CUBO A Mathematical Journal

Vol.10, N
o
¯ 03, (161–170). October 2008

Existence and Uniqueness of Pseudo Almost

Automorphic Solutions to Some Classes of

Partial Evolution Equations

J. Blot, D. Pennequin

Université Paris 1 Panthéon-Sorbonne, Laboratoire Marin Mersenne,

Centre P.M.F., 90 rue de Tolbiac,

75647 PARIS Cedex 13, FRANCE

emails: blot@univ-paris1.fr, pennequi@univ-paris1.fr

and

Gaston M. N’Guérékata

Department of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore, M.D. 21251 – USA

email: Gaston.N’Guerekata@morgan.edu

ABSTRACT

We are concerned in this paper with the partial differential equation d
dt

[u(t)+f(t,u(t))] =

Au(t) t ∈ R, where A is a (generally unbounded) linear operator which generates a

semigroup of bounded linear operators (T (t))t≥0. Under appropriate sufficient con-

ditions, we prove the existence and uniqueness of a pseudo almost automorphic mild

solution to the equation.

RESUMEN

Nosotros consideramos en este art́ıculo la ecuación diferencial parcial d
dt

[u(t)+f(t,u(t))] =

Au(t) t ∈ R, donde A es un (generalmente no acotado) operador lineal que genera



162 J. Blot et al. CUBO
10, 3 (2008)

un semigrupo de operadores lineales acotados (T (t))t≥0. Bajo condiciones suficientes

apropriadas, provamos la existencia y unicidad de una solución blanda pseudo casi

automorfica para tal ecuación.

Key words and phrases: Pseudo Almost automorphic function, exponentially stable semigroup,

partial differential equations.

Math. Subj. Class.: 34G10, 47A55.

1 Introduction

Since the publication of the monograph [12], the study of almost automorphic function (a concept

introduced by S. Bochner in the literature in the mid sixties as a generalization of almost periodicity

in the sense of Bohr) has regained great interest. Several extensions of the concept were introduced

including asymptotic almost automorphy by N’Guérékata ([10]), p-almost automorphy by Diagana

([2]), and Stepanov-like almost automorphy by N’Guérékata and Pankov ([14]). Recently, J. Liang

et al. have suggested the notion of pseudo almost automorphic functions, i.e. functions that can be

written uniquely as a sum of an almost automorphic function and an ergodic term, i.e. a function

with vanishing mean (cf [6], and [7], [8]). This latter turns out to be more general than asymptotic

almost automorphy. However it seems to be more complicated.

There has been a considerable interest in the existence of (these various types of) almost

automorphic solutions of evolution equations. Semigroups theory and fixed point techniques have

been frequently used for semilinear evolution equations. In [3] the authors studied the existence

and uniqueness of an almost automorphic mild solution to the equation

d

dt
[u(t) + f(t,u(t))] = Au(t) + g(t,u(t)) t ∈ R, (1.1)

where the functions f(t,u) and g(t,u) are almost automorphic in t, for each u. This latter

motivated our recent paper [15], where we study the existence and uniqueness of a pseudo almost

automorphic mild solution the semilinear evolution equations of the form

du

dt
= Au(t) + g(t,u(t)), t ∈ R, (1.2)

where A is an unbounded sectorial operator with not necessarily dense domain in a Banach space

X and g : R × Xα → X, where Xα, α ∈ (0, 1), is any intermediate Banach space between D(A) and

X.

In this paper, we study pseudo almost automorphic solutions to perturbations to Equations:

du

dt
= Au(t) t ∈ R, (1.3)

consisting of the class of abstract partial evolution equations of the form



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Existence and Uniqueness of Pseudo ... 163

d

dt
[u(t) + f(t,u(t))] = Au(t) t ∈ R, (1.4)

where A is the infinitesimal generator of an exponentially stable C0-semigroup acting on X, B,C

are two densely defined closed linear operators on X, and f is continuous functions.

Under some appropriate assumptions, we establish the existence and uniqueness of an almost

automorphic (mild) solution to Eq. (1.4) using the Banach fixed-point principle.

We start this work by presenting some properties of pseuso almost automorphic functions in

Section 2 including an application to a Volterra-like integral equation. Our main result (Theoreme

3.3) is presented in Section 3.

2 Preliminaries

In this work, (X,‖·‖) will stand for a Banach space. The collection of all bounded linear operators

from X is denoted by B(X) — this is a Banach space when it is equipped with its natural norm

‖A‖B(X) := sup
x∈X,x 6=0

‖Ax‖X
‖x‖X

.

The fields of real and complex numbers, are respectively denoted by C and R. We let BC(R, X)

denote the space of all X-valued bounded continuous functions R → X– it is a Banach space when

equipped with the sup norm ‖u‖∞ := sup
t∈R

‖u(t)‖X for each u ∈ B(R, X).

We will use the following well-known concepts in the sequel.

Definition 2.1. A continuous function f : R 7→ 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 ⊂ (s
′

n)n∈N such that

g(t) := lim
n7→∞

f(t + sn)

is well-defined for each t ∈ R, and

f(t) = lim
n7→∞

g(t − sn)

for each t ∈ R.

Similarly,

Definition 2.2. A continuous function f : R × X 7→ X is said to be almost automorphic in t ∈ R

for each u ∈ X if every sequence of real numbers (σn)n∈N contains a subsequence (sn)n∈N such that

g(t,u) := lim
n7→∞

f(t + sn,u)

is well defined for each t ∈ R and each u ∈ X and,

f(t,u) = lim
n7→∞

g(t − sn,u)

exists for each t ∈ R and u ∈ X.



164 J. Blot et al. CUBO
10, 3 (2008)

The following natural properties hold: If f,h : R 7→ X are almost automorphic functions and

if λ ∈ R, then f + h, λf, and fλ are almost automorphic, where fλ(t) := f(t + λ). Moreover,

R(f) := {f(t), t ∈ R} is relatively compact.

Since the range of an almost automorphic function f is relatively compact on X, then it is

bounded. Almost automorphic functions constitute a Banach space AA(X) when it is endowed

with the sup norm. This naturally generalizes the concept of (Bochner) almost periodic functions.

Definition 2.3. Let X be a Banach space.

1. A bounded continuous function with vanishing mean value can be defined as

AA0(R, X) =

{

φ ∈ BC(R, X) : lim
T →∞

1

2T

∫ T

−T

‖φ(σ)‖dσ = 0

}

.

2. Similarly we define AA0(R × X, X) to be the collection of all functions f : t 7→ f(t,x) ∈

BC(R × X, X) satisfying

lim
T →∞

1

2T

∫ T

−T

‖f(σ,x)‖dσ = 0

uniformly for x in any bounded subset of X.

Now we describe the sets PAA(R, X) and PAA(R × X, X) of pseudo almost automorphic

functions:

PAA(R, X) =

{

f = g + φ ∈ BC(R, X),

g ∈ AA(R, X) and φ ∈ AA0(R, X)

}

;

PAA(R × X, X) =

{

f = g + φ ∈ BC(R × X, X),

g ∈ AA(R × X, X) and φ ∈ AA0(R × X, X)

}

.

In both cases above, g and φ are called respectively the principal and the ergodic terms of f.

We have the following elementary properties of pseudo almost automorphic functions.

Theorem 2.4. ( [8] Theorem 2.2).

PAA(R, X) is a Banach space under the supremum norm.

Let now f,h : R → R and consider the convolution

(f ⋆ h)(t) :=

∫

R

f(s)h(t − s)ds, t ∈ R,

if the integral exists.

Remark 2.5. The operator J : PAA(R, X) → PAA(R, X) such that (Jx)(t) := x(−t) is well-defined

and linear. Moreover it is an isometry and J2 = I.



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Existence and Uniqueness of Pseudo ... 165

Remark 2.6. The operator Ta defined by (Tax)(t) := x(t + a) for a fixed a ∈ R leaves PAA(R, X)

invariant.

Let us now discuss conditions which do ensure the pseudo almost automorphy of the con-

volution function f ⋆ h of f with h where f is pseudo almost automorphic and h is a Lebesgue

mesurable function satisfying additional assumptions.

Let f : R → X and h : R → R; the convolution function (if it does exist) of f with h denoted

f ⋆ h is defined by:

(f ⋆ h)(t) :=

∫

R

f(σ)h(t − σ)dσ =

∫

R

f(t − σ)h(σ)dσ = (h ⋆ f)(t), for all t ∈ R.

Hence, if f ⋆ h is well-defined, then f ⋆ h = h ⋆ f.

Let ϕ ∈ L1 and λ ∈ C. It is well-known that the operator Aϕ,λ defined by

Aϕ,λu = λu + ϕ ⋆ u (2.1)

acts continuously in BC(R, X) i.e., there exists K > 0 such that

‖Aϕ,λu‖BC(R,X) ≤ K‖u‖BC(R,X),∀u ∈ BC(R, X) (2.2)

Moreover Aϕ,λ leaves BC(R, X) invariant.

Now denote M := {PAP(R, X),PAA(R, X)} where PAP(R,X) is the Banach space of all

pseudo almost periodic functions f : R → X. Then we have.

Theorem 2.7. For Ω ∈ M,

Aϕ,λ(Ω) ⊂ Ω.

Proof. . It is an immediate consequence of the remarks above.

Application: A Volterra-like equation

Consider the equation

x(t) = g(t) +

∫

+∞

−∞

a(t − σ)x(σ)dσ, t ∈ R, (2.3)

where g : R → R is a continuous function and a ∈ L1(R).

Theorem 2.8. Suppose g ∈ PAA(R, X) and ‖a‖L1 < 1. Then (2.3) above has a unique pseudo

almost automorphic solution.

Proof. It is clear that the operator

x ∈ PAA(R, X) →

∫

+∞

−∞

a(t − σ)x(σ)dσ ∈ PAA(R, X)



166 J. Blot et al. CUBO
10, 3 (2008)

is well-defined. Now consider Γ : PAA(R, X) → PAA(R, X) such that

(Γx)(t) = g(t) +

∫

+∞

−∞

a(t − σ)x(σ)dσ, t ∈ R.

We can easily show that

‖(Γx) − (Γy)‖ ≤ ‖a‖L1‖x − y‖.

The conclusion is immediate by the principle of contraction.

3 Main results

This section is devoted to the proof of the main result of the paper, that is, the existence and

uniqueness of an almost automorphic (mild) solution to Eq. (1.4). For that we need to establish

a few preliminary results.

Definition 3.1. A function u ∈ BC(R, X) is said to be a mild solution to Eq. (1.4) if the function

s → AT (t − s)f(s,u(s)) is integrable on (−∞, t) for each t ∈ R and

u(t) = −f(t,u(t)) −

∫ t

−∞

AT (t − s)f(s,u(s))ds

for each t ∈ R.

We now make the following assumptions.

(H.1) The operator A is the infinitesimal generator of an exponentially stable semigroup (T (t))t≥0

such that there exist constants M > 0 and δ > 0 with

‖T (t)‖B(X) ≤ Me
−δt, ∀t ≥ 0.

Furthermore, the function σ → AT (σ) defined from (0,∞) into B(X) is strongly (Lebesgue)

measurable and there exist a function γ : (0,∞) → [0,∞) such that sups≥s0 γ(s) < ∞ for

any s0 > 0, and a constant ω > 0 with ρ :=

∫

∞

0

e−ωsγ(s)ds < ∞ such that

‖AT (s)‖B(X) ≤ e
−ωs

.γ(s), s > 0.

(H.2) The function f : R × X 7→ X, (t,u) 7→ f(t,u) is jointly continuous and

‖f(t,u) − f(t,v)‖X ≤ k(t) .‖u − v‖, and

for all t ∈ R, and ∀u,v ∈ X. Here k ∈ L1(R, R+).

(H.3) f = g + ψ ∈ PAA(R × X, X), where g and ψ are the principal and the ergodic terms of f

respectively and f(t,u) and g(t,u) are uniformly continuous on every bounded subset K ⊂ X

uniformly in t ∈ R.



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Existence and Uniqueness of Pseudo ... 167

Lemma 3.2. Suppose that assumptions (H.1)-(H.2)-(H.3) hold. Define the nonlinear operator Λ1

by: For each ξ ∈ PAA(X),

(Λ1ξ)(t) =

∫ t

−∞

AT (t − s)f(s,ξ(s))ds

Then Λ1 maps PAA(X) into itself.

Proof. Set h defined by: h(.) = f(.,ξ(.)). Since h ∈ PAA(R, X) using [6, Theorem 2.4] with

assumption (H.3), we can write h = β + φ where β is the principal part and φ the ergodic term of

h. Using the same argument as in [11], we can prove that t 7→
∫ t

−∞
AT (t − s)β(s)ds is in AA(X).

Now, set:

ν(t) = −

∫ t

−∞

AT (t − s)φ(s)ds.

We have:
1

2T

∫ T

−T

‖ν(t)‖Xdt ≤
1

2T

∫ T

−T

∫ t

−∞

‖A(t − s)φ(s)‖Xdsdt

≤
1

2T

∫ T

−T

∫ t

−∞

e−ω(t−s)γ(t − s)‖φ(s)‖Xdsdt.

Let’s write:
1

2T

∫ T

−T

∫ t

−∞

e−ω(t−s)γ(t − s)‖φ(s)‖Xdsdt = I1 + I2,

where:

I1 =
1

2T

∫ T

−T

∫

−T

−∞

e−ω(t−s)γ(t − s)‖φ(s)‖Xdsdt

and

I2 =
1

2T

∫ T

−T

∫ t

−T

e−ω(t−s)γ(t − s)‖φ(s)‖Xdsdt.

We prove know that I1 → 0 and I2 → 0 as T → ∞.

Indeed, for I1, let s0 > 0 and set M(s0) = sups≥s0 γ(s), and K = supt∈R ‖φ(t)‖X. We have:

I1 ≤ K
1

2T

∫ T

−T

∫

−T

−∞

e−ω(t−s)γ(t − s)dsdt =
K

2T

∫ ∫

D

e−ω(t−s)γ(t − s)dsdt,

where D = {(s,t) ∈ R2, |t| ≤ T, s ≤ −T}. We introduce also:

D1 = {(s,t) ∈ D,t − s ≥ s0}, D2 = D \ D1.

We have:
∫ ∫

D1

e−ω(t−s)γ(t − s)dsdt ≤ M(s0)

∫ ∫

D1

e−ω(t−s)dsdt

≤ M(s0)

∫ ∫

D

e
−ω(t−s)

dsdt =
M(s0)e

−ωT

ω

∫ T

−T

e
−ωt

dt ≤ 2T
M(s0)e

−ωT

ω
.



168 J. Blot et al. CUBO
10, 3 (2008)

Moreover,
∫ ∫

D2

e−ω(t−s)γ(t − s)dsdt ≤

∫

−T

−T −s0

∫ s+s0

−T

e−ω(t−s)γ(t − s)dtds

≤

∫

−T

−T −s0

∫ s0

−T −s

e−ωσγ(σ)dσds

≤

∫

−T

−T −s0

∫ s0

0

e
−ωσ

γ(σ)dσds

≤ s0

∫ s0

0

e−ωσγ(σ)dσ.

So, for any T ≥ 1, we have:

I1 ≤
K

2T

(

2T
M(s0)e

−ωT

ω
+ s0

∫ s0

0

e−ωσγ(σ)dσ

)

≤ K

(

e−ωT
M(s0)

ω
+ s0

∫ s0

0

e−ωσγ(σ)dσ

)

.

Let ǫ > 0. We can find s0 > 0 such that Ks0
∫ s0

0
e−ωσγ(σ)dσ < ε/2. Let us take such an s0. After,

for T sufficiently large, Ke−ωT
M(s0)

ω
< ǫ/2, and so, for sufficiently large T , I1 ≤ ǫ.

Now, we consider I2. We have:

I2 =
1

2T

∫ T

−T

‖φ(s)‖Xds

∫ T

s

e
−ω(t−s)

γ(t − s)dt

≤
1

2T

∫ T

−T

‖φ(s)‖Xds

∫ T −s

0

e−ωσγ(σ)dσ

≤ ρ
1

2T

∫ T

−T

‖φ(s)‖Xds → 0 as T → ∞.

Now we are ready to state and prove the following.

Theorem 3.3. Suppose that assumptions (H.1)-(H.2) hold. Then Eq. (1.4) has a unique pseudo

almost automorphic (mild) solution if

Proof. Define the nonlinear operator Γ : AA(X) 7→ AA(X) by:

Γ(u) : t 7→ −f(t,u(t)) −

∫ t

−∞

AT (t − s)f(s,u(s))ds.

We have:

‖Γ(u)(t) − Γ(v)(t)‖X ≤ ‖f(t,u(t)) − f(t,v(t))‖X +

∫ t

−∞

‖AT (t − s)(f(s,u(s)) − f(s,v(s)))‖Xds



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Existence and Uniqueness of Pseudo ... 169

≤ k(t)‖u(t) − v(t)‖X +

∫ t

−∞

e−ω(t−s)γ(t − s)k(s)‖u(x) − v(s)‖Xds

≤

[

k(t) +

∫ t

−∞

e−ω(t−s)γ(t − s)k(s)ds

]

‖u − v‖∞

≤ (1 + ρ)‖k‖∞‖u − v‖∞.

So, we obtain:

‖Γ(u) − Γ(v)‖∞ ≤ (1 + ρ)‖k‖∞‖u − v‖∞,

and we can conclude using the Banach’s fixed point principle.

Acknowledgements. This paper was written when the second author was visiting the ”Lab-

oratoire Marin Mersenne” of the University of Paris 1 Panthéon-Sorbonne in June 2008. He is

grateful to Professors Blot and Pennequin for their invitation.

Received: June 2008. Revised: August 2008.

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