C:/Documents and Settings/Profesor/Escritorio/Cubo/Cubo/Cubo/Cubo 2011/Cubo2011-13-01/Vol13 n\2721/Art N\2606 .dvi


CUBO A Mathematical Journal

Vol.13, No¯ 01, (73–101). March 2011

Existence of Pseudo Almost Automorphic Solutions to a
Nonautonomous Heat Equation

Toka Diagana

Department of Mathematics,

Howard University,

2441 6th Street NW, Washington, DC 20005 - USA.

email: tokadiag@gmail.com

ABSTRACT

In this paper, upon making some suitable assumptions, we obtain the existence of

pseudo-almost automorphic solutions to a nonautonomous heat equation with gradient

coefficients.

RESUMEN

En este trabajo, al hacer algunos supuestos adecuados, se obtiene la existencia de pseudo

soluciones automorfas a una ecuacin del calor no autnoma con coeficientes degradados.

Keywords: Pseudo almost periodicity; almost automorphic; pseudo almost automorphic; Sp-

pseudo almost automorphic; Sp-almost automorphic; Sp-pseudo almost periodic; Acquistapace

and Terreni conditions; intermediate space; exponential dichotomy.

AMS Subject Classification: 43A60; 34G20.

1 Introduction

Fix p > 1 and α,β ∈ R with 0 ≤ α < β < 1. This paper is concerned with the existence of pseudo-

almost automorphic solutions to the the class of abstract nonautonomous differential equations

given by

d

dt

[
u(t) + f

(
t,Bu(t)

)]
= A(t)u(t) + g

(
t,Cu(t)

)
, t ∈ R, (1.1)



74 Toka Diagana CUBO
13, 1 (2011)

where A(t) for t ∈ R is a family of closed linear operators on their domains D(A(t)) satisfying

the well-known Acquistapace-Terreni conditions, B,C are (possibly unbounded) linear operators,

and f : R × X 7→ Xtβ and g : R × X 7→ X are S
p-pseudo-almost automorphic functions in t ∈ R

uniformly in the second variable. In view of above, there exists an evolution family U = {U(t,s)}t≥s

associated with the family of operators A(t). Assuming that U = {U(t,s)}t≥s is exponentially

dichotomic (hyperbolic) and under some additional assumptions, it will be shown that Eq. (1.1)

has a unique pseudo-almost automorphic solution. It is worth mentioning that the main result of

this paper (Theorem 5.5) generalizes, to some extent, most of known results on (pseudo) almost

automorphic solutions to autonomous and nonautonomous differential equations, especially those

in [9, 22, 48].

Let Ω ⊂ RN (N ≥ 1) be a bounded subset with regular boundary Γ = ∂Ω and let X = L2(Ω)

be the space of square integrable functions equipped with its natural topology. To illustrate our

main result, we study the existence of pseudo-almost automorphic solutions to the nonautonomous

heat equation with gradient coefficients given by





∂

∂t

[
ϕ + F

(
t,∇ϕ

)]
= a(t,x)∆ϕ + G

(
t,∇ϕ

)
, in R × Ω

ϕ = 0, on R × Γ

(1.2)

where the diffusion coefficient a : R × Ω 7→ R, and F,G : R × X1/2 7→ L
2(Ω) are Sp-pseudo-almost

automorphic functions.

The concept of pseudo-almost automorphy, which is the central tool here, was introduced in

the literature a few years ago by Liang et al. [34, 47] and is a generalization of both the classical

almost automorphy due to Bochner [8] and that of pseudo almost periodicity due to Zhang [18].

Such a concept has recently generated several developments and extensions. For the most recent

developments, we refer the reader to [17, 25, 34, 47, 35]. More recently, in Diagana [17], the concept

of Sp-pseudo almost automorphy (or Stepanov-like pseudo almost automorphy) was introduced,

which in fact is a natural generalization of the notion of pseudo almost automorphy.

In this paper, we make extensive use of the concept of Sp-pseudo almost automorphy combined

with intermediate space techniques to investigate the existence of pseudo-almost automorphic

solutions Eq. (1.1) and then to the N-dimensional heat equation Eq. (1.2). The literature related

to those intermediate spaces is very extensive, in particular, we refer the reader to the excellent

book by Lunardi [33], which contains a comprehensive presentation on that topic and related issues.

Existence results related to almost periodic, asymptotically almost periodic, pseudo almost

periodic and almost automorphic solutions to Eq. (1.1) in the autonomous case have recently been

established in [19], [20], [22], [23], [28], [29], and [30], respectively. Though to the best of our

knowledge, the existence of pseudo almost automorphic solutions to Eq. (1.1) in the case when

the coefficients F,G are Sp-pseudo almost automorphic is an untreated original problem, which



CUBO
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Pseudo Almost Automorphic Solutions 75

constitutes one of the main motivations of the present paper.

2 Preliminaries

Let (X,‖ · ‖), (Y,‖ · ‖Y) 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

‖ · ‖∞ 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).

The notation B(X, Y) stands for the Banach space of bounded linear operators from X into Y

equipped with its natural topology; in particular, this is simply denoted B(X) whenever X = Y.

Definition 2.1. [41] The Bochner transform fb(t,s), t ∈ R, s ∈ [0, 1] of a function f : R 7→ X is

defined by fb(t,s) := f(t + s).

Remark 2.2. (i) A function ϕ(t,s), t ∈ R, s ∈ [0, 1], is the Bochner transform of a certain function

f, ϕ(t,s) = fb(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 fb = hb + ϕb. Moreover, (λf)b = λfb for each scalar λ.

Definition 2.3. 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 2.4. Let p ∈ [1,∞). The space BSp(X) of all Stepanov bounded functions, with the ex-

ponent p, consists of all measurable functions f : R 7→ X such that fb belongs to L∞
(
R; Lp((0, 1), X)

)
.

This is a Banach space with the norm

‖f‖Sp := ‖f
b‖L∞(R,Lp) = sup

t∈R

(∫ t+1

t

‖f(τ)‖p dτ

)1/p
.

2.1 Sp-Pseudo Almost Periodicity

Definition 2.5. A function f ∈ C(R, X) is called (Bohr) almost periodic if for each ε > 0 there

exists l(ε) > 0 such that every interval of length l(ε) contains a number τ with the property that

‖f(t + τ) − f(t)‖ < ε for each t ∈ R.

The number τ above is called an ε-translation number of f, and the collection of all such

functions will be denoted AP(X).



76 Toka Diagana CUBO
13, 1 (2011)

Definition 2.6. A function F ∈ C(R × Y, X) is called (Bohr) almost periodic in t ∈ R uniformly

in y ∈ K where K ⊂ Y is any compact subset K ⊂ Y if for each ε > 0 there exists l(ε) such that

every interval of length l(ε) contains a number τ with the property that

‖F(t + τ,y) − F(t,y)‖ < ε for each t ∈ R, y ∈ K.

The collection of those functions is denoted by AP(R × Y).

Define the classes of functions PAP0(X) and PAP0(R × X) respectively as follows:

PAP0(X) :=

{
u ∈ BC(R, X) : lim

T →∞

1

2T

∫ T

−T

‖u(s)‖ds = 0

}
,

and PAP0(R × Y) is the collection of all functions F ∈ BC(R × Y, X) such that

lim
T →∞

1

2T

∫ T

−T

‖F(t,u)‖dt = 0

uniformly in u ∈ Y.

Definition 2.7. [18] A function f ∈ BC(R, X) is called pseudo almost periodic if it can be

expressed as f = h + ϕ, where h ∈ AP(X) and ϕ ∈ PAP0(X). The collection of such functions

will be denoted by PAP(X).

Definition 2.8. [18] A function F ∈ C(R × Y, X) is said to be pseudo almost periodic if it can

be expressed as F = G + Φ, where G ∈ AP(R × Y) and φ ∈ PAP0(R × Y). The collection of such

functions will be denoted by PAP(R × Y).

Define AA0(R × Y) as the collection of all functions F ∈ BC(R × Y, X) such that

lim
T →∞

1

2T

∫ T

−T

‖F(t,u)‖dt = 0

uniformly in u ∈ K, where K ⊂ Y is any bounded subset.

Obviously,

PAP0(R × Y) ⊂ AA0(R × Y).

A weaker version of Definition 2.8 is the following:

Definition 2.9. A function F ∈ C(R × Y, X) is said to be B-pseudo almost periodic if it can be

expressed as F = G + Φ, where G ∈ AP(R × Y) and φ ∈ AA0(R × Y). The collection of such

functions will be denoted by BPAP(R × Y).



CUBO
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Pseudo Almost Automorphic Solutions 77

Definition 2.10. [15, 16] A function f ∈ BSp(X) is called Sp-pseudo almost periodic (or Stepanov-

like pseudo almost periodic) if it can be expressed as f = h+ϕ, where hb ∈ AP
(
Lp((0, 1), X)

)
and

ϕb ∈ PAP0
(
Lp((0, 1), X)

)
. The collection of such functions will be denoted by PAP p(X).

In other words, a function f ∈ L
p
loc(R, X) is said to be S

p-pseudo almost periodic if its

Bochner transform fb : R → Lp((0, 1), X) is pseudo almost periodic in the sense that there

exist two functions h,ϕ : R 7→ X such that f = h + ϕ, where hb ∈ AP(Lp((0, 1), X)) and

ϕb ∈ PAP0(L
p((0, 1), X)).

To define the notion of Sp-pseudo almost automorphy for functions of the form F : R×Y 7→ Y,

we need to define the Sp-pseudo almost periodicity for these functions as follows:

Definition 2.11. A function F : R × Y 7→ X, (t,u) 7→ F(t,u) with F(·,u) ∈ L
p
loc(R, X) for each

u ∈ X, is said to be Sp-pseudo almost periodic if there exist two functions H, Φ : R × Y 7→ X such

that F = H + Φ, where Hb ∈ AP(R × Lp((0, 1), X)) and Φb ∈ AA0(R × L
p((0, 1), X)).

The collection of those Sp-pseudo almost periodic functions F : R × Y 7→ X will be denoted

PAP p(R × Y).

2.2 Sp-Almost Automorphy

The notion of Sp-almost automorphy is a new notion due to N’Guérékata and Pankov [40].

Definition 2.12. (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
n→∞

f(t + sn)

is well defined for each t ∈ R, and

lim
n→∞

g(t − sn) = f(t)

for each t ∈ R.

Remark 2.13. The function g in Definition 2.12 is measurable, but not necessarily continuous.

Moreover, if g is continuous, then f is uniformly continuous. If the convergence above is uniform

in t ∈ R, then f is almost periodic. Denote by AA(X) the collection of all almost automorphic

functions R → X. Note that AA(X) equipped with the sup norm, ‖·‖∞, turns out to be a Banach

space.

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

2.12 are uniform on compact intervals, i.e. in the Fréchet 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).



78 Toka Diagana CUBO
13, 1 (2011)

Definition 2.14. [40] The space ASp(X) of Stepanov-like almost automorphic functions (or Sp-

almost automorphic) consists of all f ∈ BSp(X) such that fb ∈ AA
(
Lp(0, 1; X)

)
. That is, a

function f ∈ L
p
loc(R; X) is said to be S

p-almost automorphic if its Bochner transform fb : 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 ∈ L
p
loc(R; X) such that

[∫ t+1

t

‖f(sn + s) − g(s)‖
pds

]1/p
→ 0, and

[∫ t+1

t

‖g(s − sn) − f(s)‖
pds

]1/p
→ 0

as n → ∞ pointwise on R.

Remark 2.15. It is clear that if 1 ≤ p < q < ∞ and f ∈ L
q
loc(R; X) is S

q-almost automorphic,

then f is Sp-almost automorphic. Also if f ∈ AA(X), then f is Sp-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 2.16. A function F : R × Y 7→ X, (t,u) 7→ F(t,u) with F(·,u) ∈ L
p
loc(R; X) for each

u ∈ Y, is said to be Sp-almost automorphic in t ∈ R uniformly in u ∈ Y if t 7→ F(t,u) is Sp-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) ∈ L
p
loc(R; X) such that

[∫ t+1

t

‖F(sn + s,u) − G(s,u)‖
pds

]1/p
→ 0, and

[∫ t+1

t

‖G(s − sn,u) − F(s,u)‖
pds

]1/p
→ 0

as n → ∞ pointwise on R for each u ∈ Y.

The collection of those Sp-almost automorphic functions F : R × Y 7→ X will be denoted by

ASp(R × Y).

2.3 Pseudo Almost Automorphy

The notion of pseudo almost automorphy is a new notion due to Liang, Xiao and Zhang [47, 35].

Definition 2.17. A function f ∈ C(R, X) is called pseudo almost automorphic if it can be

expressed as f = h + ϕ, where h ∈ AA(X) and ϕ ∈ PAP0(X). The collection of such functions

will be denoted by PAA(X).

Obviously, the following inclusions hold:

AP(X) ⊂ PAP(X) ⊂ PAA(X) and AP(X) ⊂ AA(X) ⊂ PAA(X).



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Pseudo Almost Automorphic Solutions 79

Definition 2.18. A function F ∈ C(R × Y, X) is said to be pseudo almost automorphic if it can

be expressed as F = G + Φ, where G ∈ AA(R × Y) and ϕ ∈ AA0(R × Y). The collection of such

functions will be denoted by PAA(R × Y).

A substantial result is the next theorem, which is due to Liang et al. [47].

Theorem 2.19. [47] The space PAA(X) equipped with the sup norm ‖ · ‖∞ is a Banach space.

We also have the following composition result.

Theorem 2.20. [47] If f : R × Y 7→ X belongs to PAA(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 PAA(X) provided ϕ ∈ PAA(Y).

3 Sp-Pseudo Almost Automorphy

This section is devoted to the notion of Sp-pseudo almost automorphy. Such a concept is completely

new and is due to Diagana [17].

Definition 3.1. [17] A function f ∈ BSp(X) is called Sp-pseudo almost automorphic (or Stepanov-

like pseudo almost automorphic) if it can be expressed as

f = h + ϕ,

where hb ∈ AA
(
Lp((0, 1), X)

)
and ϕb ∈ PAP0

(
Lp((0, 1), X)

)
. The collection of such functions will

be denoted by PAAp(X).

Clearly, a function f ∈ L
p
loc(R, X) is said to be S

p-pseudo almost automorphic if its Bochner

transform fb : R → Lp((0, 1), X) is pseudo almost automorphic in the sense that there exist

two functions h,ϕ : R 7→ X such that f = h + ϕ, where hb ∈ AA(Lp((0, 1), X)) and ϕb ∈

PAP0(L
p((0, 1), X)).

Theorem 3.2. [17] If f ∈ PAA(X), then f ∈ PAAp(X) for each 1 ≤ p < ∞. In other words,

PAA(X) ⊂ PAAp(X).

Obviously, the following inclusions hold:

AP(X) ⊂ PAP(X) ⊂ PAA(X) ⊂ PAAp(X),

AP(X) ⊂ AA(X) ⊂ PAA(X) ⊂ PAAp(X).

Theorem 3.3. [17] The space PAAp(X) equipped with the norm ‖ · ‖Sp is a Banach space.



80 Toka Diagana CUBO
13, 1 (2011)

Definition 3.4. A function F : R × Y 7→ X, (t,u) 7→ F(t,u) with F(·,u) ∈ Lp(R, X) for each

u ∈ Y, is said to be Sp-pseudo almost automorphic if there exists two functions H, Φ : R×Y 7→ X

such that

F = H + Φ,

where Hb ∈ AA(R × Lp((0, 1), X)) and Φb ∈ AA0(R × L
p((0, 1), X)). The collection of those

Sp-pseudo almost automorphic functions will be denoted by PAAp(R × Y).

We have the following composition theorems.

Theorem 3.5. Let F : R × X 7→ X be a Sp-pseudo almost automorphic function. Suppose that

F(t,u) is Lipschitzian in u ∈ X uniformly in t ∈ R, that is there exists L > 0 such

‖F(t,u) − F(t,v)‖ ≤ L.‖u − v‖ (3.1)

for all t ∈ R, (u,v) ∈ X × X.

If φ ∈ PAAp(X), then Γ : R → X defined by Γ(·) := F(·,φ(·)) belongs to PAAp(X).

Proof. Let F = H + Φ, where Hb ∈ AA(R × Lp((0, 1), X)) and Φb ∈ AA0(R × L
p((0, 1), X)).

Similarly, let φ = φ1 + φ2, where φ
b
1 ∈ AA(L

p((0, 1), X)) and φb2 ∈ PAP0(L
p((0, 1), X)), that is,

lim
T →∞

1

2T

∫ T

−T

(∫ t+1

t

‖ϕ2(σ)‖
pdσ

)1/p
dt = 0 (3.2)

for all t ∈ R.

It is obvious to see that F b(·,φ(·)) : R 7→ Lp((0, 1), X). Now decompose F b as follows

F b(·,φ(·)) = Hb(·,φ1(·)) + F
b(·,φ(·)) − Hb(·,φ1(·))

= Hb(·,φ1(·)) + F
b(·,φ(·)) − F b(·,φ1(·)) + Φ

b(·,φ1(·)).

Using the theorem of composition of almost automorphic functions, it is easy to see that

Hb(·,φ1(·)) ∈ AA(L
p((0, 1), X)). Now, set

Gb(·) := F b(·,φ(·)) − F b(·,φ1(·)).

Clearly, Gb(·) ∈ PAP0(L
p((0, 1), X)). Indeed, we have

∫ t+1

t

‖G(σ)‖pdσ =

∫ t+1

t

‖F(σ,φ(σ)) − F(σ,φ1(σ))‖
pdσ

≤ Lp
∫ t+1

t

‖φ(σ) − φ1(σ)‖
pdσ

= Lp
∫ t+1

t

‖φ2(σ)‖
pdσ,



CUBO
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Pseudo Almost Automorphic Solutions 81

and hence for T > 0,

1

2T

∫ T

−T

(∫ t+1

t

‖G(σ)‖pdσ

)1/p
dt ≤

L

2T

∫ T

−T

(∫ t+1

t

‖φ2(σ)‖
pdσ

)1/p
dt.

Now using Eq. (3.2), it follows that

lim
T →∞

1

2T

∫ T

−T

(∫ t+1

t

‖G(σ)‖pdσ

)1/p
dt = 0.

Using the theorem of composition of functions of PAP(Lp((0, 1), X)) (see [18]) it is easy to

see that Φb(·,φ1(·)) ∈ PAP0(L
p((0, 1), X)).

Theorem 3.6. Let F = H + Φ : R × X 7→ X be a Sp-pseudo almost automorphic function, where

Hb ∈ AA(R × Lp((0, 1), X)) and Φb ∈ AA0(R × L
p((0, 1), X)). Suppose that F(t,u) and Φ(t,x)

are uniformly continuous in every bounded subset K ⊂ X uniformly for t ∈ R. If g ∈ PAAp(X),

then Γ : R → X defined by Γ(·) := F(·,g(·)) belongs to PAAp(X).

Proof. Let F = H + Φ, where Hb ∈ AA(R × Lp((0, 1), X)) and Φb ∈ AA0(R × L
p((0, 1), X)).

Similarly, let g = φ1 + φ2, where φ
b
1 ∈ AA(L

p((0, 1), X)) and φb2 ∈ PAP0(L
p((0, 1), X)).

It is obvious to see that F b(·,g(·)) : R 7→ Lp((0, 1), X). Now decompose F b as follows

F b(·,g(·)) = Hb(·,φ1(·)) + F
b(·,g(·)) − Hb(·,φ1(·))

= Hb(·,φ1(·)) + F
b(·,g(·)) − F b(·,φ1(·)) + Φ

b(·,φ1(·)).

Using the theorem of composition of almost automorphic functions, it is easy to see that

Hb(·,φ1(·)) ∈ AA(L
p((0, 1), X)). Now, set

Gb(·) := F b(·,g(·)) − F b(·,φ1(·)).

We claim that Gb(·) ∈ PAP0(L
p((0, 1), X)). First of all, note that the uniformly continuity of

F on bounded subsets K ⊂ X yields the uniform continuity of its Bohr transform F b on bounded

subsets of X. Since both g,φ1 are bounded functions it follows that there exists K ⊂ X a bounded

subset such that g(σ),φ1(σ) ∈ K for each σ ∈ R. Now from the uniform continuity of F
b on

bounded subsets of X, it obviously follows that F b is uniformly continuous on K uniformly for

each t ∈ R. Therefore for every ε > 0 there exists δ > 0 such that for all X,Y ∈ K with

‖X − Y ‖ < δ yield

‖F b(σ,X) − F b(σ,Y )‖ < ε for all σ ∈ R.

Using the proof of the composition theorem [47, Theorem 2.4] (applied to F b) it follows

lim
T →∞

1

2T

∫ T

−T

(∫ t+1

t

‖G(σ)‖pdσ

)1/p
dt = 0.

Using the theorem of composition [47, Theorem 2.4] for functions of PAP0(L
p((0, 1), X)) it is easy

to see that Φb(·,φ1(·)) ∈ PAP0(L
p((0, 1), X)).



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4 Evolution Families

This section is devoted to some preliminary results needed in the sequel. We basically use the

same setting as in [7] with slight adjustments. Throughout the rest of this paper, (X,‖ · ‖) stands

for a Banach space, A(t) for t ∈ R is a family of closed linear operators on D = D(A(t)), which is

independent of t, satisfying the so-called Acquistapace and Terreni conditions (Hypothesis (H.1)).

Moreover, the operators A(t) are not necessarily densely defined. The linear operators B,C are

(possibly unbounded) defined on X such that A(t) + B + C is not trivial for each t ∈ R. The

functions, f : R × X 7→ Xtβ (0 < α < β < 1), g : R × X 7→ X are respectively jointly continuous

satisfying some additional assumptions.

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.

Moreover, one sets R(λ,L) := (λI − L)−1 for all λ ∈ ρ(A). Furthermore, we set Q = I − P for a

projection P .

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 ∈
(

π
2
,π
)
, L > 0 and µ,ν ∈ (0, 1] with µ + ν > 1 such that

Σθ ∪
{

0
}
⊂ ρ
(
A(t) − ω

)
∋ λ,

∥∥∥R
(
λ,A(t) − ω

)∥∥∥ ≤
K

1 + |λ|
for all t ∈ R, (4.1)

and ∥∥∥
(
A(t) − ω

)
R
(
λ,A(t) − ω

)[
R
(
ω,A(t)

)
− R

(
ω,A(s)

)]∥∥∥ ≤ L |t − s|µ |λ|−ν (4.2)

for t,s ∈ R, λ ∈ Σθ :=
{
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 [4, 42] that Eq. (4.2) can be replaced with the following: There exist constants L and

0 < µ ≤ 1 such that

∥∥∥
(
A(t) − A(s)

)
R
(
ω,A(r)

)∥∥∥ ≤ L|t − s|µ, s,t,r ∈ R.



CUBO
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Pseudo Almost Automorphic Solutions 83

It should mentioned that (H.1) was introduced in the literature by Acquistapace and Terreni

in [2, 3] for ω = 0. Among other things, the Acquistapace-Terreni Conditions ensure that there

exists a unique evolution family

U = {U(t,s) : t,s ∈ R such that t ≥ s}

on X associated with A(t) such that U(t,s)X ⊆ D(A(t)) for all t,s ∈ R with t ≥ s, and

(a) U(t,s)U(s,r) = U(t,r) for t,s ∈ R such that t ≥ s ≥ r;

(b) U(t,t) = I for t ∈ R where I is the identity operator of X;

(c) (t,s) 7→ U(t,s) ∈ B(X) is continuous for t > s;

It should also be mentioned that the above-mentioned proprieties were mainly established in

[1, Theorem 2.3] and [50, Theorem 2.1], see also [3, 49].

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 set ŨQ(s,t) :=

UQ(t,s)
−1); and

(h)
∥∥∥U(t,s)P(s)

∥∥∥ ≤ Ne−δ(t−s) and
∥∥∥ŨQ(s,t)Q(t)

∥∥∥ ≤ Ne−δ(t−s) for t ≥ s and t,s ∈ R.

According to [44], 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
t,s∈R

∥∥∥A(t)A(s)−1
∥∥∥

is finite, and ∥∥∥A(t)A(s)−1 − I
∥∥∥ ≤ L0|t − s|µ

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)



84 Toka Diagana CUBO
13, 1 (2011)

and ∥∥∥A(t)eτ AQ(t)Qt
∥∥∥ ≤ ψ(−τ)

for τ > 0 and a function ψ such that R ∋ s 7→ ϕ(s) := |s|µψ(s) is integrable with

L0‖ϕ‖L1(R) < 1.

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 [4], [24], and [33] for proofs

and further information on theses interpolation spaces.

Let A be a sectorial operator on X (in assumption (H.1), replace A(t) with A) and let α ∈ (0, 1).

Define the real interpolation space

X
A
α :=

{
x ∈ X :

∥∥∥x
∥∥∥

A

α
:= supr>0

∥∥∥rα
(
A − ω

)
R
(
r,A − ω

)
x
∥∥∥ < ∞

}
,

which, by the way, is a Banach space when endowed with the norm
∥∥∥ ·
∥∥∥

A

α
. For convenience we

further write

X
A
0 := X,

∥∥∥x
∥∥∥

A

0
:=
∥∥∥x
∥∥∥, XA1 := D(A)

and ∥∥∥x
∥∥∥

A

1
:=
∥∥∥(ω − A)x

∥∥∥.

Moreover, let X̂A := D(A) of X. In particular, we have the following continuous embedding

D(A) →֒ XAβ →֒ D((ω − A)
α) →֒ XAα →֒ X̂

A →֒ X, (4.3)

for all 0 < α < β < 1, where the fractional powers are defined in the usual way.

In general, D(A) is not dense in the spaces XAα and X. However, we have the following

continuous injection

X
A
β →֒ D(A)

‖·‖Aα
(4.4)

for 0 < α < β < 1.

Given the family of linear operators A(t) for t ∈ R, satisfying (H.1), we set

X
t
α := X

A(t)
α , X̂

t := X̂A(t)

for 0 ≤ α ≤ 1 and t ∈ R, with the corresponding norms. Then the embedding in Eq. (4.3) holds

with constants independent of t ∈ R. These interpolation spaces are of class Jα ([33, Definition

1.1.1 ]) and hence there is a constant c(α) such that

∥∥∥y
∥∥∥

t

α
≤ c(α)

∥∥∥y
∥∥∥

1−α∥∥∥A(t)y
∥∥∥

α

, y ∈ D(A(t)). (4.5)

We have the following fundamental estimates for the evolution family U. Its proof was given

in [7] though for the sake of clarity, we reproduce it here.



CUBO
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Pseudo Almost Automorphic Solutions 85

Proposition 4.1. For x ∈ X, 0 ≤ α ≤ 1 and t > s, the following hold:

(i) There is a constant c(α), such that

∥∥∥U(t,s)P(s)x
∥∥∥

t

α
≤ c(α)e−

δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥. (4.6)

(ii) There is a constant m(α), such that

∥∥∥ŨQ(s,t)Q(t)x
∥∥∥

s

α
≤ m(α)e−δ(t−s)

∥∥∥x
∥∥∥. (4.7)

Proof. (i) Using (4.5) we obtain

∥∥∥U(t,s)P(s)x
∥∥∥

t

α
≤ c(α)

∥∥∥U(t,s)P(s)x
∥∥∥

1−α∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α

≤ c(α)
∥∥∥U(t,s)P(s)x

∥∥∥
1−α∥∥∥A(t)U(t,t − 1)U(t − 1,s)P(s)x

∥∥∥
α

≤ c(α)
∥∥∥U(t,s)P(s)x

∥∥∥
1−α∥∥∥A(t)U(t,t − 1)

∥∥∥
α∥∥∥U(t − 1,s)P(s)x

∥∥∥
α

≤ c(α)N′ e−δ(t−s)(1−α)e−δ(t−s−1)α
∥∥∥x
∥∥∥

≤ c(α)(t − s)−αe−
δ
2
(t−s)(t − s)αe−

δ
2
(t−s)

∥∥∥x
∥∥∥

for t − s ≥ 1 and x ∈ X.

Since (t − s)αe−
δ
2
(t−s) → 0 as t → ∞ it easily follows that

∥∥∥U(t,s)P(s)x
∥∥∥

t

α
≤ c(α)(t − s)−αe−

δ
2
(t−s)

∥∥∥x
∥∥∥.

If 0 < t − s ≤ 1, we have

∥∥∥U(t,s)P(s)x
∥∥∥

t

α
≤ c(α)

∥∥∥U(t,s)P(s)x
∥∥∥

1−α∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α

≤ c(α)
∥∥∥U(t,s)P(s)x

∥∥∥
1−α∥∥∥A(t)U(t,

t + s

2
)U(

t + s

2
,s)P(s)x

∥∥∥
α

≤ c(α)
∥∥∥U(t,s)P(s)x

∥∥∥
1−α∥∥∥A(t)U(t,

t + s

2
)
∥∥∥

α∥∥∥U(
t + s

2
,s)P(s)x

∥∥∥
α

≤ c(α)Ne−δ(t−s)(1−α)2α(t − s)−αe−
δα
2

(t−s)
∥∥∥x
∥∥∥

≤ c(α)Ne−
δ
2
(t−s)(1−α)2α(t − s)−αe−

δα
2

(t−s)
∥∥∥x
∥∥∥

≤ c(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥,

and hence

∥∥∥U(t,s)P(s)x
∥∥∥

t

α
≤ c(α)(t − s)−αe−

δ
2
(t−s)

∥∥∥x
∥∥∥ for t > s.



86 Toka Diagana CUBO
13, 1 (2011)

(ii)

∥∥∥ŨQ(s,t)Q(t)x
∥∥∥

s

α
≤ c(α)

∥∥∥ŨQ(s,t)Q(t)x
∥∥∥

1−α∥∥∥A(s)ŨQ(s,t)Q(t)x
∥∥∥

α

≤ c(α)
∥∥∥ŨQ(s,t)Q(t)x

∥∥∥
1−α∥∥∥A(s)Q(s)ŨQ(s,t)Q(t)x

∥∥∥
α

≤ c(α)
∥∥∥ŨQ(s,t)Q(t)x

∥∥∥
1−α∥∥∥A(s)Q(s)

∥∥∥
α∥∥∥ŨQ(s,t)Q(t)x

∥∥∥
α

≤ c(α)Ne−δ(t−s)(1−α)
∥∥∥A(s)Q(s)

∥∥∥
α

e−δ(t−s)α
∥∥∥x
∥∥∥

≤ m(α)e−δ(t−s)
∥∥∥x
∥∥∥.

In the last inequality we have used that
∥∥∥A(s)Q(s)

∥∥∥ ≤ c for some constant c ≥ 0, see e.g. [46,
Proposition 3.18].

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. Moreover, 0 ∈ ρ(A(t)) for each

t ∈ R and the following holds

sup
t,s∈R

∥∥∥A(s)A−1(t)
∥∥∥

B(Xα,X)
< c0. (4.8)

Remark 4.2. Note that Eq. (4.8) is satisfied in many cases in the literature. In particular, it holds

when A(t) = d(t)A where A : D(A) ⊂ X 7→ X is any closed linear operator such that 0 ∈ ρ(A)

and d : R 7→ R with inf
t∈R

|d(t)| > 0 and sup
t∈R

|d(t)| < ∞.

Hypothesis (H.3). There exists 0 ≤ α < β < 1 such that

X
t
α = Xα and X

t
β = Xβ

for all t ∈ R, with uniform equivalent norms.

If 0 ≤ α < β < 1, then we let k(α) and c′ denote respectively the bounds of the embedding

Xβ →֒ Xα and Xα →֒ X, that is, ∥∥∥u
∥∥∥

α
≤ k(α)

∥∥∥u
∥∥∥

β

for each u ∈ Xβ and

∥∥∥u
∥∥∥ ≤ c′

∥∥∥u
∥∥∥

α

for each u ∈ Xα



CUBO
13, 1 (2011)

Pseudo Almost Automorphic Solutions 87

5 Main Results

To study the existence and uniqueness of pseudo almost automorphic solutions to Eq. (1.1) we

first introduce the notion of bounded solution.

Definition 5.1. A function u : R 7→ Xα is said to be a bounded solution to Eq. (1.1) provided

that the function s → A(s)U(t,s)P(s)f(s,Bu(s)) is integrable on (−∞, t), and the function s →

A(s)U(t,s)Q(s)f(s,Bu(s)) is integrable on (t,∞) for each t ∈ R, and

u(t) = −f(t,Bu(t)) −

∫ t

−∞

A(s)U(t,s)P(s)f(s,Bu(s))ds

+

∫ ∞

t

A(s)U(t,s)Q(s)f(s,Bu(s))ds +

∫ t

−∞

U(t,s)P(s)g(s,Cu(s))ds

−

∫ ∞

t

U(t,s)Q(s)g(s,Cu(s))ds

for each ∀t ∈ R.

Throughout the rest of the paper we denote by Γ1, Γ2, Γ3, and Γ4, the nonlinear integral

operators defined by

(Γ1u)(t) :=

∫ t

−∞

A(s)U(t,s)P(s)f(s,Bu(s))ds,

(Γ2u)(t) :=

∫ ∞

t

A(s)U(t,s)Q(s)f(s,Bu(s))ds,

(Γ3u)(t) :=

∫ t

−∞

U(t,s)P(s)g(s,Cu(s))ds,

(Γ4u)(t) :=

∫ ∞

t

U(t,s)Q(s)g(s,Cu(s))ds.

In this paper we suppose that the linear operators B,C : Xα 7→ X are bounded and set

̟ := max

(∥∥∥B
∥∥∥

B(Xα,X)
,
∥∥∥C
∥∥∥

B(Xα,X)

)
.

To study Eq. (1.1), in addition to the previous assumptions, we require that p > 1,
1

p
+

1

q
= 1,

and that the following additional assumptions hold:

(H.4) R(ω,A(·))u ∈ AA(Xα) for each u ∈ X. Moreover, for any sequence of real numbers (τ
′
n)n∈N

there exist a subsequence (τn)n∈N and G(·, ·) such that

G(t,s)P(s)u = lim
n→∞

A(s + τn)U(t + τn,s + τn)P(s + τn)u

and

A(s)U(t,s)P(s)u = lim
n→∞

G(t − τn,s − τn)P(s − τn)u

for all t,s ∈ R and u ∈ Xα.



88 Toka Diagana CUBO
13, 1 (2011)

(H.5) Let 0 ≤ α < β < 1, and f : R×X 7→ Xβ , g : R×X 7→ X are S
p-pseudo almost automorphic.

Moreover, the functions f,g are uniformly Lipschitz with respect to the second argument in

the following sense: there exists K > 0 such that

∥∥∥f(t,u) − f(t,v)
∥∥∥

β
≤ K

∥∥∥u − v
∥∥∥,

and ∥∥∥g(t,u) − g(t,v)
∥∥∥ ≤ K

∥∥∥u − v
∥∥∥

for all u,v ∈ X and t ∈ R.

The proof of our main result requires the following key technical Lemma.

Lemma 5.2. Under assumptions (H.1)—(H.3), then there exist constant m(α,β),n(α) > 0 such

that

∥∥∥A(s)ŨQ(t,s)Q(s)x
∥∥∥

α
≤ m(α,β)eδ(s−t)

∥∥∥x
∥∥∥

β
for t ≤ s, (5.1)

∥∥∥A(s)U(t,s)P(s)x
∥∥∥

α
≤ n(α)(t − s)−αe−

δ
2
(t−s)

∥∥∥x
∥∥∥

β
, for t > s. (5.2)

Proof. Let x ∈ Xβ . Since the restriction of A(s) to R(Q(s)) is a bounded linear operator it follows

that

∥∥∥A(s)ŨQ(t,s)Q(s)x
∥∥∥

α
≤ ck(α)

∥∥∥ŨQ(t,s)Q(s)x
∥∥∥

β

≤ ck(α)m(β)eδ(s−t)
∥∥∥x
∥∥∥

≤ m(α,β)eδ(s−t)
∥∥∥x
∥∥∥

β

for t ≤ s by using Eq. (4.7).

Similarly, for each x ∈ Xβ , using Eq. (4.8), we obtain

∥∥∥A(s)U(t,s)P(s)x
∥∥∥

α
=

∥∥∥A(s)A(t)−1A(t)U(t,s)P(s)x
∥∥∥

α

≤
∥∥∥A(s)A(t)−1

∥∥∥
B(Xα,X)

∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α

≤ c0

∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α

for t ≥ s.

First of all, note that
∥∥∥A(t)U(t,s)

∥∥∥ ≤ K(t − s)−1 for all t,s such that 0 < t − s ≤ 1.

Now, let t − s ≥ 1. Then, using Eq. (4.6), we obtain



CUBO
13, 1 (2011)

Pseudo Almost Automorphic Solutions 89

∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α
=

∥∥∥A(t)U(t,t − 1)U(t − 1,s)P(s)x
∥∥∥

α

≤
∥∥∥A(t)U(t,t − 1)

∥∥∥
B(Xα,X)

∥∥∥U(t − 1,s)P(s)x
∥∥∥

α

≤ Kc(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

≤ KK′c(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

α

≤ KK′k(α)c(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

β

≤ n′(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

β
.

Now, let 0 < t − s ≤ 1. Again, using Eq. (4.6), we obtain

∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α
=

∥∥∥A(t)U(t,
t + s

2
)U(

t + s

2
,s)P(s)x

∥∥∥
α

≤
∥∥∥A(t)U(t,

t + s

2
)
∥∥∥

B(Xα,X)

∥∥∥U(
t + s

2
,s)P(s)x

∥∥∥
α

≤ Kc(α)e−
δ
4
(t−s)2α(t − s)−α‖x‖

≤ KK′c(α)e−
δ
4
(t−s)2α(t − s)−α

∥∥∥x
∥∥∥

α

≤ KK′k(α)c(α)e−
δ
4
(t−s)2α(t − s)−α

∥∥∥x
∥∥∥

β

≤ n′′(α)e−
δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

β
.

Therefore, ∥∥∥A(t)U(t,s)P(s)x
∥∥∥

α
≤ n(α)e−

δ
2
(t−s)(t − s)−α

∥∥∥x
∥∥∥

β

for all t,s ∈ R with t ≥ s.

Lemma 5.3. Under assumptions (H.1)—(H.5) and if

N(α,q,δ) :=

∞∑

n=1

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q
< ∞,

then the integral operators Γ3 and Γ4 defined above map PAA(Xα) into itself.

Proof. Let u ∈ PAAp(Xα). Since C ∈ B(Xα, X) then Cu(·) ∈ PAA
p(X). Setting h(t) =

g(t,C(u(t)) and using the theorem of composition of Sp-pseudo almost automorphic functions (The-

orem 3.5) it follows that h ∈ PAAp(X). Now write h = φ+ζ where φb belongs to AA(Lp(0, 1), Xα))

and ζb ∈ PAP0(L
p(0, 1), Xα)). Define for all n = 1, 2, ..., the sequence of integral operators



90 Toka Diagana CUBO
13, 1 (2011)

Φn(t) =

∫ n

n−1

U(t,t − s)P(t − s)φ(t − s)ds, and Ψn(t) =

∫ n

n−1

U(t,t − s)P(t − s)ζ(t − s)ds

for each t ∈ R.

Letting r = t − s it follows that

Φn(t) =

∫ t−n+1

t−n

U(t,r)P(r)φ(r)dr,

and hence from the Hölder’s inequality and the estimate Eq. (4.6) it follows that

∥∥∥Φn(t)
∥∥∥

α
≤

∫ t−n+1

t−n

c(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥φ(r)
∥∥∥dr

≤ c′
∫ t−n+1

t−n

c(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥φ(r)
∥∥∥

α
dr

≤ c′c(α)

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q∥∥∥φ
∥∥∥
Sp
.

From the assumption that N(α,q,δ) is finite we then deduce from Weirstrass Theorem that the

series D(t) :=

∞∑

n=1

Φn(t) is uniformly convergent on R. Moreover, D ∈ C(R, Xα) and

∥∥∥D(t)
∥∥∥

α
≤

∞∑

n=1

∥∥∥Φn(t)
∥∥∥

α
≤ c′c(α)N(α,q,δ)

∥∥∥φ
∥∥∥
Sp

for all t ∈ R.

Let us show that Φn ∈ AA(Xα) for each n = 1, 2, 3, ... Indeed, since φ ∈ AS
p(Xα), for every

sequence of real numbers (τ′n)n∈N there exist a subsequence (τnk )k∈N and a function φ̂ such that

∫ t+1

t

∥∥∥φ̂(s) − φ(s + τnk )
∥∥∥

p

α
ds → 0

and ∫ t+1

t

∥∥∥φ̂(s − τnk ) − φ(s)
∥∥∥

p

α
ds → 0

as k → ∞ pointwise in R.

Define for all n = 1, 2, 3, ..., the sequence of integral operators

Φ̂n(t) =

∫ n

n−1

U(t,t − s)P(t − s)φ̂(t − s)ds

for all t ∈ R.

Now



CUBO
13, 1 (2011)

Pseudo Almost Automorphic Solutions 91

Φ(t + τnk ) − Φ̂(t) =

∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)φ(t + τnk − s)ds

−

∫ n

n−1

U(t,t − s)P(t − s)φ̂(t − s)ds

=

∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)φ(t + τnk − s)ds

+

∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)φ̂(t − s)ds

−

∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)φ̂(t − s)ds

−

∫ n

n−1

U(t,t − s)P(t − s)φ̂(t − s)ds

=

∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)
[
φ(t + τnk − s) − φ̂(t − s)

]
ds

+

∫ n

n−1

[
U(t,t + τnk − s)P(t + τnk − s) − U(t,t − s)P(t − s)

]
φ̂(t − s)ds.

Using Lebesgue Dominated Convergence Theorem, one can easily see that

∥∥∥
∫ n

n−1

U(t,t + τnk − s)P(t + τnk − s)
[
φ(t + τnk − s) − φ̂(t − s)

]
ds
∥∥∥

α
→ 0 as k → ∞, t ∈ R.

Similarly, using [9, Proposition 3.3] it follows that

∥∥∥
∫ n

n−1

[
U(t,t + τnk − s)P(t + τnk − s) − U(t,t − s)P(t − s)

]
φ̂(t − s)ds

∥∥∥
α
→ 0 as k → ∞, t ∈ R.

Thus

Φ̂n(t) = lim
k→∞

Φn(t + τnk ), t ∈ R.

Similarly, one can easily see that

Φn(t) = lim
k→∞

Φ̂n(t − τnk )

for all t ∈ R and n = 1, 2, 3, ... Therefore the sequence Φn ∈ AA(Xα) for each n = 1, 2, ... and

hence its uniform limit D ∈ AA(Xα).

To complete the proof for Γ3, we have to show that Ψn ∈ PAP0(Xα).

Letting r = t − s it follows that

Φn(t) =

∫ t−n+1

t−n

U(t,r)P(r)ζ(r)dr,



92 Toka Diagana CUBO
13, 1 (2011)

and hence from the Hölder’s inequality and the estimate Eq. (4.6) it follows that

∥∥∥Φn(t)
∥∥∥

α
≤

∫ t−n+1

t−n

c(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥ζ(r)
∥∥∥dr

≤ c′
∫ t−n+1

t−n

c(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥ζ(r)
∥∥∥

α
dr

≤ c′c(α)

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q[∫ t−n+1

t−n

∥∥∥ζ(s)
∥∥∥

p

α
ds

]1/p
,

and hence Ψn ∈ PAP0(Xα) as ζ
b ∈ PAP0(L

p((0, 1), Xα). From the assumption that N(α,q,δ)

is finite we then deduce from Weirstrass Theorem that the series E(t) :=

∞∑

n=1

Ψn(t) is uniformly

convergent on R. Moreover, E ∈ C(R, Xα) and

∥∥∥E(t)
∥∥∥

α
≤

∞∑

n=1

∥∥∥Φn(t)
∥∥∥

α
≤ c′c(α)N(α,q,δ)

∥∥∥φ
∥∥∥
Sp

for all t ∈ R.

Consequently, the uniform limit E(t) =

∞∑

n=1

Ψn(t) belongs to PAP0(Xα). Therefore Γ3 maps

PAA(Xα) into itself.

The proof for Γ4 is similar to that of Γ3 except that we make use of approximate Eq. (4.7)

rather than Eq. (4.6).

Lemma 5.4. Under assumptions (H.1)—(H.5) and if

N(α,q,δ) :=

∞∑

n=1

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q
< ∞,

then the integral operators Γ1 and Γ2 defined above map PAA(Xα) into itself.

Proof. Let u ∈ PAAp(Xα). Since B ∈ B(Xα, X) then Bu(·) ∈ PAA
p(X). Setting h(t) =

f(t,Bu(t)) and using the theorem of composition of Sp-pseudo almost automorphic functions (The-

orem 3.5) it follows that h ∈ PAAp(X). Now write h = φ+ζ where φb belongs to AA(Lp(0, 1), Xα))

and ζb ∈ PAP0(L
p(0, 1), Xα)). Define for all n = 1, 2, ..., the sequence of integral operators

Φn(t) =

∫ n

n−1

A(t − s)U(t,t − s)P(t − s)φ(t − s)ds

and

Ψn(t) =

∫ n

n−1

A(t − s)U(t,t − s)P(t − s)ζ(t − s)ds



CUBO
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Pseudo Almost Automorphic Solutions 93

for each t ∈ R.

Letting r = t − s it follows that

Φn(t) =

∫ t−n+1

t−n

A(r)U(t,r)P(r)φ(r)dr,

and hence from the Hölder’s inequality and the estimate Eq. (5.2) it follows that

∥∥∥Φn(t)
∥∥∥

α
≤

∫ t−n+1

t−n

n(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥φ(r)
∥∥∥

β
dr

≤ c′
∫ t−n+1

t−n

n(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥φ(r)
∥∥∥

β
dr

≤ c′n(α)

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q∥∥∥φ
∥∥∥
Sp
.

From the assumption that N(α,q,δ) is finite we then deduce from Weirstrass Theorem that the

series D(t) :=

∞∑

n=1

Φn(t) is uniformly convergent on R. Moreover, D ∈ C(R, Xα) and

∥∥∥D(t)
∥∥∥

α
≤

∞∑

n=1

∥∥∥Φn(t)
∥∥∥

α
≤ c′n(α)N(α,q,δ)

∥∥∥φ
∥∥∥
Sp

for all t ∈ R.

Let us show that Φ ∈ AA(Xα). Indeed, since φ ∈ AS
p(Xα), for every sequence of real numbers

(τ′n)n∈N there exist a subsequence (τnk )k∈N and a function φ̂ such that
∫ t+1

t

∥∥∥φ̂(s) − φ(s + τnk )
∥∥∥

p

α
ds → 0

and ∫ t+1

t

∥∥∥φ̂(s − τnk ) − φ(s)
∥∥∥

p

α
ds → 0

as k → ∞ pointwise in R.

Define for all n = 1, 2, 3, ..., the sequence of integral operators

Φ̂n(t) =

∫ n

n−1

A(t − s)U(t,t − s)P(t − s)φ̂(t − s)ds

for all t ∈ R.

Now

Φ(t + τnk ) − Φ̂(t) =

∫ n

n−1

A(t + τnk − s)U(t,t + τnk − s)P(t + τnk − s)φ(t + τnk − s)ds

−

∫ n

n−1

A(t − s)U(t,t − s)P(t − s)φ̂(t − s)ds

= I1k,n(t) + I
2
k,n(t),



94 Toka Diagana CUBO
13, 1 (2011)

where

I1k,n(t) =

∫ n

n−1

A(t + τnk − s)U(t,t + τnk − s)P(t + τnk − s)
[
φ(t + τnk − s) − φ̂(t − s)

]
ds

and

I2k,n(t) =

∫ n

n−1

[
A(t+τnk −s)U(t,t+τnk −s)P(t+τnk −s) −A(t−s)U(t,t−s)P(t−s)

]
φ̂(t−s)ds.

Using Lebesgue Dominated Convergence Theorem, one can easily see that

∥∥∥
∫ n

n−1

A(t + τnk − s)U(t,t + τnk − s)P(t + τnk − s)
[
φ(t + τnk − s) − φ̂(t − s)

]
ds
∥∥∥

α
→ 0

as k → ∞ for each t ∈ R.

Similarly, using (H.4) it follows that

∥∥∥
∫ n

n−1

[
A(t + τnk −s)U(t,t + τnk −s)P(t + τnk −s) −A(t−s)U(t,t−s)P(t−s)

]
φ̂(t−s)ds

∥∥∥
α
→ 0

as k → ∞ for each t ∈ R.

Thus

Φ̂n(t) = lim
k→∞

Φn(t + τnk ), t ∈ R.

Similarly, one can easily see that

Φn(t) = lim
k→∞

Φ̂n(t − τnk )

for all t ∈ R and n = 1, 2, 3, ... Therefore the sequence Φn ∈ AA(Xα) for each n = 1, 2, ... and

hence its uniform limit E ∈ AA(Xα).

To complete the proof for Γ1, we have to show that Ψn ∈ PAP0(Xα).

Letting r = t − s it follows that

Φn(t) =

∫ t−n+1

t−n

A(r)U(t,r)P(r)ζ(r)dr,

and hence from the Hölder’s inequality and the estimate Eq. (4.6) it follows that

∥∥∥Φn(t)
∥∥∥

α
≤

∫ t−n+1

t−n

n(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥ζ(r)
∥∥∥dr

≤ c′
∫ t−n+1

t−n

n(α)e−
δ
2
(t−r)(t − r)−α

∥∥∥ζ(r)
∥∥∥

α
dr

≤ c′n(α)

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q[∫ t−n+1

t−n

∥∥∥ζ(s)
∥∥∥

p

α
ds

]1/p
,



CUBO
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Pseudo Almost Automorphic Solutions 95

and hence Ψn ∈ PAP0(Xα) as ζ
b ∈ PAP0(L

p((0, 1), Xα). From the assumption that N(α,q,δ)

is finite we then deduce from Weirstrass Theorem that the series E(t) :=

∞∑

n=1

Ψn(t) is uniformly

convergent on R. Moreover, E ∈ C(R, Xα) and

∥∥∥E(t)
∥∥∥

α
≤

∞∑

n=1

∥∥∥Φn(t)
∥∥∥

α
≤ c′n(α)N(α,q,δ)

∥∥∥φ
∥∥∥
Sp

for all t ∈ R. Consequently, the uniform limit E(t) =

∞∑

n=1

Ψn(t) belongs to PAP0(Xα). Therefore

Γ1 maps PAA(Xα) into itself.

The proof for Γ2u(·) is similar to that of Γ1u(·) except that one makes use of Eq. (5.1) rather

than Eq. (5.2).

Theorem 5.5. Suppose assumptions (H.1)—(H.5) hold. Moreover, suppose

N(α,q,δ) :=

∞∑

n=1

[∫ n

n−1

e−q
δ
2
ss−qαds

]1/q
< ∞.

Then the evolution equation (1.1) has a unique pseudo-almost automorphic mild solution whenever

K is small enough.

Proof. Consider the nonlinear operator M defined on PAA(Xα) by

Mu(t) = −f(t,Bu(t)) −

∫ t

−∞

A(s)U(t,s)P(s)f(s,Bu(s))ds

+

∫ ∞

t

A(s)U(t,s)Q(s)f(s,Bu(s))ds +

∫ t

−∞

U(t,s)P(s)g(s,Cu(s))ds

−

∫ ∞

t

U(t,s)Q(s)g(s,Cu(s))ds

for each t ∈ R.

As we have previously seen, for every u ∈ PAA(Xα), f(·,Bu(·)) ∈ PAA(Xβ ) ⊂ PAA(Xα).

In view of Lemma 5.3 and Lemma 5.4, it follows that M maps PAA(Xα) into itself. To complete

the proof one has to show that M has a unique fixed-point.

Let v,w ∈ PAA(Xα). It is not difficult to see that
∥∥∥Mv − Mw

∥∥∥
∞,α

≤ KΘ .
∥∥∥v − w

∥∥∥
∞,α

,

where Θ := ̟

[
k(α) + δ−1

(
m(α,β) + m(α)

)
+
(
n(α) + c(α)

)
21−α Γ(1 − α)δα−1

]
, and hence if K

is small enough, then Eq. (1.1) has a unique solution, which obviously is its only pseudo-almost

automorphic solution.



96 Toka Diagana CUBO
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Example 5.6. Let Ω ⊂ RN (N ≥ 1) be an open bounded subset with regular boundary Γ = ∂Ω

and let X = L2(Ω) equipped with its natural topology ‖ · ‖L2(Ω).

Define the linear operator appearing in Eq. (1.2) as follows:

A(t)u = a(t,x)∆u for all u ∈ D(A(t)) = H10(Ω) ∩ H
2(Ω),

where a : R × Ω 7→ R is a jointly continuous function, almost automorphic and satisfies the

following assumptions:

(H.6) inf
t∈R,x∈Ω

a(t,x) = m0 > 0, and

(H.7) there exists L > 0 and 0 < µ ≤ 1 such that

|a(t,x) − a(s,x)| ≤ L|s − t|µ

for all t,s ∈ R uniformly in x ∈ Ω.

First of all, note that in view of the above, sup
t∈R,x∈Ω

a(t,x) < ∞. Also, a classical example of a

function a satisfying the above-mentioned assumptions is for instance

aγ (t,x) = 3 + sin |x|t + sin γ|x|t,

where |x| = (x21 + ... + x
2
N )

1/2 for each x = (x1,x2, ...,xN ) ∈ Ω and γ ∈ R \ Q.

Under previous assumptions, it is clear that the operators A(t) defined above are invertible

and satisfy Acquistapace-Terreni conditions. Moreover, it can be easily shown that

R
(
ω,a(·,x)∆

)
ϕ =

1

a(·,x)
R
(

ω

a(·,x)
, ∆
)
ϕ ∈ AA(X1/2)

for each ϕ ∈ L2(Ω) with ∥∥∥R
(
ω,a∆

)∥∥∥
B(L2(Ω))

≤
const.

|ω|
.

Furthermore, assumptions (H.1)—(H.4) are fulfilled.

We require the following assumption:

(H.8) Let 1
2
< β < 1, and F,G : R × X1/2 7→ Xβ be S

p-pseudo-almost automorphic functions in

t ∈ R uniformly in u ∈ X1/2. Moreover, the functions F,G are uniformly Lipschitz with

respect to the second argument in the following sense: there exists K′ > 0 such that
∥∥∥F(t,ϕ) − F(t,ψ)

∥∥∥
β
≤ K′

∥∥∥ϕ − ψ
∥∥∥

L2(Ω)
,

and ∥∥∥G(t,ϕ) − G(t,ψ)
∥∥∥

L2(Ω)
≤ K′

∥∥∥ϕ − ψ
∥∥∥

L2(Ω)

for all ϕ,ψ ∈ L2(Ω) and t ∈ R.



CUBO
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Pseudo Almost Automorphic Solutions 97

We have

Theorem 5.7. Under previous assumptions including (H.6)-(H.8), then Eq. (1.2) has a unique

solution ϕ ∈ PAA(X1/2) whenever K
′ is small enough.

Classical examples of the above-mentioned functions F,G : R × X1/2 7→ L
2(Ω) are given as

follows:

F(t,ϕ) =
Ke(t)

1 + |∇ϕ|
and

G(t,ϕ) =
Kh(t)

1 + |∇ϕ|

where the functions e,h : R 7→ R are Sp-pseudo-almost automorphic.

In this particular case, the corresponding reaction-diffusion equation, that is,





∂

∂t

[
ϕ +

Ke(t)

1 + |∇ϕ|

]
= a(t,x)∆ϕ +

Kh(t)

1 + |∇ϕ|
, in R × Ω

ϕ = 0, on R × Γ

has a unique solution ϕ ∈ PAA(X1/2) whenever K is small enough.

Received: October 2009. Revised: November 2009.

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