Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 45, pp. 1–21.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE

FUNCTIONS AND APPLICATIONS

MARKO KOSTIĆ

Communicated by Jerome A. Goldstein

Abstract. In this article, we analyze multi-dimensional Bohr (B, c)-almost

periodic type functions. The main structural characterizations for the intro-
duced classes of Bohr (B, c)-almost periodic type functions are established.

Several applications of our abstract theoretical results to the abstract Volterra

integro-differential equations in Banach spaces are provided, as well.

1. Introduction and preliminaries

The notion of almost periodicity was introduced by the Danish mathematician
H. Bohr [6] around 1924-1926 and later generalized by many others (for more details
about the subject, we refer the reader to the research monographs [5, 10, 11, 14,
15, 16, 17, 18, 19, 20, 22]). Let I be either R or [0,∞), and let f : I → X be a
given continuous function, where X is a complex Banach space equipped with the
norm ‖ · ‖. Given ε > 0, we call τ > 0 a ε-period for f(·) if and only if

‖f(t+ τ)− f(t)‖ ≤ ε, t ∈ I.
The set consisting of all ε-periods for f(·) is denoted by ϑ(f, ε). The function f(·)
is said to be almost periodic if and only if for each ε > 0 the set ϑ(f, ε) is relatively
dense in [0,∞), i.e., there exists l > 0 such that any subinterval of [0,∞) of length
l intersects ϑ(f, ε).

As emphasized in [7], the theory of almost periodic functions of several real vari-
ables has not attracted so much attention of the authors by now. In support of our
investigation of the multi-dimensional c-almost periodicity, we would like to present
the following illustrative examples (the notion and notation will be explained in the
next section):

Example 1.1 ((cf. also [7])). Suppose that a closed linear operator A generates a
strongly continuous semigroup (T (t))t≥0 on a Banach space X consisting of certain
complex-valued functions defined on Rn. Under some assumptions, the function

u(t, x) =
(
T (t)u0

)
(x) +

∫ t

0

[T (t− s)f(s)](x) ds, t ≥ 0, x ∈ Rn

2020 Mathematics Subject Classification. 42A75, 43A60, 47D99.
Key words and phrases. Bohr (B, c)-almost periodic type functions;

(B, c)-uniformly recurrent type functions; abstract Volterra integro-differential equations.
©2022. This work is licensed under a CC BY 4.0 license.

Submitted Janaury 15, 2021. Published July 2, 2022.

1



2 M. KOSTIĆ EJDE-2022/45

is a unique classical solution of the abstract Cauchy problem

ut(t, x) = Au(t, x) + F (t, x), t ≥ 0, x ∈ Rn; u(0, x) = u0(x),

where F (t, x) := [f(t)](x), t ≥ 0, x ∈ Rn. For a large class of strongly continuous
semigroups (for example, this holds for the Gaussian semigroup on Rn; see e.g.,
[4, Example 3.7.6]), there exists a kernel (t, y) 7→ E(t, y), t > 0, y ∈ Rn which is
integrable on any set [0, T ]× Rn (T > 0) and satisfies

[T (t)f(s)](x) =

∫
Rn

F (s, x− y)E(t, y) dy, t > 0, s ≥ 0, x ∈ Rn.

Fix a positive real number t0 > 0. As in the case that c = 1, the c-almost periodic

behaviour of function x 7→ ut0(x) ≡
∫ t0

0
[T (t0 − s)f(s)](x) ds, x ∈ Rn strongly

depends on the c-almost periodic behaviour of the function F (t, x) in the space
variable x. Suppose, for example, that the function F (t, x) is Bohr c-almost periodic
with respect to the variable x ∈ Rn, uniformly in the variable t on compact subsets
of [0,∞). Then the function ut0(·) is also Bohr c-almost periodic, which follows
from the estimate

|ut0(x+ τ)− cut0(x)| ≤
∫ t0

0

∫
Rn

|F (s, x+ τ − y)− cF (s, x− y)| ·
∣∣E(t0, y)∣∣ dy ds

≤ ε
∫ t0

0

∫
Rn

|E
(
t0, y

)
| dy ds

and corresponding definitions.

Example 1.2. In this example, we observe an interesting feature of the famous
d’Alembert formula, which has been used by Zaidman [22, Example 5] in a slightly
different context (for almost periodic functions of one real variable). Let a > 0; then
it is well known that the regular solution of the wave equation utt = a2uxx in domain
{(x, t) : x ∈ R, t > 0}, equipped with the initial conditions u(x, 0) = f(x) ∈ C2(R)
and ut(x, 0) = g(x) ∈ C1(R), is given by the d’Alembert formula

u(x, t) =
1

2

[
f(x− at) + f(x+ at)

]
+

1

2a

∫ x+at

x−at
g(s) ds, x ∈ R, t > 0.

Let us suppose that the function x 7→ (f(x), g[1](x)), x ∈ R is c-almost periodic,
where g[1](·) ≡

∫ ·
0
g(s) ds. Then the solution u(x, t) can be extended to the whole

real line in the time variable and this solution is c-almost periodic in (x, t) ∈ R2. To
verify this, fix a positive real number ε > 0. Then there exists a finite real number
l > 0 such that any subinterval I of R of length l contains a point τ ∈ I such that

|f(x+ τ)− cf(x)|+
∣∣g[1](x+ τ)− cg[1](x)

∣∣ < ε, x ∈ R. (1.1)

Furthermore, we have (x, t, τ1, τ2 ∈ R):∣∣u(x+ τ1, t+ τ2
)
− cu(x, t)

∣∣
≤ 1

2

∣∣f((x− at) + (τ1 − aτ2)
)
− cf(x− at)

∣∣
+

1

2

∣∣f((x+ at) + (τ1 + aτ2)
)
− cf([x+ at+ (τ1 + aτ2)]− (τ1 + aτ2))

∣∣
+

1

2a

∣∣g[1]
(
(x− at) + (τ1 − aτ2)

)
− cg[1](x− at)

∣∣
+

1

2a

∣∣g[1]
(
(x+ at)− (τ1 − aτ2)

)
− cg[1](x+ at)

∣∣.
(1.2)



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 3

Let (t1, t2) ∈ R2. Then the interval [−t1 − at2 − (l/2),−t1 − at2 + (l/2)] contains
the point τ ′ and the interval [t1 − at2 − (l/2), t1 − at2 + (l/2)] contains the point
τ ′′ such that the equation (1.1) holds with the number τ replaced therein with any
of the numbers τ ′, τ ′′. Setting τ1 := (τ ′′− τ ′)/2 and τ2 := (−τ1− τ2)/2a, it can be
easily shown that |τ1 − t1| ≤ l/2 and |τ2 − t2| ≤ l/2a, so that the final conclusion
simply follows from the corresponding definition and (1.2).

The notion of (ω, c)-periodicity and various generalizations of this concept have
recently been introduced and investigated by Alvarez, Gómez, Pinto [1] and Al-
varez, Castillo, Pinto [2, 3]. In [13], our joint paper with Khalladi, Rahmani, Pinto
and Velinov, we have recently introduced and analyzed the classes of c-almost peri-
odic functions, c-uniformly recurrent functions, semi-c-periodic functions and their
Stepanov generalizations, where c ∈ C and |c| = 1. On the other hand, in [7], we
have recently analyzed various notions of multi-dimensional almost periodic type
functions. The main aim of this paper is to continue the research studies [7] and
[13] by investigating various notions of multi-dimensional c-almost periodic type
functions and related applications, where c ∈ C \ {0}. For simplicity, we will not
consider the corresponding Stepanov classes here (see [15] for more details).

Notation and terminology. We assume henceforth that (X, ‖ ·‖), (Y, ‖ ·‖Y ) and
(Z, ‖ · ‖Z) are complex Banach spaces and n ∈ N; usually, B denotes the collection
of all bounded subsets of X or all compact subsets of X. Set BX := {y ∈ X :
(∃B ∈ B), y ∈ B}. We will always assume henceforth that BX = X, i.e., that for
each x ∈ X there exists B ∈ B such that x ∈ B. By L(X,Y ) we denote the Banach
algebra of all bounded linear operators from X into Y ; L(X,X) ≡ L(X). By B◦

and ∂B we denote the interior and the boundary of a subset B of a topological
space X, respectively.

The symbol C(I : X) stands for the space of all X-valued continuous functions
defined on the domain I. By Cb(I : X) (respectively, BUC(I : X)) we denote the
subspace of C(I : X) consisting of all bounded (respectively, all bounded uniformly
continuous functions). Both Cb(I : X) and BUC(I : X) are Banach spaces with the
sup-norm ‖·‖∞. This also holds for the space C0(I : X) consisting of all continuous
functions f : I → X such that lim|t|→+∞ f(t) = 0. If t0 ∈ Rn and ε > 0, then we
set B(t0, ε) := {t ∈ Rn : |t − t0| ≤ ε}, where | · | denotes the Euclidean norm in
Rn. Set Nn := {1, . . . , n} and S1 := {z ∈ C : |z| = 1}. If any component of the
tuple t = (t1, t2, . . . , tn) ∈ Rn is strictly positive, then we simply write t > 0.

Now we briefly explain the organization and main ideas of this paper. In Subsec-
tion 1.1, we recall the basic definitions and results about almost periodic functions
in Rn. If ∅ 6= I ⊆ Rn, I + I ⊆ I and F : I × X → Y is a continuous function,
then the notions of Bohr (B, c)-almost periodicity and (B, c)-uniform recurrence for
F (·; ·) are introduced in Definition 2.1. If the region I satisfies certain conditions,
F : I × X → Y is Bohr (B, c)-almost periodic and B is any family of compact
subsets of X, then some sufficient conditions ensuring that for each set B ∈ B we
have that the set {F (t;x) : t ∈ I, x ∈ B} is relatively compact in Y are given
in Proposition 2.2 (see also Proposition 2.4, where we analyze the compositions
of Bohr (B, c)-almost periodic/(B, c)-uniformly recurrent functions with uniformly
continuous functions φ : Y → Z).

The notion introduced in Definition 2.1 is reexamined and extended in Defini-
tion 2.6, where we introduce the notions of Bohr (B, I ′, c)-almost periodicity and



4 M. KOSTIĆ EJDE-2022/45

(B, I ′, c)-uniform recurrence (∅ 6= I ′ ⊆ I ⊆ Rn). Example 2.8, although very sim-
ple and elaborate, shows that the statement of [13, Proposition 2.6] fails to be
true for multi-dimensional (B, I ′, c)-uniformly recurrent functions, in general. An
important extension of [13, Proposition 2.17] is proved in Proposition 2.9, where
condition I + I ′ = I is crucial for proving the fact that we always have c = ±1 pro-
vided the existence of a (B, I ′, c)-uniformly recurrent non-zero function F : I → R
(if F (t) ≥ 0 for all t ∈ I, then c = 1); see also Example 2.10. Proposition 2.9 is later
employed in the proof of Proposition 2.11, where it is shown that, if the function
F : I × X → Y is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent),
I + I ′ = I and F (·; ·) 6= 0, then |c| = 1.

The first example of a multi-dimensional almost anti-recurrent function F : Rn →
R (c = −1) which is not almost periodic is presented in Example 2.12(iii)-(b). Af-
ter that, in Proposition 2.13, we transfer the statement of [13, Proposition 2.9]
for multi-dimensional Bohr (B, c)-almost periodic type functions (see also Corol-
lary 2.14 and Proposition 2.16 for similar results). The convolution invariance of
Bohr (B, c)-almost periodic type functions, invariance of Bohr c-almost periodicity
and composition theorem for Bohr (B, c)-almost periodic type functions are inves-
tigated in Proposition 2.17, Proposition 2.18 and Theorem 2.19, respectively. The
main structural profilations of D-asymptotically c-almost periodic type functions
are given in Subsection 2.1. In this subsection, we state and prove our main re-
sults, Theorem 2.27 (in which we analyze certain relations between the classes of
I-asymptotically Bohr c-almost periodic functions of type 1 and I-asymptotically
Bohr c-almost periodic functions) and Theorem 2.28 (in which we analyze the ex-
tensions of Bohr (I ′, c)-almost periodic functions and (I ′, c)-uniformly recurrent
functions). The final section of paper is reserved for applications of our abstract
theoretical results.

1.1. Almost periodic functions on Rn. Suppose that F : Rn → X is a continu-
ous function. Let us recall that F (·) is said to be almost periodic if and only if for
each ε > 0 there exists l > 0 such that for each t0 ∈ Rn there exists τ ∈ B(t0, l)
such that

‖F (t + τ )− F (t)‖ ≤ ε, t ∈ Rn.
This is equivalent to saying that for any sequence (bn) in Rn there exists a sub-
sequence (an) of (bn) such that (F (· + an)) converges in Cb(Rn : X). The vector
space of all almost periodic functions F : Rn → X is denoted by AP (Rn : X). Any
almost periodic function F : Rn → X is bounded and AP (Rn : X) is the Banach
space equipped with the sup-norm.

Any trigonometric polynomial in Rn is almost periodic and a continuous func-
tion F (·) is almost periodic if and only if there exists a sequence of trigonometric
polynomials in Rn which converges uniformly to F (·); let us recall that a trigono-
metric polynomial in Rn is any linear combination of functions like t 7→ ei〈λ,t〉,
t ∈ Rn, where λ ∈ Rn and 〈·, ·〉 denotes the usual inner product in Rn. Any almost
periodic function F : Rn → X is also uniformly continuous, the mean value

M(F ) := lim
T→+∞

1

(2T )n

∫
s+KT

F (t) dt

exists and it does not depend on s ∈ Rn; here, KT := {t = (t1, t2, . . . , tn) ∈ Rn :
|ti| ≤ T for 1 ≤ i ≤ n}. We define the Bohr-Fourier coefficient Fλ ∈ X by

Fλ := M
(
e−i〈λ,·〉F (·)

)
, λ ∈ Rn,



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 5

and the Bohr spectrum σ(F ) of F by

σ(F ) :=
{
λ ∈ Rn : Fλ 6= 0

}
.

It is well known that σ(F ) is at most a countable set. By APΛ(Rn : X) we denote
the set consisting of all almost periodic functions F : Rn → X such that σ(F ) ⊆ Λ.
As is well known, for every almost periodic function F ∈ APΛ(Rn : X), we can
always find a sequence (Pk) of trigonometric polynomials in Rn which uniformly
converges to F (·) on Rn and satisfies that σ(Pk) ⊆ Λ for all k ∈ N; see e.g., [20,
Chapter 1, Section 2.3].

2. Bohr (B, c)-almost periodic type functions

The main aim of this section is to analyze Bohr (B, c)-almost periodic type func-
tions depending of several real variables, where B denotes a non-empty collection
of non-empty subsets of X and c ∈ C \ {0}. We will consider the following notion,
which can be also analyzed on general topological (semi-)groups; see the references
quoted in [7] for more details concerning this problematic:

Definition 2.1. Suppose that ∅ 6= I ⊆ Rn, F : I×X → Y is a continuous function
and I + I ⊆ I. Then we say that:

(i) F (·; ·) is Bohr (B, c)-almost periodic if and only if for every B ∈ B and ε > 0
there exists l > 0 such that for each t0 ∈ I there exists τ ∈ B(t0, l) ∩ I
such that

‖F (t + τ ;x)− cF (t;x)‖Y ≤ ε, t ∈ I, x ∈ B.

(ii) F (·; ·) is (B, c)-uniformly recurrent if and only if for every B ∈ B there
exists a sequence (τ k) in I such that limk→+∞ |τ k| = +∞ and

lim
k→+∞

sup
t∈I;x∈B

‖F (t + τ k;x)− cF (t;x)‖Y = 0.

If X ∈ B, then it is also said that F (·; ·) is Bohr c-almost periodic (c-uniformly
recurrent); if c = 1, then we also say that F (·; ·) is Bohr B-almost periodic (B-
uniformly recurrent) [Bohr almost periodic (uniformly recurrent)].

Unless stated otherwise, we will assume that ∅ 6= I ⊆ Rn henceforth. It is clear
that any Bohr ((B, c)-)almost periodic function is ((B, c)-)uniformly recurrent; in
general, the converse statement does not hold ([15]). In [13, Proposition 2.2], we
have proved that any Bohr almost periodic function f : I → Y is bounded, provided
that I = [0,∞) or I = R. In the multi-dimensional case, the things become more
complicated and the best we can do is to prove the following extension of the above-
mentioned result following the method proposed in the proof of [7, Proposition 2.16],
which is applicable in the case that I = [0,∞)n or I = Rn :

Proposition 2.2. Suppose that ∅ 6= I ⊆ Rn, I + I ⊆ I, I is closed, F : I×X → Y
is Bohr (B, c)-almost periodic and B is any family of compact subsets of X. If

(∀l > 0) (∃t0 ∈ I) (∃k > 0)(∀t ∈ I)(∃t′0 ∈ I)

(∀t′′0 ∈ B(t′0, l) ∩ I) t− t′′0 ∈ B(t0, kl) ∩ I,

then for each B ∈ B we have that the set {F (t;x) : t ∈ I, x ∈ B} is relatively
compact in Y ; in particular, supt∈I;x∈B ‖F (t;x)‖Y <∞.

We continue by providing the following illustrative example.



6 M. KOSTIĆ EJDE-2022/45

Example 2.3 ((see also [13, Example 2.15])). Suppose that ϕ ∈ (−π, π] \ {0}, θ ∈
(−π, π], µ ∈ Rn \ {0} and c = eiθ. Then the trigonometric polynomial t→ ei〈µ,t〉,
t ∈ Rn is c-almost periodic. Towards see this, set S := {j ∈ Nn : µj 6= 0} and
l := max{2π|µj |−1 : j ∈ S}. Let ε > 0 be fixed. Then we have (t ∈ Rn; τ ∈ Rn):∣∣ei〈µ,t+τ〉 − eiθei〈µ,t〉∣∣ =

∣∣ei[µ1τ1+µ2τ2+···+µnτn−θ] − 1
∣∣

= 2
∣∣∣sin(µ1τ1 + µ2τ2 + · · ·+ µnτn − θ

2

)∣∣∣,
and therefore

∣∣ei〈µ,t+τ〉 − eiθei〈µ,t〉∣∣ ≤ ε, t ∈ Rn if and only if there exists k ∈ Z
such that

µ1τ1 + µ2τ2 + · · ·+ µnτn − θ ∈
[
− arcsin(ε/2) + kπ, arcsin(ε/2) + kπ

]
.

In particular, if there exists k ∈ Z such that µ1τ1 +µ2τ2 + · · ·+µnτn = kπ+θ, then
we have |ei〈µ,t+τ〉 − eiθei〈µ,t〉| ≤ ε, t ∈ Rn. But, we can simply prove that for each
t0 ∈ Rn there exists a point τ ∈ B(t0, l) such that µ1τ1 +µ2τ2 + · · ·+µnτn = kπ+θ
for some k ∈ Z, which simply implies the required.

Using a slight modification of the proof of [18, Property 4, p. 3], we can conclude
the following:

Proposition 2.4. Suppose that F : I×X → Y is Bohr (B, c)-almost periodic/(B, c)-

uniformly recurrent, and φ : Y → Z is uniformly continuous on R(F ) and satisfies
that φ(cy) = cφ(y) for all y ∈ Y . Then φ ◦ F : I × X → Z is Bohr (B, c)-almost
periodic/(B, c)-uniformly recurrent.

The conclusions clarified in the next illustrative example follow from the argu-
ments similar to those employed in [7, Example 2.13]:

Example 2.5. (i) Suppose that Fj : X → Y is a continuous function, for each B in

B we have supx∈B ‖Fj(x)‖Y <∞ and the mapping t 7→ (
∫ t

0
f1(s) ds, . . . ,

∫ t
0
fn(s) ds)

belongs to Cn, t ≥ 0 is c-almost periodic (1 ≤ j ≤ n). Set

F
(
t1, . . . , tn+1;x

)
:=

n∑
j=1

∫ tj+1

tj

fj(s) ds · Fj(x)

for all x ∈ X and tj ≥ 0, 1 ≤ j ≤ n. Then the mapping F : [0,∞)n+1 ×X → Y is
Bohr (B, c)-almost periodic.

(ii) Suppose that F : X → Y is a continuous function, for each B ∈ B we have
supx∈B ‖F (x)‖Y <∞ and the complex-valued mapping t 7→ fj(t), t ≥ 0 is c-almost
periodic, resp. bounded and c-uniformly recurrent (1 ≤ j ≤ n). Set

F
(
t1, . . . , tn;x

)
:=

n∏
j=1

fj
(
tj
)
· F (x)

for all x ∈ X and tj ≥ 0, 1 ≤ j ≤ n. Then the mapping F : [0,∞)n ×X → Y is
Bohr (B, c)-almost periodic, resp. (B, c)-uniformly recurrent.

(iii) Suppose that G : [0,∞)n → C is c-almost periodic, resp. bounded and
c-uniformly recurrent, F : [0,∞) × X → Y is Bohr B-almost periodic, resp. B-
uniformly recurrent, and for each set B ∈ B we have supt≥0;x∈B ‖F (t;x)‖Y < ∞.
Set

F
(
t1, . . . , tn+1;x

)
:= G

(
t1, . . . , tn

)
· F
(
tn+1;x

)



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 7

for all x ∈ X and tj ≥ 0, 1 ≤ j ≤ n+1. Then the mapping F : [0,∞)n+1×X → Y is
Bohr (B, c)-almost periodic, resp. (B, c)-uniformly recurrent (see also [7, Proposition
2.19, Example 2.22], which can be modified in a similar fashion).

The notion introduced in Definition 2.1 can be extended as follows:

Definition 2.6. Suppose that ∅ 6= I ′ ⊆ I ⊆ Rn, F : I × X → Y is a continuous
function and I + I ′ ⊆ I. Then we say that:

(i) F (·; ·) is Bohr (B, I ′, c)-almost periodic if and only if for every B ∈ B and
ε > 0 there exists l > 0 such that for each t0 ∈ I ′ there exists τ ∈ B(t0, l)∩I ′
such that

‖F (t + τ ;x)− cF (t;x)‖Y ≤ ε, t ∈ I, x ∈ B. (2.1)

(ii) F (·; ·) is (B, I ′, c)-uniformly recurrent if and only if for every B ∈ B there
exists a sequence (τ k) in I ′ such that limk→+∞ |τ k| = +∞ and

lim
k→+∞

sup
t∈I;x∈B

‖F (t + τ k;x)− cF (t;x)‖Y = 0. (2.2)

If X ∈ B, then it is also said that F (·; ·) is Bohr (I ′, c)-almost periodic ((I ′, c)-
uniformly recurrent).

Remark 2.7. (i) Let |c| = 1 and F : R→ Y be a continuous function. Then F (·)
is c-almost periodic (c-uniformly recurrent) in the sense of [13] if and only if F (·)
is Bohr ((0,∞), c)-almost periodic (((0,∞), c)-uniformly recurrent) in the sense of
Definition 2.6. Albeit we will not consider here the general question concerning the
existence of larger sets I ′′ ⊇ I ′ for which a given a Bohr (B, I ′, c)-almost periodic
function F (·; ·) is also (B, I ′′, c)-almost periodic (the only exception is the proof
of Theorem 2.28), we would like to note that any Bohr ((0,∞), c)-almost periodic
function is already Bohr (R, c)-almost periodic. This is clear if arg(c)/π /∈ Q since
we can apply then [13, Proposition 2.11(i)] in order to see that the function F (·) is
also Bohr ((0,∞), c−1)-almost periodic and therefore, given ε > 0 in advance, we
can collect all positive (ε, c)-periods of function F (·) and all negative values of all
positive (ε, c−1)-periods of function F (·) (with the meaning clear), obtaining thus
a relatively dense set in R consisting solely of (ε, c)-periods of F (·). The situation
is similar if arg(c)/π ∈ Q because then there exists m ∈ N such that cm+1 = 1 so
that cm = c−1 and we can collect all positive (ε, c)-periods of function F (·) and
all negatives of all positive (ε/m, c)-periods of function F (·) in order to obtain a
relatively dense set in R consisting solely of (ε, c)-periods of F (·); observe here only
that the assumption ‖F (t+ τ)− cF (t)‖ ≤ ε for all t ∈ R and some τ ∈ R implies

‖F (t+mτ)− cmF (t)‖
≤ ‖F (t+mτ)− cF (t+ (m− 1)τ)‖+ |c|‖F (t+ (m− 1)τ)− cF (t+ (m− 2)τ)‖

+ · · ·+ |c|m−2‖F (t+ 2τ)− cF (t+ τ)‖+ |c|m−1‖F (t+ τ)− cF (t)‖ ≤ mε,
(2.3)

for all t ∈ R.
(ii) Condition ∅ 6= I ′ ⊆ I is a bit unnecessary and intended for considerations of

regions I for which 0 ∈ I; more precisely, the assumption I + I ′ ⊆ I is mandatory
and implies that for each t0 ∈ I we have I ′ ⊆ I − t0 (take, for example I = [1,∞)
and I ′ = [0,∞); then we do not have I ′ ⊆ I but the notion introduced in Definition
2.6 is meaningful).



8 M. KOSTIĆ EJDE-2022/45

(iii) The main structural properties of functions introduced in Definition 2.1 and
Definition 2.6, clarified in [13, Proposition 2.28] and [13, Theorem 2.13], continue
to hold with appropriate modifications. For example, the introduced spaces of
functions are translation invariant, in a certain sense, with respect to the both
variables.

Clearly, the notion from Definition 2.1 is recovered by plugging I ′ = I and any
(B, I ′, c)-uniformly recurrent function is (B, I, c)-uniformly recurrent provided that
I + I ⊆ I. Concerning the statement of [13, Proposition 2.6], we would like to
present first the following instructive example.

Example 2.8. Suppose that I := {(x, y) ∈ R2 : x + y ≥ 0} (I := {(x, y) ∈ R2 :
x+ y ≥ 0}) and I ′ := {(x, y) ∈ R2 : x+ y = 1} (I ′ := {(x, y) ∈ R2 : x+ y = −1}).
Set F (x, y) := 2−x−y, (x, y) ∈ I. Then I+I ′ ⊆ I+I = I and for every (a, b) ∈ I ′ we
have F ((x, y)+(a, b)) = 2−1F (x, y), (x, y) ∈ I (F ((x, y)+(a, b)) = 2F (x, y), (x, y) ∈
I), so that F (·, ·) is both Bohr (I ′, 2−1)-almost periodic and 2−1-uniformly recurrent
(Bohr (I ′, 2)-almost periodic and 2-uniformly recurrent) but not identically equal
to zero.

Furthermore, the statement of [13, Proposition 2.7] can be simply reformulated
in our new framework and, if the function F (·; ·) is Bohr (B, I ′, c)-almost peri-
odic ((B, I ′, c)-uniformly recurrent), then the function ‖F (·; ·)‖Y is Bohr (B, I ′, |c|)-
almost periodic ((B, I ′, |c|)-uniformly recurrent). The following fact should be also
clarified: If the function F (·; ·) is (B, I ′, c)-uniformly recurrent, then for each B ∈ B
we have

sup
t∈I,x∈B

‖F (t;x)‖Y ≤ |c|−1 sup
t∈I,|t|≥a,t∈I+I′

‖F (t;x)‖Y , (2.4)

and for each x ∈ X the function F (·;x) is identically equal to zero provided that the
function F (·; ·) is (B, I ′, c)-uniformly recurrent and lim|t|→+∞,t∈I+I′ F (t;x) = 0.

Now we are able to state and prove the following extension of [13, Proposition
2.17].

Proposition 2.9. Suppose that ∅ 6= I ′ ⊆ I ⊆ Rn and I + I ′ = I. If the function
F : I → R is (B, I ′, c)-uniformly recurrent and F 6= 0, then c = ±1. Furthermore,
if F (t) ≥ 0 for all t ∈ I, then c = 1.

Proof. Since we have assumed I + I ′ = I and F 6= 0, the equation (2.4) yields
the existence of a finite real number a > 0 and a sequence (tk) in I such that
|F (tk)| > a/2 for all k ∈ N. Then the final conclusion follows by repeating verbatim
the arguments contained in the proof of [13, Proposition 2.17]. �

Remark 2.10. Suppose that c = 1/2 in Example 2.8. Then the function F (·; ·) is
real-valued so that the conclusion of Proposition 2.9 does not hold if the assumption
I + I ′ 6= I is neglected.

The most important corollary of Proposition 2.9, which extends the statement
of [13, Proposition 2.6], is stated below.

Corollary 2.11. Suppose that ∅ 6= I ′ ⊆ I ⊆ Rn, I + I ′ = I and F : I ×X → Y is
Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent). If F (·; ·) 6= 0, then
|c| = 1.



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 9

Proof. By our assumption, there exist t0 ∈ I and x ∈ X such that F (t0;x) 6= 0.
Further on, there exists B ∈ B such that x ∈ B and this simply implies that
the function Fx : I → Y is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly
recurrent) and not identically equal to zero. Therefore, the function ‖Fx(·)‖Y is
Bohr (B, I ′, |c|)-almost periodic ((B, I ′, |c|)-uniformly recurrent) and not identically
equal to zero. By Proposition 2.9, we obtain |c| = 1. �

If c = ±1, then we also say that the function F (·) is Bohr B-almost (anti-)periodic
(B-uniformly (anti-)recurrent)/Bohr (B, I ′)-almost (anti-)periodic ((B, I ′)-uniformly
(anti-)recurrent). Let us recall that there is a great number of very simple examples
showing that the notion of (B, I ′)-almost periodicity is neither stronger nor weaker
than the notion of (B, I)-almost periodicity, provided that I + I ⊆ I.

The conclusions established in the subsequent example follow similarly as in [7,
Example 2.15]:

Example 2.12. (i) Suppose that the complex-valued mapping t 7→
∫ t

0
fj(s) ds,

t ∈ R is c-almost periodic, resp. bounded and c-uniformly recurrent (1 ≤ j ≤ n).
Set

F1

(
t1, . . . , t2n

)
:=

n∏
j=1

∫ tj+n

tj

fj(s) ds and tj ∈ R, 1 ≤ j ≤ 2n.

Then the mapping F1 : R2n → C is Bohr (I ′, c)-almost periodic, resp. (I ′, c)-
uniformly recurrent, where I ′ = {(τ , τ ) : τ ∈ Rn}; furthermore, if the function

t 7→ (
∫ t

0
f1(s) ds, . . . ,

∫ t
0
fn(s) ds), t ∈ R is c-almost periodic, resp. bounded and

c-uniformly recurrent, then the function F1(·) is Bohr (I ′′, c)-almost periodic, resp.
(I ′′, c)-uniformly recurrent, where I ′′ = {(a, a, . . . , a) ∈ R2n : a ∈ R}.

(ii) Suppose that an X-valued mapping t 7→
∫ t

0
fj(s) ds, t ∈ R is c-almost peri-

odic, resp. bounded and c-uniformly recurrent, as well as that a strongly continuous
operator family (Tj(t))t∈R ⊆ L(X,Y ) is uniformly bounded (1 ≤ j ≤ n). Set

F2

(
t1, . . . , t2n

)
:=

n∑
j=1

Tj(tj − tj+n)

∫ tj+n

tj

fj(s) ds

and tj ∈ R, 1 ≤ j ≤ 2n. Then the mapping F2 : R2n → Y is Bohr (I ′, c)-almost
periodic, resp. (I ′, c)-uniformly recurrent, where I ′ = {(τ , τ ) : τ ∈ Rn}, but not
generally Bohr c-almost periodic, in the case of consideration of almost periodicity;

furthermore, if the function t 7→ (
∫ t

0
f1(s) ds, . . . ,

∫ t
0
fn(s) ds), t ∈ R is c-almost

periodic, resp. bounded and c-uniformly recurrent, then the function F2(·) is Bohr
I ′′-almost periodic, where I ′′ = {(a, a, . . . , a) ∈ R2n : a ∈ R}.

(iii) Suppose that ∅ 6= I ⊆ Rn, I0 = [0,∞) or I0 = R, a = (a1, . . . , an) ∈
Rn 6= 0 and the linear function g(t) := a1t1 + · · · + antn, t = (t1, . . . , tn) ∈ I
maps surjectively the region I onto I0. Suppose, further, that f : I0 → X is
a c-uniformly recurrent function as well as that a sequence (αk) in I0 satisfies
that limk→+∞ |αk| = +∞ and limk→+∞ supt∈I0 ‖f(t + αk) − cf(t)‖ = 0. Define

I ′ := g−1({αk : k ∈ N}) and F : I → X by F (t) := f(g(t)), t ∈ I. Then
F (·) is (I ′, c)-uniformly recurrent, and F (·) is not c-almost periodic provided that
f(·) is not c-almost periodic (note that the conclusions established in [7, Example
2.12] cannot be reformulated for the c-uniform recurrence). We will provide two
illustrative examples of the obtained conclusion as follows:



10 M. KOSTIĆ EJDE-2022/45

(a) It is worth recalling that A. Haraux and P. Souplet have proved, in [12,
Theorem 1.1], that the function f : R→ R, given by

f(t) :=

∞∑
n=1

1

n
sin2

( t

2n

)
dt, t ∈ R,

is uniformly continuous, uniformly recurrent (the sequence (τk ≡ 2kπ)k∈N
can be chosen in definition of uniform recurrence) and unbounded; in [13,
Example 2.19(i)], we have shown that f(·) is c-uniformly recurrent if and
only if c = 1. Let a = (a1, . . . , an) ∈ Rn \ {0}, let I ′ = g−1({2kπ :
k ∈ N}) and let F : Rn → R be given by F (t) := f(a1t1 + · · · + antn),
t = (t1, . . . , tn) ∈ Rn. Then the function F (·) is uniformly continuous,
unbounded, I ′-uniformly recurrent and not almost periodic ([7]); further-
more, an application of Proposition 2.9 shows that F (·) is (I ′, c)-uniformly
recurrent if and only if c = 1.

(b) In [13, Example 2.20], we have proved that the function g : R → R, given
by

f(t) := (sin t) ·
∞∑
n=1

1

n
sin2

( t

3n

)
, t ∈ R,

is c-uniformly recurrent if and only if c = ±1 (the sequence (τk ≡ 3kπ)k∈N
can be chosen in definition of uniform anti-recurrence). Let a ∈ Rn \ {0},
let I ′ = g−1({3kπ : k ∈ N}) and let F : Rn → R be defined as in (a). Then
the function F (·) is uniformly continuous, unbounded, I ′-uniformly anti-
recurrent and not almost periodic; furthermore, an application of Proposi-
tion 2.9 shows that F (·) is (I ′, c)-uniformly recurrent if and only if c = ±1.

Set lI ′ := {lt : t ∈ I ′} for all l ∈ N. The following result extends [13, Proposition
2.9] for c-almost periodic functions and c-uniformly recurrent functions.

Proposition 2.13. Suppose that l ∈ N, ∅ 6= I ′ ⊆ I ⊆ Rn, I + I ′ ⊆ I and
F : I × X → Y is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent).
Then lI ′ ⊆ I, I + lI ′ ⊆ I and F (·; ·) is Bohr (B, lI ′, cl)-almost periodic ((B, lI ′, cl)-
uniformly recurrent).

Proof. Since I ′ ⊆ I and I + I ′ ⊆ I, we inductively get that jI ′ ⊆ I and I + jI ′ ⊆ I
for all j ∈ N. Keeping this in mind, the proof simply follows from the corresponding
definitions and the identity (t ∈ I, τ ∈ I ′):

F
(
t + lτ

)
− clF (t) =

l−1∑
j=0

cj
[
F
(
t + (l − j)τ

)
− cF

(
t + (l − j − 1)τ

)]
. �

Suppose now that:

p ∈ Z \ {0}, q ∈ N, (p, q) = 1, |c| = 1, arg(c) = πp/q. (2.5)

The most important corollary of Proposition 2.13, which extends [13, Corollary
2.10], follows by plugging l = q.

Corollary 2.14. Suppose that (2.5) holds, ∅ 6= I ′ ⊆ I ⊆ Rn, I + I ′ ⊆ I and
F : I × X → Y is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent).
Then the following holds:



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 11

(i) If p is even, then F (·; ·) is Bohr (B, qI ′)-almost periodic ((B, qI ′)-uniformly
recurrent).

(ii) If p is odd, then F (·; ·) is Bohr (B, qI ′)-almost anti-periodic ((B, qI ′)-uni-
formly anti-recurrent).

Similarly we can prove the following result.

Proposition 2.15. Suppose that |c| = 1, arg(c) ∈ πQ, ∅ 6= I ′ ⊆ I ⊆ Rn, I +
I ′ ⊆ I and F : I × X → Y is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly
recurrent). Define Cc := {l ∈ N : cl = 1} and Cc,−1 := {l ∈ N : cl = −1}. If S
is any finite non-empty subset of Cc, resp. Cc,−1, and I ′S := ∪l∈SlI ′, then F (·; ·)
is Bohr (B, I ′S)-almost periodic ((B, I ′S)-uniformly recurrent), resp. Bohr (B, I ′S)-
almost anti-periodic ((B, I ′S)-uniformly anti-recurrent).

The subsequent result follows from the argumentation contained in the proof of
[13, Proposition 2.11(i)].

Proposition 2.16. Let |c| = 1 and arg(c)/π /∈ Q. If ∅ 6= I ′ ⊆ I ⊆ Rn, I + I ′ ⊆ I,
lI ′ = I ′ for all l ∈ N and F : I × X → Y is a bounded, Bohr (B, I ′, c)-almost
periodic ((B, I ′, c)-uniformly recurrent) function, then the function F (·; ·) is Bohr
(B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent) for all c′ ∈ S1.

Concerning the convolution invariance of introduced spaces of Bohr (B, c)-almost
periodic type functions, we would like to state the following result.

Proposition 2.17. Suppose that h ∈ L1(Rn), ∅ 6= I ′ ⊆ Rn and the function F (·; ·)
is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent). If

(B)b: For every B ∈ B, there exists a finite real constant cB > 0 such that
supt∈Rn,x∈B ‖F (t;x)‖Y ≤ cB,

then the function

(h ∗ F )(t;x) :=

∫
Rn

h(σ)F (t− σ;x) dσ, t ∈ Rn, x ∈ X

is Bohr (B, I ′, c)-almost periodic ((B, I ′, c)-uniformly recurrent) and satisfies (B)b.

Proof. Since h ∈ L1(Rn), the prescribed assumptions imply that the function (h ∗
F )(·; ·) is well defined and satisfies (B)b. The continuity of function (h ∗ F )(·; ·)
follows from the dominated convergence theorem, the continuity of the function
F (·; ·) and condition (B)b. Let B ∈ B and ε > 0 be fixed. Then there exists l > 0
such that for each t0 ∈ I ′ there exists τ ∈ B(t0, l) ∩ I ′ such that (2.1) holds with
I = Rn. Therefore,

‖(h ∗ F )(t + τ ;x)− c
(
h ∗ F )(t;x)‖Y

≤
∫
Rn

|h(σ)| · ‖F (t + τ − σ;x)− cF (t− σ;x)‖Y dσ,

for any t ∈ Rn and x ∈ B. This simply implies the required. �

The following result, which has recently been considered in [7] in the case that
c = 1, can be slightly extended for the Stepanov classes of c-almost periodic type
functions (see the forthcoming monograph [15] for more details):



12 M. KOSTIĆ EJDE-2022/45

Proposition 2.18. Let (R(t))t>0 ⊆ L(X,Y ) be a strongly continuous operator
family such that

∫
(0,∞)n

‖R(t)‖ dt < ∞. If f : Rn → X is c-almost periodic, then

the function F : Rn → Y , given by

F (t) :=

∫ t1

−∞

∫ t2

−∞
· · ·
∫ tn

−∞
R(t− s)f(s) ds, t ∈ Rn,

is well-defined and c-almost periodic.

Suppose now that |c| = 1. Concerning the assertion of [13, Theorem 2.24], we
will first observe that any almost periodic function F ∈ APRn\{0}(Rn : X) can be
uniformly approximated by trigonometric polynomials whose frequencies belong to
the set Rn \ {0}. If we denote by APc,0(Rn : X) the linear span of all c-almost
periodic functions F : Rn → X and by APc,0(Rn : X) its closure in AP (Rn : X),
then it follows from the above and our conclusion established in Example 2.3 that
APRn\{0}(Rn : X) ⊆ APc,0(Rn : X). But, it is not clear how to prove or disprove
the converse inclusion provided that arg(c) ∈ π ·Q.

Before we move ourselves to the next subsection, we will state and prove a com-
position theorem for multi-dimensional Bohr (B, c)-almost periodic type functions.
Suppose that F : I × X → Y and G : I × Y → Z are given functions; then the
multi-dimensional Nemytskii operator W : I ×X → Z is defined by

W (t;x) := G
(
t;F (t;x)

)
, t ∈ I, x ∈ X. (2.6)

Set R(F ) ≡ {F (t;x) : t ∈ I, x ∈ X} and suppose that there exists a finite real
constant L > 0 such that

‖G(t; y)−G
(
t; y′

)
‖Z ≤ L‖y − y′‖Y , t ∈ I, y ∈ R(F ), y′ ∈ cR(F ). (2.7)

The following result is an extension of [13, Theorem 2.28].

Theorem 2.19. Suppose that the functions F : I × X → Y and G : I × Y → Z
are continuous as well as ∅ 6= I ′ ⊆ I ⊆ Rn and (2.7) holds.

(i) Suppose further that, for every B ∈ B and ε > 0, there exists l > 0 such
that for each t0 ∈ I ′ there exists τ ∈ B(t0, l)∩ I ′ such that (2.1) holds and

‖G(t + τ ; cy)− cG(t; y)‖Z ≤ ε, t ∈ I, y ∈ R(F ). (2.8)

Then the function W (·; ·), given by (2.6), is Bohr (B, I ′, c)-almost periodic.
(ii) Suppose further that, for every B ∈ B, there exists a sequence (τ k) in I ′

such that limk→+∞ |τ k| = +∞, (2.2) holds and

lim
k→+∞

sup
t∈I;x∈B

‖G
(
t + τ k; cF (t;x)

)
− cG(t;F (t;x))‖Y = 0. (2.9)

Then the function W (·; ·), given by (2.6), is (B, I ′, c)-uniformly recurrent.

Proof. We will prove only (i). The continuity of function W (·; ·) is obvious. Then
the final conclusion follows from the assumption made, the corresponding definition
of Bohr (B, I ′, c)-almost periodicity and the next simple computation:

‖G
(
t + τ ;F (t + τ ;x)

)
−G

(
t;F (t;x)

)
‖Z

≤ ‖G
(
t + τ ;F (t + τ ;x)

)
−G

(
t + τ ; cF (t;x)

)
‖Z

+ ‖G
(
t + τ ; cF (t;x)

)
− cG

(
t;F (t;x)

)
‖Z

≤ L‖F (t + τ ;x)− cF (t;x)‖Y + ‖G
(
t + τ ; cF (t;x)

)
− cG

(
t;F (t;x)

)
‖Z ,

for any t ∈ I, τ ∈ I ′ and x ∈ X. �



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 13

2.1. D-asymptotically (B, c)-almost periodic type functions. In [7], we have
recently introduced the following notion:

Definition 2.20. Suppose that D ⊆ I ⊆ Rn and the set D is unbounded. By
C0,D,B(I ×X : Y ) we denote the vector space consisting of all continuous functions
Q : I × X → Y such that, for every B ∈ B, we have limt∈D,|t|→+∞Q(t;x) = 0,
uniformly for x ∈ B.

Definition 2.21. Suppose that the set D ⊆ I ⊆ Rn is unbounded, ∅ 6= I ′ ⊆ I ⊆ Rn
and F : I ×X → Y is a continuous function. Then we say that F (·; ·) is (strongly)
D-asymptotically Bohr (B, I ′, c)-almost periodic, resp. (strongly) D-asymptotically
(B, I ′, c)-uniformly recurrent, if and only if there exist a Bohr (B, I ′, c)-almost peri-
odic function (G : Rn×X → Y ) G : I×X → Y , resp. a (B, I ′, c)-uniformly recurrent
function (G : Rn ×X → Y ) G : I ×X → Y and a function Q ∈ C0,D,B(I ×X : Y )
such that F (t;x) = G(t;x) + Q(t;x) for all t ∈ I and x ∈ X. If I ′ = I, then we
also say that F (·; ·) is (strongly) D-asymptotically Bohr (B, c)-almost periodic, resp.
(strongly) D-asymptotically (B, c)-uniformly recurrent; if X ∈ B, then we omit the
term B from the notation introduced, with the meaning clear.

Before we go any further, we would like to present the following extension of [13,
Theorem 2.29].

Theorem 2.22. Suppose that the functions Fh : I × X → Y , F0 : I × X → Y ,
Gh : I × Y → Zand G0 : I × Y → Z are continuous, F = Fh + F0, G = Gh + G0

as well as ∅ 6= I ′ ⊆ I ⊆ Rn and (2.7) holds with the functions F (·; ·) and G(·; ·)
replaced therein with the functions Fh(·; ·) and Gh(·; ·), respectively.

(i) Suppose further that, for every B ∈ B and ε > 0, there exists l > 0 such that
for each t0 ∈ I ′ there exists τ ∈ B(t0, l)∩ I ′ such that (2.1) holds with the function
F (·; ·) replaced with the function Fh(·; ·) and (2.8) holds with the functions F (·; ·)
and G(·; ·) replaced therein with the functions Fh(·; ·) and Gh(·; ·), respectively. If
F0 ∈ C0,D,B(I ×X : Y ) and for each B ∈ B we have limt∈D,|t|→+∞G0(t;F (t;x)) =
0, uniformly for x ∈ B, then the function W (·; ·), given by (2.6), is D-asymptotically
Bohr (B, I ′, c)-almost periodic.

(ii) Suppose further that, for every B ∈ B, there exists a sequence (τ k) in I ′ such
that limk→+∞ |τ k| = +∞, (2.2) holds and (2.9) holds with the functions F (·; ·) and
G(·; ·) replaced therein with the functions Fh(·; ·) and Gh(·; ·), respectively. If F0 ∈
C0,D,B(I × X : Y ) and for each B ∈ B we have limt∈D,|t|→+∞G0(t;F (t;x)) = 0,
uniformly for x ∈ B, then the function W (·; ·), given by (2.6), is (B, I ′, c)-uniformly
recurrent.

Proof. We will outline all details of the proof of (i) for the sake of completeness.
Clearly, the following decomposition holds true:

G(·;F (·; ·)) = Gh
(
·;Fh(·; ·)

)
+ [Gh(·;F (·; ·))−Gh

(
·;Fh(·; ·)

)
] +G0(·;F (·; ·)).

From Theorem 2.19, we have that the function Gh(·;Fh(·; ·)) is Bohr (B, I ′, c)-
almost periodic. Furthermore, the prescribed assumption implies that the function
G0(·;F (·; ·)) belongs to the space C0,D,B(I ×X : Y ). This also holds for the func-
tion Gh(·;F (·; ·)) − Gh(·;Fh(·; ·)) since the function Gh(·; ·) satisfies the Lipschitz
condition with respect to the first variable and F0 ∈ C0,D,B(I ×X : Y ). �

Set, for brevity, It := (−∞, t1] × (−∞, t2] × · · · × (−∞, tn] and Dt := It ∩
D for any t = (t1, t2, . . . , tn) ∈ Rn. Concerning the convolution invariance of



14 M. KOSTIĆ EJDE-2022/45

strong D-asymptotical c-almost periodicity under the actions of finite convolution
products, we will formulate the following result (the proof is similar to the proof of
corresponding result from [7] and therefore omitted).

Proposition 2.23. Suppose that (R(t))t>0 ⊆ L(X,Y ) is a strongly continuous
operator family such that

∫
(0,∞)n

‖R(t)‖ dt < ∞. If f : I → X is strongly D-

asymptotically c-almost periodic,

lim
|t|→∞,t∈D

∫
It∩Dc

‖R(t− s)‖ ds = 0

and for each r > 0 we have

lim
|t|→∞,t∈D

∫
Dt∩B(0,r)

‖R(t− s)‖ ds = 0,

then the function

F (t) :=

∫
Dt

R(t− s)f(s) ds, t ∈ I

is strongly D-asymptotically c-almost periodic.

Assuming that D = [α1,∞) × [α2,∞) × · · · × [αn,∞) for some real numbers
α1, α2, . . . , αn, then Dt = [α1, t1] × [α2, t2] × · · · × [αn, tn]. In this case, the
function F (t) =

∫ α

t
R(t − s)f(s) ds, t ∈ I is strongly D-asymptotically c-almost

periodic, where we accept the notation∫ α

t

· =
∫ t1

α1

∫ t2

α2

· · ·
∫ tn

αn

.

Although clarified, we feel it is our duty to emphasize that our results concerning
the invariance of multi-dimensional c-almost periodicity are not so easily applicable
as the corresponding results known in the one-dimensional case, unfortunately. This
is a very unexplored theme which will be further analyzed somewhere else.

Let F (·; ·) be I-asymptotically c-uniformly recurrent, G : I × X → Y , Q ∈
C0,I,B(I ×X : Y ) and F (t;x) = G(t;x) + Q(t;x) for all t ∈ I and x ∈ X. Then,
for every x ∈ X, we have

c
{
G(t;x) : t ∈ I, x ∈ X

}
⊆
{
F (t;x) : t ∈ I, x ∈ X

}
.

The following proposition can be deduced as in the case that c = 1.

Proposition 2.24. (i) Suppose that for each integer j ∈ N the function Fj(·; ·) is
Bohr (B, c)-almost periodic ((B, c)-uniformly recurrent). If for each B ∈ B there
exists εB > 0 such that the sequence (Fj(·; ·)) converges uniformly to a function
F (·; ·) on the set B◦∪∪x∈∂BB(x, εB), then the function F (·; ·) is Bohr (B, c)-almost
periodic ((B, c)-uniformly recurrent).

(ii) Suppose that for each integer j ∈ N the function Fj(·; ·) = Gj(·; ·) +Qj(·; ·) is I-
asymptotically Bohr (B, c)-almost periodic (I-asymptotically (B, c)-uniformly recur-
rent), where Gj(·; ·) is Bohr (B, c)-almost periodic ((B, c)-uniformly recurrent) and
Qj ∈ C0,I,B(I×X : Y ). If for each B ∈ B there exists εB > 0 such that the sequence
(Fj(·; ·)) converges uniformly to a function F (·; ·) on the set B◦ ∪ ∪x∈∂BB(x, εB),
and if for each natural numbers m, k ∈ N the function Gk(·; ·) − Gm(·; ·) is Bohr
(B, c)-almost periodic ((B, c)-uniformly recurrent), then the function F (·; ·) is I-
asymptotically Bohr (B, c)-almost periodic (I-asymptotically (B, c)-uniformly recur-
rent).



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 15

Now we will introduce the following definition (for any set Λ ⊆ Rn and number
M > 0, we define ΛM := {λ ∈ Λ ; |λ| ≥M}).

Definition 2.25. Suppose that D ⊆ I ⊆ Rn and the set D is unbounded, as well
as ∅ 6= I ′ ⊆ I ⊆ Rn, F : I ×X → Y is a continuous function and I + I ′ ⊆ I. Then
we say that:

(i) F (·; ·) is D-asymptotically Bohr (B, I ′, c)-almost periodic of type 1 if and
only if for every B ∈ B and ε > 0 there exist l > 0 and M > 0 such that
for each t0 ∈ I ′ there exists τ ∈ B(t0, l) ∩ I ′ such that

‖F (t + τ ;x)− cF (t;x)‖Y ≤ ε, provided t, t + τ ∈ DM , x ∈ B. (2.10)

(ii) F (·; ·) is D-asymptotically (B, I ′, c)-uniformly recurrent of type 1 if and only
if for every B ∈ B there exist a sequence (τ k) in I ′ and a sequence (Mk) in
(0,∞) such that limk→+∞ |τ k| = limk→+∞Mk = +∞ and

lim
k→+∞

sup
t,t+τk∈DMk

;x∈B
‖F (t + τ k;x)− cF (t;x)‖Y = 0.

If I ′ = I, then we also say that F (·; ·) is D-asymptotically Bohr (B, c)-almost peri-
odic of type 1 (D-asymptotically (B, c)-uniformly recurrent of type 1); furthermore,
if X ∈ B, then it is also said that F (·; ·) is D-asymptotically Bohr (I ′, c)-almost
periodic of type 1 (D-asymptotically (I ′, c)-uniformly recurrent of type 1). If I ′ = I
and X ∈ B, then we also say that F (·; ·) is D-asymptotically Bohr c-almost periodic
of type 1 (D-asymptotically c-uniformly recurrent of type 1). As before, we remove
the prefix “D-” in the case that D = I and remove the prefix “(B, )” in the case
that X ∈ B.

Clearly, we have the following result.

Proposition 2.26. Suppose that D ⊆ I ⊆ Rn and the set D is unbounded, as well
as ∅ 6= I ′ ⊆ I ⊆ Rn, F : I × X → Y is a continuous function and I + I ′ ⊆ I.
If F (·; ·) is D-asymptotically Bohr (B, I ′, c)-almost periodic, resp. D-asymptotically
(B, I ′, c)-uniformly recurrent, then F (·; ·) is D-asymptotically Bohr (B, I ′, c)-almost
periodic of type 1, resp. D-asymptotically (B, I ′, c)-uniformly recurrent of type 1.

Concerning the converse of Proposition 2.26, we will state and prove the following
statement which can be applied in the case that I = [0,∞)n.

Theorem 2.27. Suppose that ∅ 6= I ⊆ Rn, I + I = I, I is closed and F : I → Y is
a uniformly continuous, bounded I-asymptotically Bohr c-almost periodic function
of type 1, where |c| = 1. If

(∀l > 0) (∀M > 0) (∃t0 ∈ I) (∃k > 0) (∀t ∈ IM+l)(∃t′0 ∈ I)

(∀t′′0 ∈ B(t′0, l) ∩ I) t− t′′0 ∈ B(t0, kl) ∩ IM ,

there exists L > 0 such that IkL \ I(k+1)L 6= ∅ for all k ∈ N and IM + I ⊆ IM for
all M > 0, then the function F (·) is I-asymptotically Bohr c-almost periodic.

Proof. Since we have assumed that the function F (·) is bounded and |c| = 1,
we can use the foregoing arguments in order to see that the function F (·) is I-
asymptotically Bohr almost periodic function of type 1. By [7, Theorem 2.34], it
follows that for each sequence (bk) in I there exist a subsequence (bkl) of (bk) and
a function F ∗ : I → Y such that liml→+∞ F (t + bkl) = F ∗(t), uniformly in t ∈ I.
We continue the proof by observing that for each integer k ∈ N there exist lk > 0



16 M. KOSTIĆ EJDE-2022/45

and Mk > 0 such that for each t0 ∈ I there exists τ ∈ B(t0, l) ∩ I such that (2.10)
holds with c = 1, ε = 1/k and D = I. Let τ k be any fixed element of I such that
|τ k| > Mk + k2 and (2.10) holds with c = 1, ε = 1/k and D = I (k ∈ N). Then
there exist of a subsequence (τ kl) of (τ k) and a function F ∗ : I → Y such that

lim
l→+∞

F (t + τ kl) = F ∗(t), uniformly for t ∈ I. (2.11)

The mapping F ∗(·) is clearly continuous and now we will prove that F ∗(·) is Bohr
c-almost periodic. Let ε > 0 be fixed, and let l > 0 and M > 0 be such that for
each t0 ∈ I there exists τ ∈ B(t0, l) ∩ I such that (2.10) holds with D = I and the
number ε replaced therein by ε/3. Let t ∈ I be fixed, and let l0 ∈ N be such that
|t + τ kl0 | ≥M and |t + τ + τ kl0 | ≥M . Then

‖F ∗(t + τ )− cF ∗(t)‖
≤ ‖F ∗(t + τ )− F

(
t + τ + τ kl0

)
‖+ ‖F

(
t + τ + τ kl0

)
− cF

(
t + τ kl0

)
‖

+ ‖cF
(
t + τ kl0

)
− cF ∗(t)‖

≤ 3 · (ε/3) = ε,

as required. The function t 7→ F (t)−F ∗(t), t ∈ I belongs to the space C0,I(I : Y )
due to (2.11) and the fact that F : I → Y is an I-asymptotically Bohr almost
periodic function of type 1, which completes the proof. �

For any set S ⊆ Rn and for any integer l ∈ N, we define the set Sl inductively
by S1 := S and Sl+1 := Sl + S (l = 1, 2, . . . ). Further on, we define Ω := I ′ and
ΩS := I ′ ∪ S if arg(c)/π /∈ Q. If arg(c)/π ∈ Q, then we take any non-empty finite
set of integers S1 ⊆ Z \ {0} such that cm+1 = 1 for all m ∈ S1 and any non-empty
finite set of integers S2 ⊆ N such that cl = 1 for all l ∈ S2; in this case, we set
Ω: = (I ′ ∪m∈S1

(−mI ′))l and ΩS := Ω ∪ S.
Now we are able to state and prove the following result concerning the extensions

of Bohr (I ′, c)-almost periodic functions and (I ′, c)-uniformly recurrent functions
(in [13, Proposition 2.25], we have obeyed a different approach, where we have also
considered semi-c-periodicity but not c-uniform recurrence):

Theorem 2.28. Suppose that I ′ ⊆ I ⊆ Rn, I+I ′ ⊆ I, the set I ′ is unbounded, |c| =
1, F : I → Y is a uniformly continuous, Bohr (I ′, c)-almost periodic function, resp.
a uniformly continuous, (I ′, c)-uniformly recurrent function, S ⊆ Rn is bounded
and the following condition holds:

(AP-E) For every t′ ∈ Rn, there exists a finite real number M > 0 such that
t′ + I ′M ⊆ I.

Then there exists a uniformly continuous, Bohr (ΩS , c)-almost periodic, resp. a

uniformly continuous, (ΩS , c)-uniformly recurrent, function F̃ : Rn → Y such that

F̃ (t) = F (t) for all t ∈ I; furthermore, in c-almost periodic case, the uniqueness

of such a function F̃ (·) holds provided that Rn \ ΩS is a bounded set.

Proof. We will consider only uniformly continuous, Bohr (I ′, c)-almost periodic
functions. In this case, for each natural number k ∈ N there exists a point τk ∈ I ′
such that ‖F (t+τ k)−cF (t)‖Y ≤ 1/k for all t ∈ I and k ∈ N; furthermore, since the
set I ′ is unbounded, we may assume without loss of generality that limk→+∞ |τ k| =
+∞. Hence, we have

lim
k→+∞

F (t + τ k) = cF (t), uniformly for t ∈ I. (2.12)



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 17

If t′ ∈ Rn, then there exists a finite real number M > 0 such that t′ + I ′M ⊆ I,
and now we will prove that the sequence (F (t′ + τ k))k∈N is Cauchy and therefore
convergent. Let ε > 0 be fixed; then we have the existence of a number k0 ∈ N such
that t′ + τ k ∈ I for all k ≥ k0. Suppose that k, m ≥ k0. Then we have

‖F (t′ + τ k)− F (t′ + τm)‖
≤ ‖F (t′ + τ k)− c−1F (t′ + τ k + τ)‖

+ ‖c−1F (t′ + τ k + τ)− c−1F (t′ + τm + τ)‖
+ ‖c−1F (t′ + τm + τ)− F (t′ + τm)‖,

for any τ ∈ I ′ such that t′ + τ ∈ I. Since the function F (·) is Bohr (I ′, c)-almost
periodic, we can always find such a number τ so that the first and the third addend
in the above estimates are less or equal than ε/3; for the second addend in the
above estimate, we can find a sufficiently large number k1 ≥ k0 such that

‖c−1F (t′ + τ k + τ)− c−1F (t′ + τm + τ)‖ < ε/3,

for all k, m ≥ k1 (see (2.12)). Therefore, limk→+∞ F (t′+ τ k) := F̃ (t′) exists. The

function F̃ (·) is clearly uniformly continuous because F (·) is uniformly continuous;

furthermore, by construction, we have that F̃ (t)/c = F (t) for all t ∈ I. Now we

will prove that the function F̃ (·) is Bohr (ΩS , c)-almost periodic. Let a number
ε > 0 be given. Then there exists l > 0 such that for each t0 ∈ I ′ there exists
τ ∈ B(t0, l)∩ I ′ such that ‖F (t + τ )− cF (t)‖Y ≤ ε/2 for all t ∈ I. Let t′ ∈ Rn be
fixed. For any such numbers t0 ∈ I ′ and τ ∈ B(t0, l) ∩ I ′, we have

‖F̃ (t′ + τ )− cF̃ (t′)‖Y
= ‖ lim

k→+∞

[
F (t′ + τ + τ k)− cF (t′ + τ k)

]
‖Y

≤ lim sup
k→+∞

‖F (t′ + τ + τ k)− cF (t′ + τ k)‖Y

≤ ε/2, t′ ∈ Rn.

(2.13)

If arg(c)/π /∈ Q, this clearly implies that F (·) is Bohr (Ω, c)-almost periodic and
therefore Bohr (ΩS , c)-almost periodic. If arg(c)/π ∈ Q, then we may assume
without loss of generality that the sets S1 = {m} and S2 = {l} are singletons
(this follows from the corresponding definition of Bohr (I ′, c)-almost periodicity).
Given ε > 0 in advance, we may assume that (2.13) holds with the number ε/2
replaced therein with the number ε/l|m|. By (2.3), we obtain that the number
−mτ ∈ Ω is an (ε/l, c)-period of F (·), with the meaning clear. Arguing as in the
proof of the estimate (2.3), it readily follows that any finite sum τ1 + · · ·+ τl, where
τi ∈ I ′∪m∈S1 (−mI ′) for all i ∈ Nl, is an (ε, c)-period of F (·). As above, this implies
that F (·) is Bohr (Ω, c)-almost periodic and therefore Bohr (ΩS , c)-almost periodic.

Assume, finally, that the set Rn \ΩS is bounded. Then the function F̃ (·) is Bohr
c-almost periodic and bounded by Proposition 2.2; by the foregoing, this implies
that the function F (·) is Bohr almost periodic and therefore compactly almost
automorphic. Then we can proceed as in the final part of the proof of [13, Theorem
2.36] to prove the uniqueness of extension in c-almost periodic case. �

Remark 2.29. (i) It is clear that Theorem 2.28 strengthens [7, Theorem 2.36],
where we have assumed that c = 1 and ΩS = [(I ′ ∪ (−I ′)) + (I ′ ∪ (−I ′))] ∪ S.



18 M. KOSTIĆ EJDE-2022/45

(ii) In the case that arg(c)/π /∈ Q, it is not clear whether there exists a set Ω′S ⊇ ΩS
such that the constructed function F̃ : Rn → Y is Bohr (Ω′S , c)-almost periodic.
Concerning this problematic, it is worth noting that the notion introduced in Defi-
nition 2.6 can be further extended by allowing that the set I ′ depends on the set B
and the number ε > 0. This could probably fix some things here, but we will skip
all related details for the sake of brevity.

Before proceeding further, we would like to propose the following definition.

Definition 2.30. Suppose that ∅ 6= I ⊆ Rn and I + I ⊆ I. Then we say that
I is admissible with respect to the c-almost periodic extensions if and only if for
any complex Banach space Y and for any uniformly continuous, Bohr c-almost
periodic function F : I → Y there exists a unique Bohr c-almost periodic function
F̃ : Rn → Y such that F̃ (t) = F (t) for all t ∈ I. If c = ±1, then we also say that
the region I is admissible with respect to the almost (anti-)periodic extensions.

If |c| = 1, arg(c)/π ∈ Q, (v1, . . . ,vn) is a basis of Rn and

I ′ = I =
{
α1v1 + · · ·+ αnvn : αi ≥ 0 for all i ∈ Nn

}
is a convex polyhedral in Rn, then ΩS = Rn and therefore the set I is admissible
with respect to the c-almost periodic extensions. It is very simple to construct some
sets which are not admissible with respect to the c-almost periodic extensions; for
example, the set I = [0,∞) × {0} ⊆ R2 is not admissible with respect to the
c-almost periodic extensions since there is no c-almost periodic extension of the
function F (x, y) = y, (x, y) ∈ I to the whole Euclidean space [13].

3. Examples and applications

In this section, we will present several interesting examples and applications of
our abstract theoretical results. The first and second application have recently been
considered in [7], with c = 1:

1. Let Y be one of the spaces Lp(Rn), C0(Rn) or BUC(Rn), where 1 ≤ p <∞.
Then the Gaussian semigroup

(G(t)F )(x) :=
(
4πt
)−(n/2)

∫
Rn

F (x− y)e−
|y|2
4t dy, t > 0, f ∈ Y, x ∈ Rn,

can be extended to a bounded analytic C0-semigroup of angle π/2, generated by the
Laplacian ∆Y acting with its maximal distributional domain in Y ; see [4, Example
3.7.6]. Suppose that ∅ 6= I ′ ⊆ I = Rn and F (·) is bounded Bohr (B, I ′, c)-almost
periodic, resp. bounded (B, I ′, c)-uniformly recurrent. Then for each t0 > 0 the
function Rn 3 x 7→ u(x, t0) ≡ (G(t0)F )(x) ∈ C is likewise bounded Bohr (B, I ′, c)-
almost periodic, resp. bounded (B, I ′, c)-uniformly recurrent. Towards this end,
observe that for each x, τ ∈ Rn we have:∣∣u(x+ τ, t0

)
− cu

(
x, t0

)∣∣ ≤ (4πt0)−(n/2)
∫
Rn

|F (x− y + τ)− cF (x− y)|e−
|y|2
4t0 dy.

see also Proposition 2.17. We can similarly clarify the corresponding results for the
Poisson semigroup, which is analyzed in [4, Example 3.7.9].

2. Define

E1(x, t) :=
(
πt
)−1/2

∫ x

0

e−y
2/4t dy, x ∈ R, t > 0



EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 19

and I := {(x, t) : x > 0, t > 0}. Recall that F. Trèves [21, p. 433] has proposed
the formula

u(x, t) =
1

2

∫ x

−x

∂E1

∂y
(y, t)u0(x− y) dy −

∫ t

0

∂E1

∂t
(x, t− s)g(s) ds, x > 0, t > 0,

(3.1)
for the solution of the mixed initial value problem

ut(x, t) = uxx(x, t), x > 0, t > 0;

u(x, 0) = u0(x), x > 0, u(0, t) = g(t), t > 0.
(3.2)

Consider the case in which g(t) ≡ 0. Suppose that 0 < T <∞ and the function u0 :
[0,∞)→ C is bounded Bohr (I0, c)-almost periodic, resp. bounded (I0, c)-uniformly
recurrent, for a certain non-empty subset I0 of [0,∞). Set I ′ := I0 × (0, T ). If D is
any unbounded subset of I which has the property that

lim
|(x,t)|→+∞,(x,t)∈D

min
( x2

4(t+ T )
, t
)

= +∞,

then the solution u(x, t) of (3.2) is D-asymptotically (I ′, c)-almost periodic of type
1, resp. D-asymptotically (I ′, c)-uniformly recurrent of type 1. This can be achieved
by a careful inspection of the argumentation given in [7, Section 3, point 2.].

3. (cf. also [13, Theorem 3.1]) Let (τk) be a sequence in Rn, limk→+∞ |τk| = +∞
and

BUR(τk);c(Rn : X) :=
{
F : Rn → X is bounded, continuous and

lim
k→+∞

sup
t∈R
‖F (t+ τk)− cf(t)‖∞ = 0

}
.

Equipped with the metric d(·, ·) := ‖·−·‖∞, BUR(τk);c(Rn : X) becomes a complete
metric space. Define I ′ := {τk : k ∈ N} and consider the following Hammerstein
integral equation of convolution type on Rn (see e.g., [8, Section 4.3, pp. 170-180]):

y(t) =

∫
Rn

k(t− s)G(s, y(s)) ds, t ∈ Rn, (3.3)

where G : Rn×X → X is (B, I ′, c)-uniformly recurrent with B being the collection
of all bounded subsets of X. Suppose, further, that the set {G(t, B) : t ∈ Rn} is
bounded for any bounded subset B of X as well as that there exists a finite real
constant L > 0 such that (2.7) holds with X = Z = Y , for every y, y′ ∈ Rn,
and (2.9) holds with the term F (t;x) replaced with the term y(t) for any function
y ∈ BUC(τk);c(Rn : X). Applying Proposition 2.17 and Theorem 2.19(ii), we obtain
that the mapping

BUR(τk);c(Rn : X) 3 y 7→
∫
Rn

k(· − s)G(s, y(s)) ds ∈ BUR(τk);c(Rn : X)

is well defined. If we additionally assume that L
∫
Rn |k(t)| dt < 1, then an applica-

tion of the Banach contraction principle shows that there exists a unique solution
of (3.3) which belongs to the space BUR(τk);c(Rn : X).

We can similarly analyze the integral equation

y(t) =

∫
Rn

G(t, s, y(s)) ds, t ∈ Rn,



20 M. KOSTIĆ EJDE-2022/45

provided that G : R2n × X → X satisfies certain assumptions and there exists a
constant L ∈ (0, 1) such that

‖G(t, s, x)−G(t, s, y)‖ ≤ L‖x− y‖, t, s ∈ Rn; x, y ∈ X.

Details can be left to the interested readers.
We close the paper with the observation that the class of multi-dimensional

(ω, c)-periodic type functions will be considered in our forthcoming paper [16].

Acknowledgements. The author was partially supported by grant 451-03-68/
2020/14/200156 of Ministry of Science and Technological Development, Republic of
Serbia. The author would like to express his sincere thanks to Prof. A. Chávez, M.
T. Khalladi, M. Pinto, A. Rahmani and D. Velinov for many stimulating discussions
during this investigation. The research articles [7] and [13], among many others,
are written in a collaboration with these mathematicians.

References

[1] E. Alvarez, A. Gómez, M. Pinto; (ω, c)-Periodic functions and mild solution to abstract
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1–8.

[2] E. Alvarez, S. Castillo, M. Pinto; (ω, c)-Pseudo periodic functioins, first order Cauchy prob-
lem and Lasota-Wazewska model with ergodic and unbounded oscillating production of red

cells, Bound. Value Probl. 106 (2019), 1–20.

[3] E. Alvarez, S. Castillo, M. Pinto; (ω, c)-Asymptotically periodic functions, first-order Cauchy
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Math. Methods Appl. Sci. 43 (2020), 305–319.
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and Cauchy Problems, Birkhäuser/Springer Basel AG, Basel, 2001.

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H6 (1925), 101–214; HT (1926), 237–281.

[7] A. Chávez, K. Khalil, M. Kostić, M. Pinto; Multi-dimensional almost periodic type functions
and applications, preprint. arXiv:2012.00543.

[8] C. Corduneanu; Integral Equations and Applications, Cambridge University Press, Cam-

bridge, 1991.
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and abstract integral equations, J. Math. Anal. Appl. 435 (2016), 1382–1399.
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Spaces, Springer-Verlag, New York, 2013.

[11] A. M. Fink; Almost Periodic Differential Equations, Springer-Verlag, Berlin, 1974.

[12] A. Haraux, P. Souplet; An example of uniformly recurrent function which is not almost
periodic, J. Fourier Anal. Appl. 10 (2004), 217–220.

[13] M. T. Khalladi, M. Kostić, A. Rahmani, M. Pinto, D.Velinov; c-Almost periodic type functions
and applications, Nonauton. Dyn. Syst. 7 (2020), 176–193.

[14] M. Kostić; Almost Periodic and Almost Automorphic Type Solutions to Integro-Differential

Equations, W. de Gruyter, Berlin, 2019.
[15] M. Kostić; Selected Topics in Almost Periodicity, W. de Gruyter, Berlin, 2022.

[16] M. Kostić; Multi-dimensional (ω, c)-periodic type functions, Nonauton. Dyn. Syst. 8 (2021),

136–151.
[17] M. Levitan; Almost Periodic Functions, G.I.T.T.L., Moscow, 1953 (in Russian).

[18] B. M. Levitan, V. V. Zhikov; Almost Periodic Functions and Differential Equations, Cam-

bridge Univ. Press, London, 1982.
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Kluwer Acad. Publ, Dordrecht, 2001.

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EJDE-2022/45 MULTI-DIMENSIONAL c-ALMOST PERIODIC TYPE FUNCTIONS 21

[21] F. Trèves; Basic Linear Partial Differential Equations, Dover Publications, New York, 2006

(republication of work published by Academic Press, New York, 1975).

[22] S. Zaidman; Almost-Periodic Functions in Abstract Spaces, Pitman Research Notes in Math,
Vol. 126, Pitman, Boston, 1985.

Marko Kostić

Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovića 6, 21125

Novi Sad, Serbia
Email address: marco.s@verat.net


	1. Introduction and preliminaries
	Notation and terminology
	1.1. Almost periodic functions on Rn

	2. Bohr (B,c)-almost periodic type functions
	2.1. D-asymptotically (B,c)-almost periodic type functions

	3. Examples and applications
	Acknowledgements

	References