Citation: Kosti´ c, M.; Chaouchi, B.;
Du, W.-S.; Velinov,D. Generalized
ρ-Almost Periodic Sequences and
Applications. Fractal Fract. 2023, 7,
410. https://doi.org/10.3390/
fractalfract7050410
Academic Editor: Gani Stamov
Received: 25 April 2023
Revised: 14 May 2023
Accepted: 16 May 2023
Published: 18 May 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
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4.0/).
fractal and fractional
Article
Generalized ρ-Almost Periodic Sequences and Applications
Marko Kosti´ c
1
, Belkacem Chaouchi
2
, Wei-Shih Du
3,
* and Daniel Velinov
4
1
Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovi´ ca 6, 21125 Novi Sad, Serbia;
marco.s@verat.net
2
Laboratory de l’Energie et des Systèmes Intelligents, Khemis Miliana University,
Khemis Miliana 44225, Algeria; chaouchicukm@gmail.com
3
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 82444, Taiwan
4
Department for Mathematics and Informatics, Faculty of Civil Engineering, Ss. Cyril and Methodius
University in Skopje, Partizanski Odredi 24, P.O. Box 560, 1000 Skopje, North Macedonia;
velinovd@gf.ukim.edu.mk
* Correspondence: wsdu@mail.nknu.edu.tw
Abstract: In this paper, we analyze the Bohr ρ-almost periodic type sequences and the generalized
ρ-almost periodic type sequences of the form F : I × X → Y, where ∅ = I ⊆ Z
n
, X and Y are
complex Banach spaces and ρ is a general binary relation on Y. We provide many structural results,
observations and open problems about the introduced classes of ρ-almost periodic sequences. Certain
applications of the established theoretical results to the abstract Volterra integro-difference equations
are also given.
Keywords: generalized ρ-almost periodic sequence; generalized ρ-almost periodic function;
abstract Volterra integro-difference equation; abstract impulsive Volterra integro-differential equation;
Banach space
MSC: 42A75; 43A60; 47D99
1. Introduction and Preliminaries
The notion of almost periodicity was introduced by the Danish mathematician H.
Bohr around 1924–1926 and later generalized by many others. Let I =[0, ∞) or I = R, let
( X, ‖·‖) be a complex Banach space and let f : I → X be a continuous function. Given
ǫ > 0, we call τ > 0 an ǫ-period for f (·) if and only if ‖ f (t + τ) − f (t)‖≤ ǫ, t ∈ I ; the
set of all ǫ-periods for f (·) is denoted by ϑ( f , ǫ). It is said that f (·) is almost periodic if
and only if for each ǫ > 0 the set ϑ( f , ǫ) is relatively dense in I , which means that there
exists l > 0 such that any subinterval of I of length l meets ϑ( f , ǫ). For further information
concerning almost periodic functions and their applications, the interested reader may
consult the research monographs [1–9].
An X-valued sequence ( x
k
)
k∈Z
[( x
k
)
k∈N
] is called (Bohr) almost periodic if and only if,
for every ǫ > 0, there exists a natural number K
0
(ǫ) such that among any K
0
(ǫ) consecutive
integers in Z [N], there exists at least one integer τ ∈ Z [τ ∈ N] satisfying that
x
k+τ
− x
k
≤ ǫ, k ∈ Z [k ∈ N];
as in the case of functions, this number is said to be an ǫ-period of sequence ( x
k
). Any
almost periodic X-valued sequence is bounded and its range is relatively compact in X. The
equivalent concept of Bochner almost periodicity of X-valued sequences can be introduced
as well; see, e.g., ([10] Theorem 70, pp. 185–186) and ([10] Theorems 71–73, pp. 186–188).
It is well known that a sequence ( x
k
)
k∈Z
in X is almost periodic if and only if there exists
an almost periodic function f : R → X such that x
k
= f (k) for all k ∈ Z; see, e.g., the
proof of ([11] Theorem 2) given in the scalar-valued case. It is not difficult to prove that,
Fractal Fract. 2023, 7, 410. https://doi.org/10.3390/fractalfract7050410 https://www.mdpi.com/journal/fractalfract