)
≤ L t − s λ
(
)
for t, s ∈ R, λ ∈ Sθ := λ ∈ C \ {0} : arg λ ≤ θ .
Note that in the particular case when A(t) has a constant domain D = D(A(t)),
it is well-known [6, 38] that Eq. (2.2) can be replaced with the following: There
exist constants L and 0 < µ ≤ 1 such that
µ
(2.3)
A(t) − A(s) R ω, A(r)
≤ L t − s , s, t, r ∈ R.
It should mentioned that (H.1) was introduced in the literature by Acquistapace
and Terreni in [2, 3] for ω = 0. Among other things, it ensures that there exists a
unique evolution family U = U (t, s) on X associated with A(t) satisfying
(a) U (t, s)U (s, r) = U (t, r);
(b) U (t, t) = I for t ≥ s ≥ r in R;
(c) (t, s) 7→ U (t, s) ∈ B(X) is continuous for t > s;
∂U
(t, s) = A(t)U (t, s) and
(d) U (·, s) ∈ C 1 ((s, ∞), B(X)),
∂t
A(t)k U (t, s)
≤ K (t − s)−k
for 0 < t − s ≤ 1, k = 0, 1, 0 ≤ α < µ, x ∈ D((ω − A(s))α ), and a constant
C depending only on the constants appearing in (H.1); and
(e) ∂s+ U (t, s)x = −U (t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈
D(A(s)).
It should also be mentioned that the above-mentioned proprieties were mainly
established in [1, Theorem 2.3] and [49, Theorem 2.1], see also [3, 48]. In that case
we say that A(·) generates the evolution family U (·, ·).
One says that an evolution family U has an exponential dichotomy (or is hyperbolic) if there are projections P (t) (t ∈ R) that are uniformly bounded and strongly
continuous in t and constants δ > 0 and N ≥ 1 such that
(f) U (t, s)P (s) = P (t)U (t, s);
(g) the restriction UQ (t, s) : Q(s)X → Q(t)X of U (t, s) is invertible (we then
eQ (s, t) := UQ (t, s)−1 ); and
set U
EJQTDE, 2010 No. 22, p. 5
e
−δ(t−s)
(h)
U (t, s)P (s)
≤ N e−δ(t−s) and
U
for t ≥ s and
Q (s, t)Q(t)
≤ N e
t, s ∈ R.
According to [40], the following sufficient conditions are required for A(t) to have
exponential dichotomy.
(i) Let (A(t), D(t))t∈R be generators of analytic semigroups on X of the same
type. Suppose that D(A(t)) ≡ D(A(0)), A(t) is invertible,
sup
A(t)A(s)−1
t,s∈R
is finite, and
µ
A(t)A(s)−1 − I
≤ L0 t − s
for t, s ∈ R and constants L0 ≥ 0 and 0 < µ ≤ 1.
(j) The semigroups (eτ A(t) )τ ≥0 , t ∈ R, are hyperbolic with projection Pt and
constants N, δ > 0. Moreover, let
A(t)eτ A(t) Pt
≤ ψ(τ )
and
A(t)eτ AQ (t) Qt
≤ ψ(−τ )
µ
for τ > 0 and a function ψ such that R ∋ s 7→ ϕ(s) := s ψ(s) is integrable
< 1.
with L0
ϕ
1
L (R)
This setting requires some estimates related to U (t, s). For that, we introduce
the interpolation spaces for A(t). We refer the reader to the following excellent
books [6], [23], and [29] for proofs and further information on theses interpolation
spaces.
Let A be a sectorial operator on X (for that, in assumption (H.1), replace A(t)
with A) and let α ∈ (0, 1). Define the real interpolation space
(
)
A
α
A
Xα := x ∈ X :
x
:= supr>0
r A − ω R r, A − ω x
< ∞ ,
α
A
which, by the way, is a Banach space when endowed with the norm
·
. For
α
convenience we further write
A
A
XA
0 := X,
x
:=
x
, X1 := D(A)
0
and
A
x
:=
(ω − A)x
.
1
ˆ A := D(A) of X. In particular, we have the following continuous
Moreover, let X
embedding
(2.4)
α
A
ˆA
D(A) ֒→ XA
β ֒→ D((ω − A) ) ֒→ Xα ֒→ X ֒→ X,
for all 0 < α < β < 1, where the fractional powers are defined in the usual way.
EJQTDE, 2010 No. 22, p. 6
In general, D(A) is not dense in the spaces XA
α and X. However, we have the
following continuous injection
XA
β ֒→ D(A)
(2.5)
k·kA
α
for 0 < α < β < 1.
Given the family of linear operators A(t) for t ∈ R, satisfying (H.1), we set
ˆ t := X
ˆ A(t)
Xt := XA(t) , X
α
α
for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embedding in
Eq. (2.4) holds with constants independent of t ∈ R. These interpolation spaces
are of class Jα ([29, Definition 1.1.1 ]) and hence there is a constant c(α) such that
t
1−α
α
(2.6)
y
≤ c(α)
y
A(t)y
, y ∈ D(A(t)).
α
We have the following fundamental estimates for the evolution family U.
Proposition 2.1. [14] For x ∈ X, 0 ≤ α ≤ 1 and t > s, the following hold:
(i) There is a constant c(α), such that
t
δ
(2.7)
U (t, s)P (s)x
≤ c(α)e− 2 (t−s) (t − s)−α
x
.
α
(ii) There is a constant m(α), such that
(2.8)
s
e
UQ (s, t)Q(t)x
≤ m(α)e−δ(t−s)
x
.
α
In addition to above, we also need the following assumptions:
Hypothesis (H.2). The evolution family U generated by A(·) has an exponential
dichotomy with constants N, δ > 0 and dichotomy projections P (t) for t ∈ R.
Hypothesis (H.3). There exist α, β with 0 ≤ α < β < 1 and such that
Xtα = Xα and Xtβ = Xβ
for all t ∈ R, with uniform equivalent norms.
2.2. Stepanov Almost Automorphic Functions. Let (X, k · k), (Y, k · kY ) be
two Banach spaces. Let BC(R, X) (respectively, BC(R × Y, X)) denote the collection of all X-valued bounded continuous functions (respectively, the class of jointly
bounded continuous functions F : R × Y 7→ X). The space BC(R, X) equipped
with the sup norm k · k∞ is a Banach space. Furthermore, C(R, Y) (respectively,
C(R × Y, X)) denotes the class of continuous functions from R into Y (respectively,
the class of jointly continuous functions F : R × Y 7→ X).
Definition 2.2. [37] The Bochner transform f b (t, s), t ∈ R, s ∈ [0, 1] of a function
f : R 7→ X is defined by f b (t, s) := f (t + s).
Remark 2.3. (i) A function ϕ(t, s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a
certain function f , ϕ(t, s) = f b (t, s) , if and only if ϕ(t + τ, s − τ ) = ϕ(s, t) for all
t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s].
(ii) Note that if f = h + ϕ, then f b = hb + ϕb . Moreover, (λf )b = λf b for each
scalar λ.
EJQTDE, 2010 No. 22, p. 7
Definition 2.4. The Bochner transform F b (t, s, u), t ∈ R, s ∈ [0, 1], u ∈ X of a
function F (t, u) on R × X, with values in X, is defined by
F b (t, s, u) := F (t + s, u)
for each u ∈ X.
Definition 2.5. Let p ∈ [1, ∞). The space BS p (X) of all Stepanov bounded
functions, with the exponent p, consists of all measurable functions f : R 7→ X such
that f b belongs to L∞ R; Lp ((0, 1), X) . This is a Banach space with the norm
Z t+1
1/p
kf kS p := kf b kL∞ (R,Lp ) = sup
kf (τ )kp dτ
.
t∈R
t
p
2.3. S -Almost Automorphy.
Definition 2.6. (Bochner) A function f ∈ C(R, X) is said to be almost automorphic if for every sequence of real numbers (s′n )n∈N , there exists a subsequence
(sn )n∈N such that
g(t) := lim f (t + sn )
n→∞
is well defined for each t ∈ R, and
lim g(t − sn ) = f (t)
n→∞
for each t ∈ R.
The collection of all almost automorphic functions from R to X will be denoted
AA(X).
Similarly
Definition 2.7. (Bochner) A function F ∈ C(R × Y, X) is said to be almost automorphic if for every sequence of real numbers (s′n )n∈N , there exists a subsequence
(sn )n∈N such that
G(t, u) := lim F (t + sn , u)
n→∞
is well defined for each t ∈ R, and
lim G(t − sn , u) = F (t, u)
n→∞
for each t ∈ R uniformly in u ∈ Y.
The collection of all almost automorphic functions from R × Y to X will be
denoted AA(R × Y).
We have the following composition result:
Theorem 2.8. [34] Suppose F : R × Y 7→ X belongs to AA(R × Y) and that the
mapping x 7→ f (t, x) is Lipschitz in the sense that there exists L ≥ 0 such that
F (t, x) − F (t, y)
≤ L
x − y
for all x, y ∈ Y uniformly in t ∈ R.
Then, then the function defined by G(t) = F (t, ϕ(t)) belongs to AA(X) provided
ϕ ∈ AA(Y).
EJQTDE, 2010 No. 22, p. 8
We also have the following composition result, which is a straightforward consequence of the composition of pseudo almost automorphic functions obtained in
[43].
Theorem 2.9. [43] If F : R × Y 7→ X belongs to AA(R × Y) and if x 7→ F (t, x)
is uniformly continuous on any bounded subset K of Y for each t ∈ R, then the
function defined by h(t) = F (t, ϕ(t)) belongs to AA(X) provided ϕ ∈ AA(Y).
We will denote by AAu (X) the closed subspace of all functions f ∈ AA(X) with
g ∈ C(R, X). Equivalently, f ∈ AAu (X) if and only if f is almost automorphic
and the convergence in Definition 2.7 are uniform on compact intervals, i.e. in
the Fr´echet space C(R, X). Indeed, if f is almost automorphic, then, its range is
relatively compact. Obviously, the following inclusions hold:
AP (X) ⊂ AAu (X) ⊂ AA(X) ⊂ BC(X),
where AP (X) is the Banach space of almost periodic functions from R to X.
Definition 2.10. [36] The space AS p (X) of Stepanov almost automorphic functions (or S p -almost
automorphic) consists of all f ∈ BS p (X) such that f b ∈
p
AA L (0, 1; X) . That is, a function f ∈ Lploc (R; X) is said to be S p -almost automorphic if its Bochner transform f b : R → Lp (0, 1; X) is almost automorphic in
the sense that for every sequence of real numbers (s′n )n∈N , there exists a subsequence (sn )n∈N and a function g ∈ Lploc (R; X) such that
Z t+1
1/p
kf (sn + s) − g(s)kp ds
→ 0, and
t
Z
t+1
kg(s − sn ) − f (s)kp ds
t
as n → ∞ pointwise on R.
1/p
→0
Remark 2.11. It is clear that if 1 ≤ p < q < ∞ and f ∈ Lqloc (R; X) is S q -almost
automorphic, then f is S p -almost automorphic. Also if f ∈ AA(X), then f is S p almost automorphic for any 1 ≤ p < ∞. Moreover, it is clear that f ∈ AAu (X) if
and only if f b ∈ AA(L∞ (0, 1; X)). Thus, AAu (X) can be considered as AS ∞ (X).
Definition 2.12. A function F : R × Y 7→ X, (t, u) 7→ F (t, u) with F (·, u) ∈
Lploc (R; X) for each u ∈ Y, is said to be S p -almost automorphic in t ∈ R uniformly
in u ∈ Y if t 7→ F (t, u) is S p -almost automorphic for each u ∈ Y, that is, for
every sequence of real numbers (s′n )n∈N , there exists a subsequence (sn )n∈N and a
function G(·, u) ∈ Lploc (R; X) such that
Z t+1
1/p
p
kF (sn + s, u) − G(s, u)k ds
→ 0, and
t
Z
t+1
p
kG(s − sn , u) − F (s, u)k ds
t
as n → ∞ pointwise on R for each u ∈ Y.
1/p
→0
EJQTDE, 2010 No. 22, p. 9
The collection of those S p -almost automorphic functions F : R × Y 7→ X will be
denoted by AS p (R × Y).
We have the following straightforward composition theorems, which generalize
Theorem 2.8 and Theorem 2.9, respectively:
Theorem 2.13. Let F : R× Y 7→ X be a S p -almost automorphic function. Suppose
that u 7→ F (t, u) is Lipschz in the sense that there exists L ≥ 0 such
F (t, u) − F (t, v)
≤ L
u − v
Y
for all t ∈ R, (u, v) ∈ Y × Y.
If φ ∈ AS p (Y), then Γ : R → X defined by Γ(·) := F (·, φ(·)) belongs to AS p (X).
Theorem 2.14. Let F : R × Y 7→ X be a S p -almost automorphic function, where.
Suppose that F (t, u) is uniformly continuous in every bounded subset K ⊂ X uniformly for t ∈ R. If g ∈ AS p (Y), then Γ : R → X defined by Γ(·) := F (·, g(·))
belongs to AS p (X).
3. Main results
Consider the nonautonomous differential equation
u′ (t) = A(t)u(t) + F (t, u(t)), t ∈ R,
(3.1)
where F : R × Xα 7→ X is S p -almost automorphic.
Definition 3.1. A function u : R 7→ Xα is said to be a bounded solution to Eq.
(3.1) provided that
Z ∞
Z t
(3.2) u(t) =
UQ (t, s)Q(s)F (s, u(s))ds.
U (t, s)P (s)F (s, u(s))ds −
−∞
t
for all t ∈ R.
Throughout the rest of the paper, we set S1 u(t) := S11 u(t) − S12 u(t), where
Z t
Z ∞
UQ (t, s)Q(s)F (s, u(s))ds.
S11 u(t) :=
U (t, s)P (s)F (s, u(s))ds, S12 u(t) :=
t
−∞
for all t ∈ R.
To study Eq. (3.1), in addition to the previous assumptions, we require that
1 1
p > 1, + = 1, and that the following assumptions hold:
p q
(H.4) R(ω, A(·)) ∈ B(R, AA(Xα )).
(H.5) The function F : R × X 7→ Xβ is S p -almost automorphic in t ∈ R uniformly
in u ∈ X. Moreover, F is Lipschitz in the following sense: there exists
L > 0 for which
F (t, u) − F (t, v)
≤ L
u − v
β
for all u, v ∈ X and t ∈ R.
EJQTDE, 2010 No. 22, p. 10
Lemma 3.2. Under assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5) and if
#1/q
"Z
∞
n
X
δ
(3.3)
< ∞,
e−q 2 s s−qα ds
N (α, q, δ) :=
n−1
n=1
then the integral operator S1 defined above maps AA(Xα ) into itself.
Proof. Let u ∈ AA(Xα ). Setting ϕ(t) := F (t, u(t)) and using Theorem 2.13 it
follows that ϕ ∈ AS p (Xβ ). The next step consists of showing that S1 ∈ AA(Xα ).
Define for all n = 1, 2, ..., the sequence of integral operators
Φn (t) =
Z
n
U (t, t − s)P (t − s)ϕ(t − s)ds
n−1
for each t ∈ R.
Letting r = t − s it follows that
Z t−n+1
Φn (t) =
U (t, r)P (r)ϕ(r)dr,
t−n
and hence from the H¨
older’s inequality and the estimate Eq. (2.7) it follows that
Z t−n+1
δ
c(α)e− 2 (t−r) (t − r)−α
φ(r)
dr
≤
Φn (t)
α
≤
c
t−n
Z t−n+1
cc′
≤
δ
c(α)e− 2 (t−r) (t − r)−α
φ(r)
dr
t−n
Z t−n+1
t−n
"Z
≤
n
q(α)
α
δ
c(α)e− 2 (t−r) (t − r)−α
ϕ(r)
dr
β
δ
e−q 2 s s−qα ds
n−1
#1/q
ϕ
Sp
.
Using Eq. (3.3), we then deduce from Weirstrass Theorem that the series defined
by
∞
X
D(t) :=
Φn (t)
n=1
is uniformly convergent on R. Moreover, D ∈ C(R, Xα ) and
∞
X
Φn (t)
≤ q(α)N (α, q, δ)
φ
D(t)
≤
α
n=1
Sp
α
for all t ∈ R.
Let us show that Φn ∈ AA(Xα ) for each n = 1, 2, 3, ... Indeed, since ϕ ∈
AS p (Xβ ) ⊂ AS p (Xα ), for every sequence of real numbers (τn′ )n∈N there exist a
subsequence (τnk )k∈N and a function ϕ
b such that
Z t+1
p
b − ϕ(s + τnk )
ds → 0
ϕ(s)
t
α
EJQTDE, 2010 No. 22, p. 11
and
Z
t+1
t
p
b − τnk ) − ϕ(s)
ds → 0
ϕ(s
α
as k → ∞ pointwise in R.
Define for all n = 1, 2, 3, ..., the sequence of integral operators
Z n
b
Φn (t) =
U (t, t − s)P (t − s)ϕ(t
b − s)ds
n−1
for all t ∈ R.
Now
b
Φ(t + τnk ) − Φ(t)
=
−
=
+
−
−
=
Z
n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
n
+
Z
n−1
U (t, t + τnk − s)P (t + τnk − s)ϕ(t + τnk − s)ds
U (t, t − s)P (t − s)ϕ(t
b − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t + τnk − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t
b − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t
b − s)ds
U (t, t − s)P (t − s)ϕ(t
b − s)ds
h
i
U (t, t + τnk − s)P (t + τnk − s) ϕ(t + τnk − s) − ϕ(t
b − s) ds
h
i
U (t, t + τnk − s)P (t + τnk − s) − U (t, t − s)P (t − s) ϕ(t
b − s)ds.
Using Lebesgue Dominated Convergence Theorem, one can easily see that
Z n
h
i
U (t, t+τnk −s)P (t+τnk −s) ϕ(t+τnk −s)−ϕ(t−s)
b
ds
→ 0 as k → ∞, t ∈ R.
α
n−1
Similarly, using [15] it follows that
Z n h
i
U (t, t+τnk −s)P (t+τnk −s)−U (t, t−s)P (t−s) ϕ(t−s)ds
b
→ 0 as k → ∞, t ∈ R.
α
n−1
Thus
b n (t) = lim Φn (t + τn ), t ∈ R.
Φ
k
k→∞
Similarly, one can easily see that
b n (t − τn )
Φn (t) = lim Φ
k
k→∞
for all t ∈ R and n = 1, 2, 3, ... Therefore the sequence Φn ∈ AA(Xα ) for each
n = 1, 2, ... and hence D ∈ AA(Xα ). Consequently t 7→ S11 (t) belong to AA(Xα ).
The proof for t 7→ S12 (t) is similar to that of t 7→ S11 (t) and hence omitted.
EJQTDE, 2010 No. 22, p. 12
In view of the above, it follows that S1 ∈ AA(Xα ).
Lemma 3.3. The integral operator S1 defined above is a contraction whenever L
is small enough.
Proof. Let v, w ∈ AA(Xα ). Now,
S11 v(t) − S11 w(t)
≤
α
≤
Z
c
t
−∞
Z t
δ
c(α)(t − s)−α e− 2 (t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
−∞
≤
≤
δ
c(α)(t − s)−α e− 2 (t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
Lcc(α)
Z
Lc′ cc(α)
β
t
−∞
Z t
δ
(t − s)−α e− 2 (t−s)
v(s) − w(s)
ds
−∞
Similarly,
S12 v(t) − S12 w(t)
α
Z
δ
(t − s)−α e− 2 (t−s)
v(s) − w(s)
ds.
α
m(α)e−δ(t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
t
Z ∞
e−δ(t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
≤ cm(α)
β
t
Z ∞
e−δ(t−s)
v(s) − w(s)
ds
≤ Lcm(α)
t
Z ∞
′
e−δ(t−s)
v(s) − w(s)
ds.
≤ Lcc m(α)
≤
∞
α
t
Consequently,
S1 v − S1 w
∞,α
≤ Lcc′ c(α)Γ(1 − α)(2δ −1 )1−α + m(α)δ −1
v − w
∞,α
and hence S1 is a contraction whenever L is small enough.
Theorem 3.4. Suppose assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5) and Eq. (3.3)
hold and that L is small enough, then the nonautonmous differential equation Eq.
(3.1) has a unique almost automorphic solution u satisfying u = S1 u
Proof. The proof makes use of Lemma 3.2, Lemma 3.3, and the Bananch fixed-point
principle.
4. Almost Automorphic Solutions to Some Higher-Order
Differential Equations
We have previously seen that each u ∈ H can be written in terms of the sequence
γl
∞ X
∞
X
X
of orthogonal projections En as follows: u =
hu, ekl iekl =
El u. Moreover,
l=1 k=1
l=1
EJQTDE, 2010 No. 22, p. 13
for each u ∈ D(A), Au =
u1
.
un
!
∞
X
l=1
λl
γl
X
hu, ekl iekl =
k=1
∞
X
λl El u. Therefore, for all z :=
l=1
∈ D = D(A(t)) = D(A) × Hn−1 , we obtain the following
A(t)z =
=
0
IH
0
0
.... 0
0
0
IH
....
0
.
.
.
IH
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
−a0 (t)A
−a1 (t)IH
.
.
.
.
−an−1 (t)IH
u2
u3
.
.
.
.
.
−a0 (t)Au1 − a1 (t)u2 ... − an−1 (t)un
u1
u2
u3
.
.
.
.
.
un
∞
X
El u2
l=1
∞
X
El u3
l=1
.
=
.
.
∞
n−1
∞
−a (t) X λ E u − X a (t) X E u
0
l l 0
k
l k+1
l=1
k=1
EJQTDE, 2010 No. 22, p. 14
l=1
∞
X
=
l=1
=
∞
X
l=1
0
1
0
0
0
0
1
....
.... 0
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
.
−a0 (t)λl
−a1 (t)
.
.
.
.
−an−1 (t)
Al (t)Pl z, where
El
0
0
.
.
Pl :=
.
.
.
.
.
0
0
.
.
.
.
.
.
.
.
0
.
0
0
0 .... 0
El
0
0 .... 0
0
El
0 .... 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
El
0
0
0
.... 0
El
0
0
.... 0
0
El
0
.... 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, l ≥ 1,
.
.
.
.
El
EJQTDE, 2010 No. 22, p. 15
u1
u2
u3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
El
un
and
(4.1)
0
0
.
Al (t) :=
.
.
−a0 (t)λl
0
1
0
.... 0
0
1 ....
0
.
.
1
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
−a1 (t)
.
, l ≥ 1.
−an−1 (t)
π
From Eq. (1.3) it easily follows that there exists ω ∈
, π such that if we
2
define
(
)
Sω = z ∈ C \ {0} : arg z ≤ ω ,
.
.
.
then
Sω ∪ {0} ⊂ ρ(A(t)).
On the other hand, one can show without difficulty that Al (t) = Kl−1 (t)Jl (t)Kl (t),
where Jl (t), Kl (t) are respectively given by
and
l
ρ1 (t)
0
0
.
Jl (t) =
.
.
.
.
0
0
0
....
ρl2 (t)
0
0
....
0
ρl3 (t) 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
....
0
.
.
, l ≥ 1
.
.
.
.
.
.
l
. ρn (t)
EJQTDE, 2010 No. 22, p. 16
1
ρl1
l 2
(ρ1 )
l 3
(ρ1 )
Kl (t) =
.
.
.
l n−1
(ρ1 )
1
1
1 ....
ρl2
ρl3
.
....
(ρl2 )2
(ρl3 )2
.
....
(ρl2 )3
(ρl3 )3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(ρl2 )n−1
(ρl3 )n−1
.
.
1
l 2
(ρn )
l 3
(ρn )
.
.
.
.
l n−1
(ρn )
(ρln )2
For λ ∈ Sω and z ∈ X, one has
R(λ, A(t))z =
∞
X
(λ − Al (t))−1 Pl z
l=1
=
∞
X
Kl (t)(λ − Jl (t)Pl )−1 Kl−1 (t)Pl z.
l=1
Hence,
∞
2
2 X
A(t))z
≤
R(λ,
Kl (t)Pl (λ − Jl (t)Pl )−1 Kl−1 (t)Pl
≤
l=1
∞
X
l=1
B(X)
2
Kl (t)Pl
B(X)
2
(λ − Jl (t)Pl )−1
B(X)
2
Pl z
2
−1
Kl (t)Pl
B(X)
2
Pl z
.
EJQTDE, 2010 No. 22, p. 17
u1
u2
u3
.
Moreover, for z :=
. ∈ X, we obtain
.
.
.
un
n
2
2 X
2
l 2(k−1)
El uk
Kl (t)Pl z
=
El u1
+
ρk (t)
k=2
≤
!
n
X
l 2(k−1)
2
1+
ρk (t)
z
.
k=2
Let dln (t) :=
n
X
l 2(k−1)
> 0. Thus, there exists C1 > 0 such that
ρk (t)
k=2
Kl (t)Pl z
≤ C1 dln (t)
z
for all l ≥ 1 and t ∈ R.
u1
u2
u3
.
−1
Using induction, one can compute Kl (t) and show that for z :=
. ∈ X,
.
.
.
there is C2 > 0 such that
un
C2
−1
Kl (t)Pl z
≤ l
z
for all l ≥ 1 and t ∈ R.
dn (t)
EJQTDE, 2010 No. 22, p. 18
Now, for z ∈ X, we have
u
2
1
1
0
0
.
0
λ − ρl
1
u 2
1
0
0
0
.
0
l
u
λ − ρ2
3
2
1
.
(λ − Jl Pl )−1 z
=
0
0
0
.
0
l
λ
−
ρ
3
.
.
.
.
.
.
.
.
1
0
0
0
.
λ − ρln
un
2
2
2
1
1
1
≤
un
.
u1
+
u2
+ ... +
l
l
l
2
2
2
|λ − ρn |
|λ − ρ1 |
|λ − ρ2 |
Let λ0 > 0. Define the function
1 + λ
.
η(λ) :=
λ − ρlk
It is clear that η is continuous and bounded on the closed set
(
)
Σ := λ ∈ C : λ ≤ λ0 , arg λ ≤ ω .
On the other hand, it is clear that η is bounded for λ > λ0 . Thus η is bounded
on Sω . If we take
1 + λ
: λ ∈ Sω , l ≥ 1 ; k = 1, 2, ..., n, t ∈ R .
N = sup
λ − ρl
k
Therefore,
Consequently,
(λ − Jl Pl )−1 z
≤
N
kzk,
1 + λ
R(λ, A(t))
≤
K
1 + λ
λ ∈ Sω .
for all λ ∈ Sω and t ∈ R.
First of all, note that the domain D = D(A(t)) is independent of t. Thus to
check that Eq. (2.2) is satisfied it is enough to check that Eq. (2.3) holds. For
EJQTDE, 2010 No. 22, p. 19
that, note that the operator A(t) is invertible with
A(t)−1
a1 (t) −1
− a0 (t) A
IH
0
=
.
.
0
−
a2 (t) −1
A
a0 (t)
...
−
an−1 (t) −1
A
a0 (t)
−
1
A−1
a0 (t)
0
...
0
0
IH
...0
0
0
.
.
.
.
.
.
.
.
0
.
.
IH
0
.
0
for all t ∈ R. Hence, for t, s, r ∈ R, computing A(t) − A(s) A(r)−1 and assuming
that there exist Lk ≥ 0 (k = 0, 1, 2, ..., n − 1) and µ ∈ (0, 1] such that
(4.2)
µ
ak (t) − ak (s) ≤ Lk t − s , k = 0, 1, 2, ..., n − 1
it easily follows that there exists C > 0 such that
µ
(A(t) − A(s))A(r)−1 z
≤ C t − s
z
.
n
o
In summary, the family of operators A(t)
t∈R
satisfy Acquistpace-Terreni con-
ditions. Consequently, there exists an evolution family U (t, s) associated with it.
Let us now check that U (t, s) has exponential dichotomy. For that, we will have to
check that (i)-(j) hold. First of all note that For every t ∈ R, the family of linear
operators A(t) generate an analytic semigroup (eτ A(t) )τ ≥0 on X given by
e
τ A(t)
z=
∞
X
Kl (t)−1 Pl eτ Jl Pl Kl (t)Pl z, z ∈ X.
l=1
On the other hand, we have
∞
X
τ A(t)
z
=
e
Kl (t)−1 Pl
l=1
B(X)
τ Jl
e Pl
B(X)
Kl (t)Pl
B(X)
Pl z
,
EJQTDE, 2010 No. 22, p. 20
u1
u2
u
with for each z =
3 ,
.
un
2
τ Jl
e Pl z
=
≤
≤
Therefore
(4.3)
l
2
eρ1 τ E
0
0 0 0 ...
0
u1
l
l
u2
0
eρ2 τ El 0 0 0 ...
0
u3
.
.
. . . .
.
.
.
. . . .
. .
l
un
0
0
... 0 0 0 eλn τ El
2
2
l
2
l
l
ρ1 τ
e El u1
+
eρ2 τ El u2
+ .... +
eρn τ El un
2
e−2δ0 τ
z
.
τ A(t)
e
≤ Ce−δ0 τ ,
τ ≥ 0.
Using the continuity of ak (k = 0, ..., n − 1) and the equality
R(λ, A(t)) − R(λ, A(s)) = R(λ, A(t))(A(t) − A(s))R(λ, A(s)),
it follows that the mapping J ∋ t 7→ R(λ, A(t)) is strongly continuous for λ ∈
Sω where J ⊂ R is an arbitrary compact interval. Therefore, A(t) satisfies the
assumptions of [40, Corollary 2.3], and thus the evolution family (U (t, s))t≥s is
exponentially stable.
It is clear that (H.2) holds. It remains to check assumption (H.4). For that
we need to show that A−1 (·) ∈ AA(B(X)). Since t 7→ ak (t) (k = 0, 1, 2, ..., n − 1),
ak (t)
and t 7→ a0 (t)−1 are almost automorphic it follows that t 7→ dk (t) = −
a0 (t)
(k = 1, 2, ..., n− 1) is almost automorphic, too. Therefore, for every sequence of real
numbers (s′n )n∈N , there exists a subsequence (sn )n∈N such that for m = 0, 1, ..., n−1,
−1
a
˜−1
0 (t) := lim a0 (t + sn )
n→∞
is well defined for each t ∈ R, and
lim a
˜−1 (t
n→∞ 0
− sn ) = a−1
0 (t)
for each t ∈ R, and
d˜m (t) := lim dm (t + sn )
n→∞
is well defined for each t ∈ R, and
lim d˜m (t − sn ) = dm (t)
n→∞
for each t ∈ R.
EJQTDE, 2010 No. 22, p. 21
Setting
˜
A(t)
=
... d˜n−1 (t)A−1
1
A−1
a
˜0 (t)
d˜1 (t)A−1
d˜2 (t)A−1
d˜3 (t)A−1
IH
0
0
...
0
0
0
IH
0
0
0
0
.
.
.
.
.
0
.
.
.
.
.
.
0
0
.
.
IH
0
−
for each t ∈ R, one can easily see that, for the topology of B(X), the following hold
˜ := lim A−1 (t + sn )
A(t)
n→∞
is well defined for each t ∈ R, and
˜ − sn ) = A−1 (t)
lim A(t
n→∞
for each t ∈ R, and hence t 7→ A−1 (t) is almost automorphic with respect to
operator-topology.
It is now clear that if f satisfies (H.5) and if L is small enough, then the higherorder differential equation Eq. (1.4) has an almost automorphic solution
u1
u
2
u3 ∈ Xα = Hα × Hn−1 .
.
un
Therefore, If f = f1 + f2 satisfies (H.5) and if the Lipschitz constant of f1 is small
enough, then Eq. (1.2) has at least one almost automorphic solution u ∈ Hα .
5. Examples of Second-Order Boundary Value Problems
In this section, we provide with a few illustrative examples. Precisely, we study
the existence of almost automorphic solutions to modified versions of the so-called
(nonautonomous) Sine-Gordon equations (see [26]).
In this section, we take n = 2 and suppose a0 and a1 , in addition of being almost
1
automorphic, satisfy the other previous assumptions. Moreover, we let α = and
2
1
fix β ∈
,1 .
2
EJQTDE, 2010 No. 22, p. 22
5.1. Nonautonomous Sine-Gordon Equations. Let L > 0 and and let J =
(0, L). Let H = L2 (J) be equipped with its natural topology. Our main objective
here is to study the existence of almost automorphic solutions to a slightly modified
version of the so-called Sine-Gordon equation with Dirichlet boundary conditions,
which had been studied in the literature especially by Leiva [26] in the following
form
∂u
∂2u
∂2u
(5.1)
+c
t ∈ R, x ∈ J
− d 2 + k sin u = p(t, x),
2
∂t
∂t
∂x
(5.2)
u(t, 0) = u(t, L) = 0,
t∈R
where c, d, k are positive constants, p : R × J 7→ R is continuous and bounded.
Precisely, we are interested in the following system of second-order partial differential equations
(5.3)
∂u
∂2u
∂2u
+ a1 (t, x)
− a0 (t, x) 2 = Q(t, x, u),
2
∂t
∂t
∂x
(5.4)
u(t, 0) = u(t, L) = 0,
t ∈ R, x ∈ J
t∈R
where a1 , a0 : R × J 7→ R are almost automorphic positive functions and Q :
R × J × L2 (J) 7→ L2 (J) is S p -almost automorphic for p > 1.
Let us take
Av = −v ′′ for all v ∈ D(A) = H10 (J) ∩ H2 (J)
and suppose that Q : R × J × L2 (J) 7→ Hβ0 (J) is S p -almost automorphic in t ∈ R
uniformly in x ∈ J and u ∈ L2 (J) Moreover, Q is Lipschitz in the following sense:
there exists L′′ > 0 for which
≤ L′′
u − v
Q(t, x, u) − Q(t, x, v)
β
H0 (J)
2
2
for all u, v ∈ L (J), x ∈ J and t ∈ R.
Consequently, the system Eq. (5.3) - Eq. (5.4) has unique solution u ∈ AA(H10 (J))
when K ′′ is small enough.
5.2. A Slightly Modified Version of the Nonautonomous Sine-Gordon
Equations. Let Ω ⊂ RN (N ≥ 1) be a open bounded subset with C 2 boundary Γ =
∂Ω and let H = L2 (Ω) equipped with its natural topology k · kL2 (Ω) . Here, we are
interested in a slightly modified version of the nonautonomous Sine-Gordon studied
in the previous example, that is, the system of second-order partial differential
equations given by
(5.5)
(5.6)
∂u
∂2u
+ a1 (t, x)
− a0 (t, x)∆u = R(t, x, u),
∂t2
∂t
u(t, x) = 0,
t ∈ R, x ∈ Ω
t ∈ R, x ∈ ∂Ω
where a1 , a0 : R × Ω 7→ R are almost automorphic positive functions, and R :
R × Ω × L2 (Ω) 7→ L2 (Ω) is S p -almost automorphic for p > 1.
EJQTDE, 2010 No. 22, p. 23
Define the linear operator A as follows:
Au = −∆u for all u ∈ D(A) = H10 (Ω) ∩ H2 (Ω).
For each µ ∈ (0, 1), we take Hµ = D((−∆)µ ) = Hµ0 (Ω) ∩ H2µ (Ω) equipped with its
µ-norm k · kµ .
Suppose that R : R × Ω × L2 (Ω) 7→ Hβ0 (Ω) is S p -almost automorphic in t ∈ R
uniformly in x ∈ Ω and u ∈ L2 (Ω). Moreover, R is Lipschitz in the following sense:
there exists L′′′ > 0 for which
R(t, x, u) − R(t, x, v)
≤ L′′′
u − v
β
2
2
for all u, v ∈ L (Ω), x ∈ Ω and t ∈ R.
Therefore, the system Eq. (5.5) - Eq. (5.6) has a unique solution u ∈ AA(H10 (Ω))
when L′′′ is small enough.
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Lecture Notes in Mathematics, Vo
2010, No. 22, 1-26; http://www.math.u-szeged.hu/ejqtde/
EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS TO
SOME CLASSES OF NONAUTONOMOUS HIGHER-ORDER
DIFFERENTIAL EQUATIONS
TOKA DIAGANA
Abstract. In this paper, we obtain the existence of almost automorphic solutions to some classes of nonautonomous higher order abstract differential
equations with Stepanov almost automorphic forcing terms. A few illustrative
examples are discussed at the very end of the paper.
1. Introduction
The main motivation of this paper comes from the work of Andres, Bersani, and
Radov´
a [8], in which the existence (and uniqueness) of almost periodic solutions
was established for the class of n-order autonomous differential equations
n
X
(1.1)
ak u(n−k) (t) = f (u) + p(t), t ∈ R,
u(n) (t) +
k=1
where f, p : R 7→ R are (Stepanov) almost periodic, f is Lipschitz, and ak ∈ R
for k = 1, ..., n are given real constants such that the real part of each root of the
characteristic polynomial associated with the (linear) differential operator on the
left-hand side of Eq. (1.1), that is,
Q(λ) := λn +
n
X
ak λn−k
k=1
is at least nonzero.
The method utilized in [8] makes extensive use of a very complicated representation formula for solutions to Eq. (1.1). For details on that representation formula,
we refer the reader to [9] and [10] and the references therein.
Let H be a Hilbert space. In this paper, we study a more general equation than
Eq. (1.1). Namely, using similar techniques as in [14, 27], we study and obtain
some reasonable sufficient conditions, which do guarantee the existence of almost
automorphic solutions to the class of nonautonomous n-order differential equations
(1.2)
u(n) (t) +
n−1
X
ak (t)u(k) (t) + a0 (t)Au(t) = f (t, u), t ∈ R,
k=1
where A : D(A) ⊂ H 7→ H is a (possibly unbounded) self-adjoint linear operator on
H whose spectrum consists of isolated eigenvalues
0 < λ1 < λ2 < ... < λl → ∞ as l → ∞
1991 Mathematics Subject Classification. 43A60; 34B05; 34C27; 42A75; 47D06; 35L90.
Key words and phrases. exponential dichotomy; Acquistapace and Terreni conditions; evolution families; almost automorphic; Stepanov almost automorphic, nonautonomous higher-order
differential equation.
EJQTDE, 2010 No. 22, p. 1
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the
corresponding eigenspace, the functions ak : R 7→ R (k = 0, 1, ..., n − 1) are almost
automorphic with
inf a0 (t) = γ0 > 0,
t∈R
and the function f : R×H 7→ H is Stepanov almost automorphic in the first variable
uniformly in the second variable.
Consider the time-dependent polynomial defined by
Qlt (ρ) := ρn +
n−1
X
ak (t)ρk + λl a0 (t)
k=1
and denote its roots by
ρlk (t) = µlk (t) + iνkl (t),
k = 1, 2, ..., n, l ≥ 1, and t ∈ R.
In the rest of this paper, we suppose that there exists δ0 > 0 such that
#
"
l
l
l
(1.3)
sup
max µ1 (t), µ2 (t), ..., µn (t) ≤ −δ0 < 0.
l≥1,t∈R
To deal with Eq. (1.2), the main idea consists of rewriting it as a nonautonomous
first-order differential equation on Xn = H × H × H.... × H (n-times) involving the
family of n × n-operator matrices {A(t)}t∈R .
Indeed, assuming that u is differentiable n times and setting
u
u′
u′′
z :=
u(3)
.
,
u(n−1)
then Eq. (1.2) can be rewritten in the Hilbert space Xn in the following form
(1.4)
z ′ (t) = A(t)z(t) + F (t, z(t)), t ∈ R,
where A(t) is the family of n × n-operator matrices defined by
0
IH
0
0 .... 0
0
0
IH .... 0
(1.5)
A(t) =
.
.
. IH
.
.
.
.
.
.
.
.
.
.
−a0 (t)A −a1 (t)IH .
.
.
. −an−1 (t)IH
whose domains D(A(t)) are constant in t ∈ R and are precisely given by
D = D(A) × H × H... × H := D(A) × Xn−1
EJQTDE, 2010 No. 22, p. 2
for all t ∈ R.
Moreover, the semilinear term F appearing in
some α ∈ (0, 1) by
0
0
0
F (t, z) := 0
.
.
,
f (t, u)
the real interpolation space between Xn and D(A(t)) given by Xnα =
, with
Hα := (H, D(A))α,∞ .
Xnα is
n−1
where
Hα × X
Eq. (1.4) is defined on R × Xnα for
Under some reasonable assumptions, it will be shown that the linear operator
matrices A(t) satisfy the well-known Acquistapace-Terreni conditions [3], which do
guarantee the existence of an evolution family U (t, s) associated with it. Moreover,
it will be shown that U (t, s) is exponentially stable under those assumptions.
The existence of almost automorphic solutions to higher-order differential equations is important due to their (possible) applications. For instance when n = 2,
we have thermoelastic plate equations [14, 27] or telegraph equation [31] or SineGordon equations [26]. Let us also mention that when n = 2, some contributions
on the maximal regularity, bounded, almost periodic, asymptotically almost periodic solutions to abstract second-order differential and partial differential equations
have recently been made, among them are [11], [12], [44], [45], [46], and [47]. In
[8], the existence of almost periodic solutions to higher-order differential equations
with constant coefficients in the form Eq. (1.1) was obtained in particular in the
case when the forcing term is almost periodic. However, to the best of our knowledge, the existence of almost automorphic solutions to higher-order nonautonomous
equations in the form Eq. (1.2) in the case when the forcing term is Stepanov almost
automorphic is an untreated original question, which in fact constitutes the main
motivation of the present paper.
The paper is organized as follows: Section 2 is devoted to preliminaries facts
needed in the sequel. In particular, facts related to the existence of evolution
families as well as preliminary results on intermediate spaces will be stated there.
In addition, basic definitions and classical results on (Stepanov) almost automorphic
functions are also given. In Sections 3 and 4, we prove the main result. In Section
5, we provide the reader with an example to illustrate our main result.
2. Preliminaries
Let H be a Hilbert space equipped with the norm k·k and the inner product h·, ·i.
In this paper, A : D(A) ⊂ H 7→ H stands for a self-adjoint (possibly unbounded)
linear operator on H whose spectrum consists of isolated eigenvalues
0 < λ1 < λ2 < ... < λl → ∞ as l → ∞
EJQTDE, 2010 No. 22, p. 3
with each eigenvalue having a finite multiplicity γj equals to the multiplicity of the
corresponding eigenspace.
Let {ekj } be a (complete) orthonormal sequence of eigenvectors associated with
the eigenvalues {λj }j≥1 .
)
(
∞
2
X
2
λj
Ej u
< ∞ , we have
Clearly, for each u ∈ D(A) := u ∈ H :
j=1
Au =
∞
X
λj
j=1
where Ej u =
γj
X
γj
X
hu, ekj iekj =
∞
X
λj Ej u
j=1
k=1
hu, ekj iekj .
k=1
Note that {Ej }j≥1 is a sequence of orthogonal projections on H. Moreover, each
u ∈ H can written as follows:
∞
X
Ej u.
u=
j=1
It should also be mentioned that the operator −A is the infinitesimal generator of
an analytic semigroup {T (t)}t≥0 , which is explicitly expressed in terms of those
orthogonal projections Ej by, for all u ∈ H,
T (t)u =
∞
X
e−λj t Ej u.
j=1
In addition, the fractional powers Ar (r ≥ 0) of A exist and are given by
)
(
∞
2
X
2r
r
λj
Ej u
< ∞
D(A ) = u ∈ H :
j=1
and
Ar u =
∞
X
r
λ2r
j Ej u, ∀u ∈ D(A ).
j=1
Let (X,
·
) be a Banach space. If L is a linear operator on the Banach space
X, then:
• D(L) stands for its domain;
• ρ(L) stands for its resolvent;
• σ(L) stands for its spectrum;
• N (L) stands for its null-space or kernel; and
• R(L) stands for its range.
If L
: D
= D(L) ⊂ X 7→
X
is a closed linear operator, one denotes its graph norm
by
·
. Clearly, (D,
·
) is a Banach space. Moreover, one sets
D
D
R(λ, L) := (λI − L)−1
for all λ ∈ ρ(A).
EJQTDE, 2010 No. 22, p. 4
We set Q = I − P for a projection P . If Y, Z are Banach spaces, then the
space B(Y, Z) denotes the collection of all bounded linear operators from Y into Z
equipped with its natural topology. This is simply denoted by B(Y) when Y = Z.
2.1. Evolution Families. Hypothesis (H.1). The family of closed linear operators A(t) for t ∈ R on X with domain D(A(t)) (possibly not densely defined) satisfy
the so-called
Acquistapace-Terreni conditions, that is, there exist constants ω ∈ R,
θ ∈ π2 , π , K, L ≥ 0 and µ, ν ∈ (0, 1] with µ + ν > 1 such that
n o
K
(2.1)
Sθ ∪ 0 ⊂ ρ A(t) − ω ∋ λ,
R λ, A(t) − ω
≤
1 + λ
and
h
i
µ −ν
(2.2)
A(t) − ω R λ, A(t) − ω R ω, A(t) − R ω, A(s)
≤ L t − s λ
(
)
for t, s ∈ R, λ ∈ Sθ := λ ∈ C \ {0} : arg λ ≤ θ .
Note that in the particular case when A(t) has a constant domain D = D(A(t)),
it is well-known [6, 38] that Eq. (2.2) can be replaced with the following: There
exist constants L and 0 < µ ≤ 1 such that
µ
(2.3)
A(t) − A(s) R ω, A(r)
≤ L t − s , s, t, r ∈ R.
It should mentioned that (H.1) was introduced in the literature by Acquistapace
and Terreni in [2, 3] for ω = 0. Among other things, it ensures that there exists a
unique evolution family U = U (t, s) on X associated with A(t) satisfying
(a) U (t, s)U (s, r) = U (t, r);
(b) U (t, t) = I for t ≥ s ≥ r in R;
(c) (t, s) 7→ U (t, s) ∈ B(X) is continuous for t > s;
∂U
(t, s) = A(t)U (t, s) and
(d) U (·, s) ∈ C 1 ((s, ∞), B(X)),
∂t
A(t)k U (t, s)
≤ K (t − s)−k
for 0 < t − s ≤ 1, k = 0, 1, 0 ≤ α < µ, x ∈ D((ω − A(s))α ), and a constant
C depending only on the constants appearing in (H.1); and
(e) ∂s+ U (t, s)x = −U (t, s)A(s)x for t > s and x ∈ D(A(s)) with A(s)x ∈
D(A(s)).
It should also be mentioned that the above-mentioned proprieties were mainly
established in [1, Theorem 2.3] and [49, Theorem 2.1], see also [3, 48]. In that case
we say that A(·) generates the evolution family U (·, ·).
One says that an evolution family U has an exponential dichotomy (or is hyperbolic) if there are projections P (t) (t ∈ R) that are uniformly bounded and strongly
continuous in t and constants δ > 0 and N ≥ 1 such that
(f) U (t, s)P (s) = P (t)U (t, s);
(g) the restriction UQ (t, s) : Q(s)X → Q(t)X of U (t, s) is invertible (we then
eQ (s, t) := UQ (t, s)−1 ); and
set U
EJQTDE, 2010 No. 22, p. 5
e
−δ(t−s)
(h)
U (t, s)P (s)
≤ N e−δ(t−s) and
U
for t ≥ s and
Q (s, t)Q(t)
≤ N e
t, s ∈ R.
According to [40], the following sufficient conditions are required for A(t) to have
exponential dichotomy.
(i) Let (A(t), D(t))t∈R be generators of analytic semigroups on X of the same
type. Suppose that D(A(t)) ≡ D(A(0)), A(t) is invertible,
sup
A(t)A(s)−1
t,s∈R
is finite, and
µ
A(t)A(s)−1 − I
≤ L0 t − s
for t, s ∈ R and constants L0 ≥ 0 and 0 < µ ≤ 1.
(j) The semigroups (eτ A(t) )τ ≥0 , t ∈ R, are hyperbolic with projection Pt and
constants N, δ > 0. Moreover, let
A(t)eτ A(t) Pt
≤ ψ(τ )
and
A(t)eτ AQ (t) Qt
≤ ψ(−τ )
µ
for τ > 0 and a function ψ such that R ∋ s 7→ ϕ(s) := s ψ(s) is integrable
< 1.
with L0
ϕ
1
L (R)
This setting requires some estimates related to U (t, s). For that, we introduce
the interpolation spaces for A(t). We refer the reader to the following excellent
books [6], [23], and [29] for proofs and further information on theses interpolation
spaces.
Let A be a sectorial operator on X (for that, in assumption (H.1), replace A(t)
with A) and let α ∈ (0, 1). Define the real interpolation space
(
)
A
α
A
Xα := x ∈ X :
x
:= supr>0
r A − ω R r, A − ω x
< ∞ ,
α
A
which, by the way, is a Banach space when endowed with the norm
·
. For
α
convenience we further write
A
A
XA
0 := X,
x
:=
x
, X1 := D(A)
0
and
A
x
:=
(ω − A)x
.
1
ˆ A := D(A) of X. In particular, we have the following continuous
Moreover, let X
embedding
(2.4)
α
A
ˆA
D(A) ֒→ XA
β ֒→ D((ω − A) ) ֒→ Xα ֒→ X ֒→ X,
for all 0 < α < β < 1, where the fractional powers are defined in the usual way.
EJQTDE, 2010 No. 22, p. 6
In general, D(A) is not dense in the spaces XA
α and X. However, we have the
following continuous injection
XA
β ֒→ D(A)
(2.5)
k·kA
α
for 0 < α < β < 1.
Given the family of linear operators A(t) for t ∈ R, satisfying (H.1), we set
ˆ t := X
ˆ A(t)
Xt := XA(t) , X
α
α
for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embedding in
Eq. (2.4) holds with constants independent of t ∈ R. These interpolation spaces
are of class Jα ([29, Definition 1.1.1 ]) and hence there is a constant c(α) such that
t
1−α
α
(2.6)
y
≤ c(α)
y
A(t)y
, y ∈ D(A(t)).
α
We have the following fundamental estimates for the evolution family U.
Proposition 2.1. [14] For x ∈ X, 0 ≤ α ≤ 1 and t > s, the following hold:
(i) There is a constant c(α), such that
t
δ
(2.7)
U (t, s)P (s)x
≤ c(α)e− 2 (t−s) (t − s)−α
x
.
α
(ii) There is a constant m(α), such that
(2.8)
s
e
UQ (s, t)Q(t)x
≤ m(α)e−δ(t−s)
x
.
α
In addition to above, we also need the following assumptions:
Hypothesis (H.2). The evolution family U generated by A(·) has an exponential
dichotomy with constants N, δ > 0 and dichotomy projections P (t) for t ∈ R.
Hypothesis (H.3). There exist α, β with 0 ≤ α < β < 1 and such that
Xtα = Xα and Xtβ = Xβ
for all t ∈ R, with uniform equivalent norms.
2.2. Stepanov Almost Automorphic Functions. Let (X, k · k), (Y, k · kY ) be
two Banach spaces. Let BC(R, X) (respectively, BC(R × Y, X)) denote the collection of all X-valued bounded continuous functions (respectively, the class of jointly
bounded continuous functions F : R × Y 7→ X). The space BC(R, X) equipped
with the sup norm k · k∞ is a Banach space. Furthermore, C(R, Y) (respectively,
C(R × Y, X)) denotes the class of continuous functions from R into Y (respectively,
the class of jointly continuous functions F : R × Y 7→ X).
Definition 2.2. [37] The Bochner transform f b (t, s), t ∈ R, s ∈ [0, 1] of a function
f : R 7→ X is defined by f b (t, s) := f (t + s).
Remark 2.3. (i) A function ϕ(t, s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a
certain function f , ϕ(t, s) = f b (t, s) , if and only if ϕ(t + τ, s − τ ) = ϕ(s, t) for all
t ∈ R, s ∈ [0, 1] and τ ∈ [s − 1, s].
(ii) Note that if f = h + ϕ, then f b = hb + ϕb . Moreover, (λf )b = λf b for each
scalar λ.
EJQTDE, 2010 No. 22, p. 7
Definition 2.4. The Bochner transform F b (t, s, u), t ∈ R, s ∈ [0, 1], u ∈ X of a
function F (t, u) on R × X, with values in X, is defined by
F b (t, s, u) := F (t + s, u)
for each u ∈ X.
Definition 2.5. Let p ∈ [1, ∞). The space BS p (X) of all Stepanov bounded
functions, with the exponent p, consists of all measurable functions f : R 7→ X such
that f b belongs to L∞ R; Lp ((0, 1), X) . This is a Banach space with the norm
Z t+1
1/p
kf kS p := kf b kL∞ (R,Lp ) = sup
kf (τ )kp dτ
.
t∈R
t
p
2.3. S -Almost Automorphy.
Definition 2.6. (Bochner) A function f ∈ C(R, X) is said to be almost automorphic if for every sequence of real numbers (s′n )n∈N , there exists a subsequence
(sn )n∈N such that
g(t) := lim f (t + sn )
n→∞
is well defined for each t ∈ R, and
lim g(t − sn ) = f (t)
n→∞
for each t ∈ R.
The collection of all almost automorphic functions from R to X will be denoted
AA(X).
Similarly
Definition 2.7. (Bochner) A function F ∈ C(R × Y, X) is said to be almost automorphic if for every sequence of real numbers (s′n )n∈N , there exists a subsequence
(sn )n∈N such that
G(t, u) := lim F (t + sn , u)
n→∞
is well defined for each t ∈ R, and
lim G(t − sn , u) = F (t, u)
n→∞
for each t ∈ R uniformly in u ∈ Y.
The collection of all almost automorphic functions from R × Y to X will be
denoted AA(R × Y).
We have the following composition result:
Theorem 2.8. [34] Suppose F : R × Y 7→ X belongs to AA(R × Y) and that the
mapping x 7→ f (t, x) is Lipschitz in the sense that there exists L ≥ 0 such that
F (t, x) − F (t, y)
≤ L
x − y
for all x, y ∈ Y uniformly in t ∈ R.
Then, then the function defined by G(t) = F (t, ϕ(t)) belongs to AA(X) provided
ϕ ∈ AA(Y).
EJQTDE, 2010 No. 22, p. 8
We also have the following composition result, which is a straightforward consequence of the composition of pseudo almost automorphic functions obtained in
[43].
Theorem 2.9. [43] If F : R × Y 7→ X belongs to AA(R × Y) and if x 7→ F (t, x)
is uniformly continuous on any bounded subset K of Y for each t ∈ R, then the
function defined by h(t) = F (t, ϕ(t)) belongs to AA(X) provided ϕ ∈ AA(Y).
We will denote by AAu (X) the closed subspace of all functions f ∈ AA(X) with
g ∈ C(R, X). Equivalently, f ∈ AAu (X) if and only if f is almost automorphic
and the convergence in Definition 2.7 are uniform on compact intervals, i.e. in
the Fr´echet space C(R, X). Indeed, if f is almost automorphic, then, its range is
relatively compact. Obviously, the following inclusions hold:
AP (X) ⊂ AAu (X) ⊂ AA(X) ⊂ BC(X),
where AP (X) is the Banach space of almost periodic functions from R to X.
Definition 2.10. [36] The space AS p (X) of Stepanov almost automorphic functions (or S p -almost
automorphic) consists of all f ∈ BS p (X) such that f b ∈
p
AA L (0, 1; X) . That is, a function f ∈ Lploc (R; X) is said to be S p -almost automorphic if its Bochner transform f b : R → Lp (0, 1; X) is almost automorphic in
the sense that for every sequence of real numbers (s′n )n∈N , there exists a subsequence (sn )n∈N and a function g ∈ Lploc (R; X) such that
Z t+1
1/p
kf (sn + s) − g(s)kp ds
→ 0, and
t
Z
t+1
kg(s − sn ) − f (s)kp ds
t
as n → ∞ pointwise on R.
1/p
→0
Remark 2.11. It is clear that if 1 ≤ p < q < ∞ and f ∈ Lqloc (R; X) is S q -almost
automorphic, then f is S p -almost automorphic. Also if f ∈ AA(X), then f is S p almost automorphic for any 1 ≤ p < ∞. Moreover, it is clear that f ∈ AAu (X) if
and only if f b ∈ AA(L∞ (0, 1; X)). Thus, AAu (X) can be considered as AS ∞ (X).
Definition 2.12. A function F : R × Y 7→ X, (t, u) 7→ F (t, u) with F (·, u) ∈
Lploc (R; X) for each u ∈ Y, is said to be S p -almost automorphic in t ∈ R uniformly
in u ∈ Y if t 7→ F (t, u) is S p -almost automorphic for each u ∈ Y, that is, for
every sequence of real numbers (s′n )n∈N , there exists a subsequence (sn )n∈N and a
function G(·, u) ∈ Lploc (R; X) such that
Z t+1
1/p
p
kF (sn + s, u) − G(s, u)k ds
→ 0, and
t
Z
t+1
p
kG(s − sn , u) − F (s, u)k ds
t
as n → ∞ pointwise on R for each u ∈ Y.
1/p
→0
EJQTDE, 2010 No. 22, p. 9
The collection of those S p -almost automorphic functions F : R × Y 7→ X will be
denoted by AS p (R × Y).
We have the following straightforward composition theorems, which generalize
Theorem 2.8 and Theorem 2.9, respectively:
Theorem 2.13. Let F : R× Y 7→ X be a S p -almost automorphic function. Suppose
that u 7→ F (t, u) is Lipschz in the sense that there exists L ≥ 0 such
F (t, u) − F (t, v)
≤ L
u − v
Y
for all t ∈ R, (u, v) ∈ Y × Y.
If φ ∈ AS p (Y), then Γ : R → X defined by Γ(·) := F (·, φ(·)) belongs to AS p (X).
Theorem 2.14. Let F : R × Y 7→ X be a S p -almost automorphic function, where.
Suppose that F (t, u) is uniformly continuous in every bounded subset K ⊂ X uniformly for t ∈ R. If g ∈ AS p (Y), then Γ : R → X defined by Γ(·) := F (·, g(·))
belongs to AS p (X).
3. Main results
Consider the nonautonomous differential equation
u′ (t) = A(t)u(t) + F (t, u(t)), t ∈ R,
(3.1)
where F : R × Xα 7→ X is S p -almost automorphic.
Definition 3.1. A function u : R 7→ Xα is said to be a bounded solution to Eq.
(3.1) provided that
Z ∞
Z t
(3.2) u(t) =
UQ (t, s)Q(s)F (s, u(s))ds.
U (t, s)P (s)F (s, u(s))ds −
−∞
t
for all t ∈ R.
Throughout the rest of the paper, we set S1 u(t) := S11 u(t) − S12 u(t), where
Z t
Z ∞
UQ (t, s)Q(s)F (s, u(s))ds.
S11 u(t) :=
U (t, s)P (s)F (s, u(s))ds, S12 u(t) :=
t
−∞
for all t ∈ R.
To study Eq. (3.1), in addition to the previous assumptions, we require that
1 1
p > 1, + = 1, and that the following assumptions hold:
p q
(H.4) R(ω, A(·)) ∈ B(R, AA(Xα )).
(H.5) The function F : R × X 7→ Xβ is S p -almost automorphic in t ∈ R uniformly
in u ∈ X. Moreover, F is Lipschitz in the following sense: there exists
L > 0 for which
F (t, u) − F (t, v)
≤ L
u − v
β
for all u, v ∈ X and t ∈ R.
EJQTDE, 2010 No. 22, p. 10
Lemma 3.2. Under assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5) and if
#1/q
"Z
∞
n
X
δ
(3.3)
< ∞,
e−q 2 s s−qα ds
N (α, q, δ) :=
n−1
n=1
then the integral operator S1 defined above maps AA(Xα ) into itself.
Proof. Let u ∈ AA(Xα ). Setting ϕ(t) := F (t, u(t)) and using Theorem 2.13 it
follows that ϕ ∈ AS p (Xβ ). The next step consists of showing that S1 ∈ AA(Xα ).
Define for all n = 1, 2, ..., the sequence of integral operators
Φn (t) =
Z
n
U (t, t − s)P (t − s)ϕ(t − s)ds
n−1
for each t ∈ R.
Letting r = t − s it follows that
Z t−n+1
Φn (t) =
U (t, r)P (r)ϕ(r)dr,
t−n
and hence from the H¨
older’s inequality and the estimate Eq. (2.7) it follows that
Z t−n+1
δ
c(α)e− 2 (t−r) (t − r)−α
φ(r)
dr
≤
Φn (t)
α
≤
c
t−n
Z t−n+1
cc′
≤
δ
c(α)e− 2 (t−r) (t − r)−α
φ(r)
dr
t−n
Z t−n+1
t−n
"Z
≤
n
q(α)
α
δ
c(α)e− 2 (t−r) (t − r)−α
ϕ(r)
dr
β
δ
e−q 2 s s−qα ds
n−1
#1/q
ϕ
Sp
.
Using Eq. (3.3), we then deduce from Weirstrass Theorem that the series defined
by
∞
X
D(t) :=
Φn (t)
n=1
is uniformly convergent on R. Moreover, D ∈ C(R, Xα ) and
∞
X
Φn (t)
≤ q(α)N (α, q, δ)
φ
D(t)
≤
α
n=1
Sp
α
for all t ∈ R.
Let us show that Φn ∈ AA(Xα ) for each n = 1, 2, 3, ... Indeed, since ϕ ∈
AS p (Xβ ) ⊂ AS p (Xα ), for every sequence of real numbers (τn′ )n∈N there exist a
subsequence (τnk )k∈N and a function ϕ
b such that
Z t+1
p
b − ϕ(s + τnk )
ds → 0
ϕ(s)
t
α
EJQTDE, 2010 No. 22, p. 11
and
Z
t+1
t
p
b − τnk ) − ϕ(s)
ds → 0
ϕ(s
α
as k → ∞ pointwise in R.
Define for all n = 1, 2, 3, ..., the sequence of integral operators
Z n
b
Φn (t) =
U (t, t − s)P (t − s)ϕ(t
b − s)ds
n−1
for all t ∈ R.
Now
b
Φ(t + τnk ) − Φ(t)
=
−
=
+
−
−
=
Z
n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
Z n
n−1
n
+
Z
n−1
U (t, t + τnk − s)P (t + τnk − s)ϕ(t + τnk − s)ds
U (t, t − s)P (t − s)ϕ(t
b − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t + τnk − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t
b − s)ds
U (t, t + τnk − s)P (t + τnk − s)ϕ(t
b − s)ds
U (t, t − s)P (t − s)ϕ(t
b − s)ds
h
i
U (t, t + τnk − s)P (t + τnk − s) ϕ(t + τnk − s) − ϕ(t
b − s) ds
h
i
U (t, t + τnk − s)P (t + τnk − s) − U (t, t − s)P (t − s) ϕ(t
b − s)ds.
Using Lebesgue Dominated Convergence Theorem, one can easily see that
Z n
h
i
U (t, t+τnk −s)P (t+τnk −s) ϕ(t+τnk −s)−ϕ(t−s)
b
ds
→ 0 as k → ∞, t ∈ R.
α
n−1
Similarly, using [15] it follows that
Z n h
i
U (t, t+τnk −s)P (t+τnk −s)−U (t, t−s)P (t−s) ϕ(t−s)ds
b
→ 0 as k → ∞, t ∈ R.
α
n−1
Thus
b n (t) = lim Φn (t + τn ), t ∈ R.
Φ
k
k→∞
Similarly, one can easily see that
b n (t − τn )
Φn (t) = lim Φ
k
k→∞
for all t ∈ R and n = 1, 2, 3, ... Therefore the sequence Φn ∈ AA(Xα ) for each
n = 1, 2, ... and hence D ∈ AA(Xα ). Consequently t 7→ S11 (t) belong to AA(Xα ).
The proof for t 7→ S12 (t) is similar to that of t 7→ S11 (t) and hence omitted.
EJQTDE, 2010 No. 22, p. 12
In view of the above, it follows that S1 ∈ AA(Xα ).
Lemma 3.3. The integral operator S1 defined above is a contraction whenever L
is small enough.
Proof. Let v, w ∈ AA(Xα ). Now,
S11 v(t) − S11 w(t)
≤
α
≤
Z
c
t
−∞
Z t
δ
c(α)(t − s)−α e− 2 (t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
−∞
≤
≤
δ
c(α)(t − s)−α e− 2 (t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
Lcc(α)
Z
Lc′ cc(α)
β
t
−∞
Z t
δ
(t − s)−α e− 2 (t−s)
v(s) − w(s)
ds
−∞
Similarly,
S12 v(t) − S12 w(t)
α
Z
δ
(t − s)−α e− 2 (t−s)
v(s) − w(s)
ds.
α
m(α)e−δ(t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
t
Z ∞
e−δ(t−s)
F1 (s, v(s)) − F1 (s, w(s))
ds
≤ cm(α)
β
t
Z ∞
e−δ(t−s)
v(s) − w(s)
ds
≤ Lcm(α)
t
Z ∞
′
e−δ(t−s)
v(s) − w(s)
ds.
≤ Lcc m(α)
≤
∞
α
t
Consequently,
S1 v − S1 w
∞,α
≤ Lcc′ c(α)Γ(1 − α)(2δ −1 )1−α + m(α)δ −1
v − w
∞,α
and hence S1 is a contraction whenever L is small enough.
Theorem 3.4. Suppose assumptions (H.1)-(H.2)-(H.3)-(H.4)-(H.5) and Eq. (3.3)
hold and that L is small enough, then the nonautonmous differential equation Eq.
(3.1) has a unique almost automorphic solution u satisfying u = S1 u
Proof. The proof makes use of Lemma 3.2, Lemma 3.3, and the Bananch fixed-point
principle.
4. Almost Automorphic Solutions to Some Higher-Order
Differential Equations
We have previously seen that each u ∈ H can be written in terms of the sequence
γl
∞ X
∞
X
X
of orthogonal projections En as follows: u =
hu, ekl iekl =
El u. Moreover,
l=1 k=1
l=1
EJQTDE, 2010 No. 22, p. 13
for each u ∈ D(A), Au =
u1
.
un
!
∞
X
l=1
λl
γl
X
hu, ekl iekl =
k=1
∞
X
λl El u. Therefore, for all z :=
l=1
∈ D = D(A(t)) = D(A) × Hn−1 , we obtain the following
A(t)z =
=
0
IH
0
0
.... 0
0
0
IH
....
0
.
.
.
IH
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
−a0 (t)A
−a1 (t)IH
.
.
.
.
−an−1 (t)IH
u2
u3
.
.
.
.
.
−a0 (t)Au1 − a1 (t)u2 ... − an−1 (t)un
u1
u2
u3
.
.
.
.
.
un
∞
X
El u2
l=1
∞
X
El u3
l=1
.
=
.
.
∞
n−1
∞
−a (t) X λ E u − X a (t) X E u
0
l l 0
k
l k+1
l=1
k=1
EJQTDE, 2010 No. 22, p. 14
l=1
∞
X
=
l=1
=
∞
X
l=1
0
1
0
0
0
0
1
....
.... 0
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
1
.
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
.
−a0 (t)λl
−a1 (t)
.
.
.
.
−an−1 (t)
Al (t)Pl z, where
El
0
0
.
.
Pl :=
.
.
.
.
.
0
0
.
.
.
.
.
.
.
.
0
.
0
0
0 .... 0
El
0
0 .... 0
0
El
0 .... 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
El
0
0
0
.... 0
El
0
0
.... 0
0
El
0
.... 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
, l ≥ 1,
.
.
.
.
El
EJQTDE, 2010 No. 22, p. 15
u1
u2
u3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
El
un
and
(4.1)
0
0
.
Al (t) :=
.
.
−a0 (t)λl
0
1
0
.... 0
0
1 ....
0
.
.
1
.
.
.
.
.
.
.
1
.
.
.
.
.
.
.
−a1 (t)
.
, l ≥ 1.
−an−1 (t)
π
From Eq. (1.3) it easily follows that there exists ω ∈
, π such that if we
2
define
(
)
Sω = z ∈ C \ {0} : arg z ≤ ω ,
.
.
.
then
Sω ∪ {0} ⊂ ρ(A(t)).
On the other hand, one can show without difficulty that Al (t) = Kl−1 (t)Jl (t)Kl (t),
where Jl (t), Kl (t) are respectively given by
and
l
ρ1 (t)
0
0
.
Jl (t) =
.
.
.
.
0
0
0
....
ρl2 (t)
0
0
....
0
ρl3 (t) 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
....
0
.
.
, l ≥ 1
.
.
.
.
.
.
l
. ρn (t)
EJQTDE, 2010 No. 22, p. 16
1
ρl1
l 2
(ρ1 )
l 3
(ρ1 )
Kl (t) =
.
.
.
l n−1
(ρ1 )
1
1
1 ....
ρl2
ρl3
.
....
(ρl2 )2
(ρl3 )2
.
....
(ρl2 )3
(ρl3 )3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(ρl2 )n−1
(ρl3 )n−1
.
.
1
l 2
(ρn )
l 3
(ρn )
.
.
.
.
l n−1
(ρn )
(ρln )2
For λ ∈ Sω and z ∈ X, one has
R(λ, A(t))z =
∞
X
(λ − Al (t))−1 Pl z
l=1
=
∞
X
Kl (t)(λ − Jl (t)Pl )−1 Kl−1 (t)Pl z.
l=1
Hence,
∞
2
2 X
A(t))z
≤
R(λ,
Kl (t)Pl (λ − Jl (t)Pl )−1 Kl−1 (t)Pl
≤
l=1
∞
X
l=1
B(X)
2
Kl (t)Pl
B(X)
2
(λ − Jl (t)Pl )−1
B(X)
2
Pl z
2
−1
Kl (t)Pl
B(X)
2
Pl z
.
EJQTDE, 2010 No. 22, p. 17
u1
u2
u3
.
Moreover, for z :=
. ∈ X, we obtain
.
.
.
un
n
2
2 X
2
l 2(k−1)
El uk
Kl (t)Pl z
=
El u1
+
ρk (t)
k=2
≤
!
n
X
l 2(k−1)
2
1+
ρk (t)
z
.
k=2
Let dln (t) :=
n
X
l 2(k−1)
> 0. Thus, there exists C1 > 0 such that
ρk (t)
k=2
Kl (t)Pl z
≤ C1 dln (t)
z
for all l ≥ 1 and t ∈ R.
u1
u2
u3
.
−1
Using induction, one can compute Kl (t) and show that for z :=
. ∈ X,
.
.
.
there is C2 > 0 such that
un
C2
−1
Kl (t)Pl z
≤ l
z
for all l ≥ 1 and t ∈ R.
dn (t)
EJQTDE, 2010 No. 22, p. 18
Now, for z ∈ X, we have
u
2
1
1
0
0
.
0
λ − ρl
1
u 2
1
0
0
0
.
0
l
u
λ − ρ2
3
2
1
.
(λ − Jl Pl )−1 z
=
0
0
0
.
0
l
λ
−
ρ
3
.
.
.
.
.
.
.
.
1
0
0
0
.
λ − ρln
un
2
2
2
1
1
1
≤
un
.
u1
+
u2
+ ... +
l
l
l
2
2
2
|λ − ρn |
|λ − ρ1 |
|λ − ρ2 |
Let λ0 > 0. Define the function
1 + λ
.
η(λ) :=
λ − ρlk
It is clear that η is continuous and bounded on the closed set
(
)
Σ := λ ∈ C : λ ≤ λ0 , arg λ ≤ ω .
On the other hand, it is clear that η is bounded for λ > λ0 . Thus η is bounded
on Sω . If we take
1 + λ
: λ ∈ Sω , l ≥ 1 ; k = 1, 2, ..., n, t ∈ R .
N = sup
λ − ρl
k
Therefore,
Consequently,
(λ − Jl Pl )−1 z
≤
N
kzk,
1 + λ
R(λ, A(t))
≤
K
1 + λ
λ ∈ Sω .
for all λ ∈ Sω and t ∈ R.
First of all, note that the domain D = D(A(t)) is independent of t. Thus to
check that Eq. (2.2) is satisfied it is enough to check that Eq. (2.3) holds. For
EJQTDE, 2010 No. 22, p. 19
that, note that the operator A(t) is invertible with
A(t)−1
a1 (t) −1
− a0 (t) A
IH
0
=
.
.
0
−
a2 (t) −1
A
a0 (t)
...
−
an−1 (t) −1
A
a0 (t)
−
1
A−1
a0 (t)
0
...
0
0
IH
...0
0
0
.
.
.
.
.
.
.
.
0
.
.
IH
0
.
0
for all t ∈ R. Hence, for t, s, r ∈ R, computing A(t) − A(s) A(r)−1 and assuming
that there exist Lk ≥ 0 (k = 0, 1, 2, ..., n − 1) and µ ∈ (0, 1] such that
(4.2)
µ
ak (t) − ak (s) ≤ Lk t − s , k = 0, 1, 2, ..., n − 1
it easily follows that there exists C > 0 such that
µ
(A(t) − A(s))A(r)−1 z
≤ C t − s
z
.
n
o
In summary, the family of operators A(t)
t∈R
satisfy Acquistpace-Terreni con-
ditions. Consequently, there exists an evolution family U (t, s) associated with it.
Let us now check that U (t, s) has exponential dichotomy. For that, we will have to
check that (i)-(j) hold. First of all note that For every t ∈ R, the family of linear
operators A(t) generate an analytic semigroup (eτ A(t) )τ ≥0 on X given by
e
τ A(t)
z=
∞
X
Kl (t)−1 Pl eτ Jl Pl Kl (t)Pl z, z ∈ X.
l=1
On the other hand, we have
∞
X
τ A(t)
z
=
e
Kl (t)−1 Pl
l=1
B(X)
τ Jl
e Pl
B(X)
Kl (t)Pl
B(X)
Pl z
,
EJQTDE, 2010 No. 22, p. 20
u1
u2
u
with for each z =
3 ,
.
un
2
τ Jl
e Pl z
=
≤
≤
Therefore
(4.3)
l
2
eρ1 τ E
0
0 0 0 ...
0
u1
l
l
u2
0
eρ2 τ El 0 0 0 ...
0
u3
.
.
. . . .
.
.
.
. . . .
. .
l
un
0
0
... 0 0 0 eλn τ El
2
2
l
2
l
l
ρ1 τ
e El u1
+
eρ2 τ El u2
+ .... +
eρn τ El un
2
e−2δ0 τ
z
.
τ A(t)
e
≤ Ce−δ0 τ ,
τ ≥ 0.
Using the continuity of ak (k = 0, ..., n − 1) and the equality
R(λ, A(t)) − R(λ, A(s)) = R(λ, A(t))(A(t) − A(s))R(λ, A(s)),
it follows that the mapping J ∋ t 7→ R(λ, A(t)) is strongly continuous for λ ∈
Sω where J ⊂ R is an arbitrary compact interval. Therefore, A(t) satisfies the
assumptions of [40, Corollary 2.3], and thus the evolution family (U (t, s))t≥s is
exponentially stable.
It is clear that (H.2) holds. It remains to check assumption (H.4). For that
we need to show that A−1 (·) ∈ AA(B(X)). Since t 7→ ak (t) (k = 0, 1, 2, ..., n − 1),
ak (t)
and t 7→ a0 (t)−1 are almost automorphic it follows that t 7→ dk (t) = −
a0 (t)
(k = 1, 2, ..., n− 1) is almost automorphic, too. Therefore, for every sequence of real
numbers (s′n )n∈N , there exists a subsequence (sn )n∈N such that for m = 0, 1, ..., n−1,
−1
a
˜−1
0 (t) := lim a0 (t + sn )
n→∞
is well defined for each t ∈ R, and
lim a
˜−1 (t
n→∞ 0
− sn ) = a−1
0 (t)
for each t ∈ R, and
d˜m (t) := lim dm (t + sn )
n→∞
is well defined for each t ∈ R, and
lim d˜m (t − sn ) = dm (t)
n→∞
for each t ∈ R.
EJQTDE, 2010 No. 22, p. 21
Setting
˜
A(t)
=
... d˜n−1 (t)A−1
1
A−1
a
˜0 (t)
d˜1 (t)A−1
d˜2 (t)A−1
d˜3 (t)A−1
IH
0
0
...
0
0
0
IH
0
0
0
0
.
.
.
.
.
0
.
.
.
.
.
.
0
0
.
.
IH
0
−
for each t ∈ R, one can easily see that, for the topology of B(X), the following hold
˜ := lim A−1 (t + sn )
A(t)
n→∞
is well defined for each t ∈ R, and
˜ − sn ) = A−1 (t)
lim A(t
n→∞
for each t ∈ R, and hence t 7→ A−1 (t) is almost automorphic with respect to
operator-topology.
It is now clear that if f satisfies (H.5) and if L is small enough, then the higherorder differential equation Eq. (1.4) has an almost automorphic solution
u1
u
2
u3 ∈ Xα = Hα × Hn−1 .
.
un
Therefore, If f = f1 + f2 satisfies (H.5) and if the Lipschitz constant of f1 is small
enough, then Eq. (1.2) has at least one almost automorphic solution u ∈ Hα .
5. Examples of Second-Order Boundary Value Problems
In this section, we provide with a few illustrative examples. Precisely, we study
the existence of almost automorphic solutions to modified versions of the so-called
(nonautonomous) Sine-Gordon equations (see [26]).
In this section, we take n = 2 and suppose a0 and a1 , in addition of being almost
1
automorphic, satisfy the other previous assumptions. Moreover, we let α = and
2
1
fix β ∈
,1 .
2
EJQTDE, 2010 No. 22, p. 22
5.1. Nonautonomous Sine-Gordon Equations. Let L > 0 and and let J =
(0, L). Let H = L2 (J) be equipped with its natural topology. Our main objective
here is to study the existence of almost automorphic solutions to a slightly modified
version of the so-called Sine-Gordon equation with Dirichlet boundary conditions,
which had been studied in the literature especially by Leiva [26] in the following
form
∂u
∂2u
∂2u
(5.1)
+c
t ∈ R, x ∈ J
− d 2 + k sin u = p(t, x),
2
∂t
∂t
∂x
(5.2)
u(t, 0) = u(t, L) = 0,
t∈R
where c, d, k are positive constants, p : R × J 7→ R is continuous and bounded.
Precisely, we are interested in the following system of second-order partial differential equations
(5.3)
∂u
∂2u
∂2u
+ a1 (t, x)
− a0 (t, x) 2 = Q(t, x, u),
2
∂t
∂t
∂x
(5.4)
u(t, 0) = u(t, L) = 0,
t ∈ R, x ∈ J
t∈R
where a1 , a0 : R × J 7→ R are almost automorphic positive functions and Q :
R × J × L2 (J) 7→ L2 (J) is S p -almost automorphic for p > 1.
Let us take
Av = −v ′′ for all v ∈ D(A) = H10 (J) ∩ H2 (J)
and suppose that Q : R × J × L2 (J) 7→ Hβ0 (J) is S p -almost automorphic in t ∈ R
uniformly in x ∈ J and u ∈ L2 (J) Moreover, Q is Lipschitz in the following sense:
there exists L′′ > 0 for which
≤ L′′
u − v
Q(t, x, u) − Q(t, x, v)
β
H0 (J)
2
2
for all u, v ∈ L (J), x ∈ J and t ∈ R.
Consequently, the system Eq. (5.3) - Eq. (5.4) has unique solution u ∈ AA(H10 (J))
when K ′′ is small enough.
5.2. A Slightly Modified Version of the Nonautonomous Sine-Gordon
Equations. Let Ω ⊂ RN (N ≥ 1) be a open bounded subset with C 2 boundary Γ =
∂Ω and let H = L2 (Ω) equipped with its natural topology k · kL2 (Ω) . Here, we are
interested in a slightly modified version of the nonautonomous Sine-Gordon studied
in the previous example, that is, the system of second-order partial differential
equations given by
(5.5)
(5.6)
∂u
∂2u
+ a1 (t, x)
− a0 (t, x)∆u = R(t, x, u),
∂t2
∂t
u(t, x) = 0,
t ∈ R, x ∈ Ω
t ∈ R, x ∈ ∂Ω
where a1 , a0 : R × Ω 7→ R are almost automorphic positive functions, and R :
R × Ω × L2 (Ω) 7→ L2 (Ω) is S p -almost automorphic for p > 1.
EJQTDE, 2010 No. 22, p. 23
Define the linear operator A as follows:
Au = −∆u for all u ∈ D(A) = H10 (Ω) ∩ H2 (Ω).
For each µ ∈ (0, 1), we take Hµ = D((−∆)µ ) = Hµ0 (Ω) ∩ H2µ (Ω) equipped with its
µ-norm k · kµ .
Suppose that R : R × Ω × L2 (Ω) 7→ Hβ0 (Ω) is S p -almost automorphic in t ∈ R
uniformly in x ∈ Ω and u ∈ L2 (Ω). Moreover, R is Lipschitz in the following sense:
there exists L′′′ > 0 for which
R(t, x, u) − R(t, x, v)
≤ L′′′
u − v
β
2
2
for all u, v ∈ L (Ω), x ∈ Ω and t ∈ R.
Therefore, the system Eq. (5.5) - Eq. (5.6) has a unique solution u ∈ AA(H10 (Ω))
when L′′′ is small enough.
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