CUBO A Mathematical Journal
Vol.16, No¯ 02, (149–160). June 2014

Diagana Space and The Gas Absorption Model

Najja S. Al-Islam

Department of Mathematics,

Medgar Evers College of The City University of New York,

1650 Bedford Ave., Brooklyn, N.Y. 11225 - USA

nalislam@mec.cuny.edu

ABSTRACT

Poorkarimi and Wiener established the existence of almost periodic solutions to a class

of nonlinear hyperbolic partial differential equations with delay. Al-Islam then gener-

alized the results of Poorkarimi and Weiner to the pseudo-almost periodic setting. In

this paper, the results of Al-Islam will be extended to the space of weighted pseudo

almost periodic functions, also known as Diagana Space. The class of nonlinear hyper-

bolic partial differential equations of Poorkarimi and Wiener represents a mathematical

model for the dynamics of gas absorption.

RESUMEN

Poorkarimi y Wiener establecieron la existencia de soluciones casi periódicas de una

clase de ecuaciones diferenciales parciales hiperbólicas no lineales con retraso. Luego,

Al-Islam generalizó los resultados de Poorkarimi y Wiener al caso seudo-cuasi periódico.

En este art́ıculo los resultados de Al-Islam se extenderán al espacio de funciones seudo-

cuasi periódicas con peso, también conocido como espacio de Diagana. La clase de

ecuaciones diferenciales parciales hiperbólicas no lineales de Poorkarimi y Wiener rep-

resenta un modelo matemático de la dinámica de absorción de gas.

Keywords and Phrases: almost periodic solution, weighted pseudo-almost periodic, gas absorp-

tion.

2010 AMS Mathematics Subject Classification: 35B10; 35B10; 35J60; 35L70



150 Najja S. Al-Islam CUBO
16, 2 (2014)

1 Introduction

Let L > 0. In Poorkarimi and Wiener [22], under some reasonable assumptions, the existence

of both periodic and almost periodic solutions to the nonlinear hyperbolic second-order partial

differential equation with delay given by

⎧
⎪⎪⎨

⎪⎪⎩

uxt(x, t) + a(x, t)ux(x, t) = C(x, t, u(x, ⌊t⌋))

u(0, t) = ϕ(t)

(1.1)

where a : [0, L] × R $→ R, C : [0, L] × R × R $→ R, and ϕ : R $→ R are periodic (respectively, almost
periodic) functions in the variable t and ⌊t⌋ denotes the greatest integer function: ⌊t⌋ := n for
n ≤ t < n + 1 for an integer n, was established. Extensive use of similar assumptions as in [22]
and [3] will be used to extend the above-mentioned existence results to the weighted pseudo-almost

periodic setting. Eq.(1.1) is of great interest, being that it represents a mathematical model for

the dynamics of gas absorption. Further details of the gas absorption model, Eq.(1.1), can also be

seen in [22].

The existence of almost periodic, asymptotically almost periodic, almost automorphic [21],

pseudo almost periodic [11], and more recently, weighted pseudo-almost periodic solutions to dif-

ferential equations are among the most attractive topics in the qualitative theory of differential

equations due to their applications in physics, mathematical biology, along with other areas of

science and engineering.

The concept of pseudo almost periodicity was first introduced by Zhang [23, 25, 24] and

generalizes the almost periodicity of Bohr. More details on the concept of pseudo almost periodicity

as well as its applications to differential equations, functional differential, and partial differential

equations can be easily found in the literature, especially in [1, 2, 4, 11, 7, 9, 10] and the references

therein.

The more recent generalization of Zhang almost periodicity is the weighted pseudo almost

periodicity of Diagana. The text that follows this introduction shows, as well as, compares the

properties of the Zhang and Diagana almost periodic spaces. Following the comparisons of the

Zhang and Diagana almost periodic spaces, the results contained in [3] will be generalized in the

setting of Diagana space. For more details on Diagana space, the reader should refer to [6, 14, 20].

2 Weighted Pseudo Almost Periodic Functions

Let (BC(R), ∥ · ∥∞) denote the Banach space of all bounded continuous functions ϕ : R $→ R
endowed with the sup norm defined by

∥ϕ∥∞ := sup
t∈R

|ϕ(t)|.



CUBO
16, 2 (2014)

Diagana Space and The Gas Absorption Model 151

Definition 2.1. [5, 11, 21] A continuous function g : R $→ R 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 following property

|g(t + τ) − g(t)| < ε

for each t ∈ R.

The number τ above is then called an ε-translation number of g, and the collection of those

almost periodic functions will be denoted as AP(R).

Although the concept of almost periodicity is a natural generalization of the classical period-

icity, there are almost periodic functions that are not periodic. A classical example of an almost

periodic function that is not periodic is the function defined by:

g(t) = sin t + sin(
√
7t) for each t ∈ R.

More details on properties of almost periodic functions can be found in the literature by Cor-

duneanu [5], Diagana [11], N’Guérékata [21] and the references therein.

Let U be the collection of all functions w, weights, such that w(t) > 0 for almost each t ∈ R
and w ∈ L1loc(R). Also, for each w ∈ U and r > 0,

µ(r, w) :=

∫r

−r

w(t)dt.

From the collection of weights, U, we define two subcollections of U as:

U∞ := {w ∈ U : lim
r→∞

µ(r, w) = ∞ and lim inf
t→∞

w(t) > 0},

Uw := {w ∈ U∞ : w is bounded}

One can see that the subcollections defined above can be written as: Uw ⊂ U∞ ⊂ U Define

PAP0(R) :=

{

φ ∈ BC(R) : lim
r→∞

1

2r

∫r

−r

|φ(σ)|dσ = 0

}

.

Definition 2.2. [11, 23] A function f ∈ BC(R) is called pseudo almost periodic if it can be
expressed as

f = g + ϕ,

where g ∈ AP(R) and ϕ ∈ PAP0(R).

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

Note that the functions g and ϕ appearing in Definition 2.2 are respectively called the al-

most periodic and the ergodic perturbation components of f. Furthermore, the decomposition in

Definition 2.2 is unique [23, 25, 24].



152 Najja S. Al-Islam CUBO
16, 2 (2014)

We now equip PAP(R) the collection of all pseudo almost periodic functions from R into R

with the sup norm. It is not really hard to see that (PAP(R), ∥ · ∥∞) is a closed subspace of BC(R)
and hence is a Banach space.

An example of a pseudo almost periodic function is the function f defined by

f(t) = sin t + sin t
√
2 + e−|t|

for each t ∈ R. The core of the construction of the weighted pseudo almost periodic space, is the
enrichment of the space of ergodic perturbations, PAP0(R). That is, for w ∈ U∞, the weighted
ergodic space is defined by:

PAP0(R, w) :=

{

φ ∈ BC(R) : lim
r→∞

1

µ(r, w)

∫r

−r

|φ(σ)|w(σ)dσ = 0

}

.

Definition 2.3. [14]A function f ∈ BC(R) is called weighted pseudo almost periodic if it can be
expressed as

f = g + ϕ,

where g ∈ AP(R) and ϕ ∈ PAP0(R, w). The collection of the functions defined above is Diagana
Space, and will be denoted as Dw(R).

An example of a function f ∈ Dw(R) is the function

f(t) = cos t + cos
√
2t +

1

1 + t2
,

where w(t) = 1 + t2 for each t ∈ R.

Lemma 2.4. AP(R) ⊂ PAP(R) ⊂ Dw(R).

In [19], it was shown that the decomposition f = g+ϕ, where g ∈ AP(R) and ϕ ∈ PAP0(R, w)
is not unique. Hence, one cannot define Dw(R), equipped with the sup norm, to be a Banach Space,

despite AP(R) and PAP0(R, w) being closed subspaces with respect to the sup norm. Therefore,

with the possibility of being able to construct countably many decompositions of any weighted

pseudo almost-periodic function, written as: {gn + φn, n ∈ N}, had to be resolved to ensure the
criterion of completeness for a Banach Space. To resolve this dilemma, in [20] another norm, which

will be known as the w-norm in this writing, was constructed and defined as follows:

||f||w := inf
n∈N

(||gn|| + ||φn||) = inf
n∈N

!

sup
t∈R

||gn(t)|| + sup
t∈R

||φn(t)||
"

.

|| · ||w is undoubtedly a norm on Dw(R).

Theorem 2.5. Dw(R) is a Banach Space under the norm || · ||w.



CUBO
16, 2 (2014)

Diagana Space and The Gas Absorption Model 153

Proof. The proof of the theorem can be found in [20].

Let W∞ be the set of all functions w ∈ U∞ where there exists a measurable set K ⊂ R such
that for each τ ∈ R,

lim sup
|t|→+∞, t∈K

w(t + τ)

w(t)
:= inf

m>0

#

sup
|t|>m,t∈K

w(t + τ)

w(t)

$

< ∞

and

lim
r→+∞

∫

Kτ
r

w(t)dt

µ(r, w)
= 0,

where Kτr = [−r, r] \ K + τ.

Lemma 2.6. [17] Let w ∈ W∞ and f ∈ Dw(R) and if g is its almost periodic component, then

g(R) ⊂ f(R).

Therefore, ||f||∞ ≥ ||g||∞ ≥ inf
t∈R

|g(t)| ≥ inf
t∈R

|f(t)|.

Proof. The proof of the lemma can be found in [17].

Theorem 2.7. If (fn)n∈N ⊂ Dw(R) is a sequence which converges uniformly with respect to the
w-norm to some f : R $→ R, then f is necessarily a weighted pseudo-almost periodic function.

Proof. Write fn = gn + ϕn where (gn)n ⊂ AP(R) and (ϕn)n∈N ⊂ PAP0(R, w). Suppose ||fn −
f||w → 0 as n → ∞ for some function f : R $→ R. Of course, f ∈ BC(R), as a uniform limit of a
sequence of bounded continuous functions. So to complete the proof, it needs to be shown that

f ∈ Dw(R). For that, notice that by using Lemma 2.6, it follows that

∥gn − gm∥w ≤ ∥fn − fm∥w for all n, m ∈ N.

Now letting n, m → ∞ in the previous inequality it follows that

lim
n,m→∞

∥gn − gm∥w ≤ lim
n,m→∞

∥fn − fm∥w = 0,

and hence (gn)n∈N ⊂ AP(R) is a Cauchy sequence. Since (AP(R), ∥ · ∥∞) is a Banach space, it
follows that there exists g ∈ AP(R) such that ∥gn − g∥∞ → 0 as n → ∞.



154 Najja S. Al-Islam CUBO
16, 2 (2014)

Now, fn − gn = φn → ϕ := f − g uniformly with respect to the w-norm as n → ∞. Thus,

writing

ϕ = (ϕ − ϕn) + ϕn,

it follows that:

1

µ(r, w)

∫r

−r

|φ(σ)|w(σ)dσ ≤ ∥φn − φ∥w +
1

µ(r, w)

∫r

−r

|φn(σ)|w(σ)dσ.

Let r → ∞ in the previous inequality, then

lim
r→∞

1

µ(r, w)

∫r

−r

|φ(σ)|w(σ)dσ ≤ ∥φn − φ∥w + lim
r→∞

1

µ(r, w)

∫r

−r

|φn(σ)|w(σ)dσ

= ∥φn − φ∥w.

Letting n → ∞ in the previous inequality, then

0 ≤ lim
r→∞

1

µ(r, w)

∫r

−r

|φ(σ)|w(σ)dσ ≤ lim
n→∞

∥φn − φ∥w = 0,

and hence

lim
r→∞

1

µ(r, w)

∫r

−r

|φ(σ)|w(σ)dσ = 0.

Therefore, f = g + ϕ ∈ Dw(R).

More details on properties of weighted pseudo-almost periodic functions can be found in the

literature, especially in Diagana [14].

3 Existence of Weighted Pseudo Almost Periodic Solutions

Throughout the rest of the paper, it is assumed that the function a : [0, L] × R $→ R satisfies the
following:

inf
x∈[0,L], t∈R

a(x, t) := m > 0.

Using the previous assumption, the initial value problem, Eq.(1.1), has a unique bounded

solution, which can be explicitly given by:

u(x, t) = ϕ(t) +

∫x

0

∫t

−∞

exp

{

−

∫t

τ

a(ξ, θ)dθ

}

C(ξ, τ, u(ξ, ⌊τ⌋))dτdξ (3.1)

for each (x, t) ∈ [0, L] × R.

Before exploring the existence and uniqueness of a weighted pseudo-almost periodic solution

to Eq.(1.1), consider the existence of a weighted pseudo-almost periodic solution to the first-order

partial differential equation



CUBO
16, 2 (2014)

Diagana Space and The Gas Absorption Model 155

∂V

∂t
(x, t) + a(x, t)V(x, t) = f(x, t), (3.2)

for each (x, t) ∈ [0, L] × R.

Lemma 3.1. Assume t $→ f(x, t) is weighted pseudo-almost periodic uniformly in x ∈ [0, L],
t $→ a(x, t) is almost periodic uniformly in x ∈ [0, L]. Then Eq. (4) has a unique weighted pseudo-
almost periodic solution, which can be explicitly expressed by

V(x, t) =

∫t

−∞

exp

{

−

∫t

τ

a(x, θ)dθ

}

f(x, τ)dτ. (3.3)

Moreover, V satisfies the following a priori inequality

∥V∥w ≤
1

m
∥f∥w

where the w-norm is taken in both x ∈ [0, L] and t ∈ R.

Proof. It is clear that the only bounded solution to Eq.(4) is given by Eq.(3.3). Now from the

weighted pseudo almost periodicity of t $→ f(x, t) it follows that there exist two functions g and h
with t $→ g(x, t) ∈ AP(R) for each x ∈ [0, L] and t $→ h(x, t) ∈ PAP0(R, w) for each x ∈ [0, L] such
that f = g + h. Consequently, V(x, t) = Vg(x, t) + Vh(x, t) for x ∈ [0, L] and t ∈ R, where

Vg(x, t) =

∫t

−∞

exp

{

−

∫t

τ

a(x, θ)dθ

}

g(x, τ)dτ, and

Vh(x, t) =

∫t

−∞

exp

{

−

∫t

τ

a(x, θ)dθ

}

h(x, τ)dτ.

Thus to complete the proof we must show that t $→ Vg(x, t) belongs to AP(R) and that t $→ Vh(x, t)
belongs to PAP0(R, w) uniformly in x ∈ [0, L]. The almost periodicity of t $→ Vg(x, t) (x ∈ [0, L])
was established in [22]. Thus, it remains to show that t $→ Vh(x, t) belongs to PAP0(R, w) uniformly
in x ∈ [0, L].

Now for r > 0,



156 Najja S. Al-Islam CUBO
16, 2 (2014)

1

µ(r, w)

∫r

−r

%

%

%

%

%

∫t

−∞

e−
∫
t

τ
a(x,θ)dθh(x, τ)w(τ)dτ

%

%

%

%

%

w

dt

≤
1

µ(r, w)

∫r

−r

∫t

−∞

e−
∫
t

τ
a(x,θ)dθ

%

%h(x, τ)dτ
%

%

w
w(τ)dt

≤
1

µ(r, w)

∫r

−r

&∫t

−∞

e−m(t−τ)
%

%h(x, τ)
%

%

w
dτ

'

w(τ)dt

=
1

µ(r, w)

∫r

−r

&∫+∞

0

e−ms
%

%h(x, t − s)
%

%

w
w(s)ds

'

dt

=

∫+∞

0

e−ms

&

1

µ(r, w)

∫r

−r

%

%h(x, t − s)
%

%

w
w(t)dt

'

ds,

by letting s = t − τ (ds = −dτ).

For any w ∈ W∞, it was shown in [17] that Dw(R) is translation invariant with respect to the
time variable t ∈ R. Therefore, it follows that t $→ h(x, t − s) belongs to PAP0(R, w) uniformly in
x ∈ [0, L]. That is,

lim
r→∞

1

µ(r, w)

∫r

−r

%

%h(x, t − s)
%

%

w
w(t)dt = 0

uniformly in x ∈ [0, L]. Using the Lebesgue Dominated Convergence Theorem completes the proof.

Theorem 3.2. Assume t $→ a(x, t) is almost periodic and the functions t $→ ϕ(t),
t $→ C(x, t, u(x, ⌊t⌋) are weighted pseudo-almost periodic uniformly in x ∈ [0, L]. Additionally,
assume C(x, t, u(x, ⌊t⌋) satisfies the Lipschitz condition, that is, there exists K > 0 such that

%

%C(x, t, u(x, ⌊t⌋)) − C(x, t, V(x, ⌊t⌋))
%

%

w
≤ K

%

%u(x, ⌊t⌋) − V(x, ⌊t⌋)
%

%

w

for all x ∈ [0, L] and t ∈ R. Then Eq.(1.1) has a unique weighted pseudo-almost periodic solution.

Proof. Our proof follows along the same line as that given in [22] with the appropriate modifica-

tions. Indeed, for the first approximation, let u0(x, t) ≡ 0. The next approximation is

u1(x, t) = ϕ(t) +

∫x

0

∫t

−∞

exp

{

−

∫t

τ

a(ξ, θ)dθ

}

C(ξ, τ, 0)dτdξ.

Now since V(x, t) = ∂
∂x

u1(x, t), then from

∂V

∂t
(x, t) + a(x, t)V(x, t) = C(x, t, 0)



CUBO
16, 2 (2014)

Diagana Space and The Gas Absorption Model 157

and by Lemma 3.1 the weighted pseudo-almost periodicity of V(x, t) is obtained. Now

u1(x, t) = ϕ(t) +

∫x

0

V(ξ, t)dξ.

and hence t $→ u1(x, t) is pseudo almost periodic in t uniformly with respect to x ∈ [0, L] and

u2(x, t) = ϕ(t) +

∫x

0

∫t

−∞

exp

{

−

∫t

τ

a(ξ, θ)dθ

}

C(ξ, τ, u1(τ, ξ))dτdξ.

The relation Ṽ(x, t) =
∂

∂x
u2(x, t) and the equation

∂Ṽ

∂t
(x, t) + a(x, t)Ṽ(x, t) = C(x, t, u1(x, t))

yields the weighted pseudo-almost periodicity of

u2(x, t) = ϕ(t) +

∫x

0

Ṽ(ξ, t)dξ.

This shows that all successive approximations un(t) are weighted pseudo-almost periodic functions

in t uniformly in x ∈ [0, L]. Therefore,

lim
n→∞

un(x, t) = u(x, t)

is a weighted pseudo-almost periodic function in t uniformly with respect to

x ∈ [0, L], by using Theorem 2.7.

4 Example

To illustrate the main result of this paper(Theorem 3.2), consider the following nonlinear hyperbolic

second-order partial differential equation

⎧
⎪⎪⎨

⎪⎪⎩

uxt(x, t) +
!

P(x) + sin t
"

ux(x, t) = H(t) sin(u(x, ⌊t⌋))

u(0, t) = 0 = ϕ(t)

(4.1)

where P(x) =
n∑

k=0

akx
k for x ∈ [0, 1] is a polynomial of degree n with real coefficients,

H(t) = sin t + sin t
√
2 + w(t) sin t

where

w(t) =

{
1, t ∈ [0, ∞),
e−t

2

, t ∈ (−∞, 0),



158 Najja S. Al-Islam CUBO
16, 2 (2014)

a(x, t) = P(x) + sin t,

C(x, t, u(x, t)) = H(t) sin(u(x, ⌊t⌋)),

ϕ(t) = 0

for all x ∈ [0, 1] and t ∈ R.
Suppose

inf
x∈[0,1],t∈R

a(x, t) = a0 − 1,

where a0 > 1. Then all the assumptions of Theorem 3.2 are fulfilled and therefore the next theorem

holds.

Theorem 4.1. Under previous assumptions, the hyperbolic partial differential equation Eq. (4.1)

has a unique weighted pseudo-almost periodic solution.

Received: January 2014. Revised: April 2014.

References

[1] E. Ait Dads and O. Arino, Exponential Dichotomy and Existence of Pseudo Almost Periodic

Solutions of Some Differential Equations, Nonlinear Anal. 27 (1996), no. 4, pp. 369–386.

[2] E. Ait Dads, K. Ezzinbi, and O. Arino, Pseudo Almost Periodic Solutions for Some Differential

Equations in a Banach Space, Nonlinear Anal. 28 (1997), no. 7, pp. 1141–1155.

[3] N. Al-Islam, Existence of Pseudo-Almost Periodic Solutions to Some Nonlinear Hyperbolic

Second-Order Partial Differential Equations with Delay, International Journal of Evolution

Equations. 4 (2008), no.4, pp. 375–382.

[4] B. Amir and L. Maniar, Composition of Pseudo-Almost Periodic Functions and Cauchy Prob-

lems with Operator of Nondense Domain, Ann. Math. Blaise Pascal. 6 (1999), no. 1, pp.

1–11.

[5] C. Corduneanu, Almost Periodic Functions, 2nd Edition, Chelsea-New York, (1989).

[6] M. Damak, K. Ezzinbi, and L. Souden, Weighted Pseudo-Almost Periodic Solutions For Some

Neutral Partial Functional Differential Equations, E. J. of Diff. Eq. (2012), no.47, pp. 1–13.

[7] T. Diagana, Pseudo Almost Periodic Solutions to Some Differential Equations, Nonlinear

Anal. 60 (2005), no. 7, pp. 1277–1286.

[8] T. Diagana, Weighted pseudo almost periodic functions and applications. C.R. Math. Acad.

Sci., vol.343, (2006) no. 10, 643–646.



CUBO
16, 2 (2014)

Diagana Space and The Gas Absorption Model 159

[9] T. Diagana, E. Hernández, and M. Rabello, Pseudo Almost Periodic Solutions to some Nonau-

tonomous Neutral Functional Differential Equations with Unbounded Delay. Math. Comput.

Modelling 45 (2007), no. 9-10, 1241–1252.

[10] T. Diagana and E. Hernández, Existence and uniqueness of pseudo almost periodic solutions

to some abstract partial neutral functional-differential equations and applications. J. Math.

Anal. Appl. 327 (2007), no. 2, 776–791.

[11] T. Diagana, Pseudo Almost Periodic Functions in Banach Spaces, Nova Science Publishers,

Inc., (2007).

[12] T. Diagana, Existence of weighted pseudo almost periodic solutions to some non-autonomous

differential equations. Int. J. Evol. Equ. 2 (2008), no. 4, 397–410.

[13] T. Diagana, Weighted pseudo-almost periodic solutions to some differential equations. Non-

linear Anal. 68 (2008), no. 8, 2250–2260.

[14] T. Diagana, Existence of Weighted Pseudo-Almost Periodic Solutions to Some Classes Of

Hyperbolic Evolution Equations. J. Math. Anal. Appl., No. 350 (2009), pp. 18–28

[15] T. Diagana, Weighted Pseudo Almost Periodic Solutions to Some Neutral Delay Integral

Equation of Advanced Type. Nonlinear Anal. 70 (2009), Issue 1, Pages 298-304.

[16] T. Diagana, Almost Automorphic Type and Almost Periodic Type Functions in Abstract

Spaces. Springer, Cham, (2013).

[17] H. Ding, J. Liang, and T. Xiao, Weighted pseudo almost automorphic functions and appli-

cations to evolution equations with delay. Applied Mathematics and Computation No. 219

(2013), pp. 8949-8958.

[18] H. Ding, J. Liang, and T. Xiao, Weighted pseudo almost automorphic functions and WPAA

solutions to semilinear evolution equations. J. Math. Anal. Appl. No. 409 (2014), pp. 409-427.

[19] J. Liang, T.-J. Xiao, and J. Zhang, Decomposition of weighted pseudo-almost periodic func-

tions. Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 10, pp. 3456 3461,

(2010).

[20] J. Liang, T. Xiao, and J. Zhang, Weighted Pseudo-Almost Periodic Functions and Applications

to Semilinear Evolution Equations. Abstract and Applied Analysis, 179525 (2012), 15pgs.

[21] G. M. N’Guérékata, Almost Automorphic Functions and Almost Periodic Functions in Ab-

stract Spaces, Kluwer Academic / Plenum Publishers, New York-London-Moscow, (2001).

[22] H. Poorkarimi and J. Wiener, Almost Periodic Solutions of Nonlinear Hyperbolic Equations

with Time Delay. Proceedings of the 16th Conference on Applied Mathematics (Edmond, OK,

2001), 99–102 (electronic), Electron. J. Differ. Equ. Conf., 7, Southwest Texas State Univ.,

San Marcos, TX, (2001).



160 Najja S. Al-Islam CUBO
16, 2 (2014)

[23] C. Y. Zhang, Pseudo Almost Periodic Solutions of Some Differential Equations, J. Math. Anal.

Appl. 151 (1994), pp. 62–76.

[24] C. Y. Zhang, Integration of Vector-Valued Pseudo Almost Periodic Functions, Proc. Amer.

Math. Soc. 121 (1994), pp. 167–174.

[25] C. Y. Zhang, Pseudo Almost Periodic Solutions of Some Differential Equations II, J. Math.

Anal. Appl. 192 (1995), pp. 543–561.