Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations
UDC 517.9 Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions  in Stepanov's sense.  In addition, we present numerous interesting results, such as the composition theorems and completene...
Gespeichert in:
| Datum: | 2022 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2022
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/6315 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512327789445120 |
|---|---|
| author | Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. |
| author_facet | Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. |
| author_sort | Rebey, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-01-07T13:45:39Z |
| description |
UDC 517.9
Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions  in Stepanov's sense.  In addition, we present numerous interesting results, such as the composition theorems and completeness properties for these two  spaces of the considered functions.  We also establish the  existence and  uniqueness for the  double-measure pseudo almost periodic mild solutions  in Stepanov's sense for some evolution equations. |
| doi_str_mv | 10.37863/umzh.v74i10.6315 |
| first_indexed | 2026-03-24T03:27:02Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i10.6315
UDC 517.9
A. Rebey (Business Administration Department, College of Business Administration, Majmaah Univ., Saudi Arabia and
ISMAIK, Kairouan Univ., Tunisia),
H. Ben-Elmonser (Mathematics Department, College of Science Al-Zulfi, Majmaah Univ., Saudi Arabia),
M. Eljeri (IPEIM, Monastir Univ., Tunisia),
M. Miraoui1 (IPEIK, Kairouan Univ., Tunisia)
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM
AND STEPANOV’S SENSE FOR SOME EVOLUTION EQUATIONS
ПСЕВДО-МАЙЖЕ ПЕРIОДИЧНI РОЗВ’ЯЗКИ В АЛЬФА-НОРМI
ТА В РОЗУМIННI СТЕПАНОВА ДЛЯ ДЕЯКИХ ЕВОЛЮЦIЙНИХ РIВНЯНЬ
Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions in
Stepanov’s sense. In addition, we present numerous interesting results, such as the composition theorems and completeness
properties for these two spaces of the considered functions. We also establish the existence and uniqueness for the double-
measure pseudo almost periodic mild solutions in Stepanov’s sense for some evolution equations.
Введено поняття ергодичних функцiй подвiйної мiри та псевдо-майже перiодичних функцiй подвiйної мiри в ро-
зумiннi Степанова. Крiм того, наведено багато цiкавих результатiв, що включають як теореми про композицiю,
так i властивостi повноти для цих двох просторiв розглянутих функцiй. Встановлено також iснування та єди-
нiсть псевдо-майже перiодичних слабких розв’язкiв подвiйної мiри в розумiннi Степанова для деяких еволюцiйних
рiвнянь.
1. Introduction and preliminaries. Roughly by 1924 – 1926, the Danish mathematician Bohr
[3] pioneered the almost periodic functions theory that generalize the notion of periodicity, the so-
called almost periodicity is very useful in distinct domains involving harmonic analysis, dynamical
systems, physics, etc. C. Zhang [10, 11] defined the concept of pseudo almost periodicity, as a natural
generalization of the notion of almost periodicity. Lately, T. Diagana [5] initiated the concept of
Stepanov pseudo almost periodicity as a generalization of pseudo almost periodicity.
In this paper, we prove the existence and uniqueness of double-measure pseudo almost periodic
(or (\mu , \nu )-PAP) solutions in Stepanov’s sense for the equation
d
dt
\bigl[
v(t) - g(t, v(t))
\bigr]
= - A
\bigl[
v(t) - g(t, v(t))
\bigr]
+ f(t, v(t)) for t \geq \sigma ,
v(\sigma ) = v\sigma \in X\alpha ,
(1)
such that - A is the generator of a semigroup (T (t))t\geq 0 on a Banach space (X, \| .\| ). For 0 < \alpha < 1,
the domain of the operator A\alpha is denoted by D(A\alpha ) and the Banach space X\alpha := (D(A\alpha ), \| .\| \alpha ),
where \| x\| \alpha = \| A\alpha x\| . Furthermore, we suppose that
(H0) \exists M\alpha , \omega > 0 satisfying
\| A\alpha T (t)\| \leq M\alpha
e - \omega t
t\alpha
for t > 0.
1 Corresponding author, e-mail: miraoui.mohsen@yahoo.fr.
c\bigcirc A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI, 2022
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10 1401
1402 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
For more details on the operator A\alpha we can see [4, 9].
The research aim in this paper is to study basic properties of pseudo almost periodic functions
with measure in Stepanov’s sense, some composition theorems and their extensions. Specifically,
the notions of (\mu , \nu ) pseudo almost periodicity in Stepanov’s sense and the double-measure pseudo
almost periodic mild solutions in Stepanov’s sense for equation (1) will be introduced.
Throughout this paper (X, \| .\| ), (Y, \| .\| ) be two Banach spaces. Denote by \scrB the Lebesgue
\sigma -field of \BbbR and by \scrM the set of all positive measures \mu on \scrB with \mu (\BbbR ) = +\infty and \mu ([a, b]) < \infty
\forall a, b \in \BbbR , a \leq b.
We consider the following hypothesis taken from [2].
(M0) For \mu \nu \in \scrM ,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r - \rightarrow +\infty
\mu ([ - r, r])
\nu ([ - r, r])
:= M < \infty .
(M1) For all a, b, c \in \BbbR , with 0 \leq a < b \leq c, there exist \tau 0, \alpha 0 \geq 0 satisfying
| \tau | \geq \tau 0 =\Rightarrow \mu ((a+ \tau , b+ \tau )) \geq \alpha 0\mu ([\tau , c+ \tau ]).
(M2) For all \tau \in \BbbR , there exist \beta > 0 and I = (c, d) \subset \BbbR satisfying
\mu (\{ a+ \tau : a \in A\} ) \leq \beta \mu (A) with A \in \bfB and A \cap I = \varnothing .
Firstly, we recall some useful definitions:
1. AP (\BbbR , X) is the set of all continuous functions f \in X\BbbR such that \forall \varepsilon > 0 \exists l(\varepsilon ) > 0
\forall (a, b) \subset \BbbR : b - a < \varepsilon \Rightarrow \exists \tau \in (a, b) satisfying
\| f(t+ \tau ) - f(t)\| < \varepsilon \forall t \in \BbbR .
Such functions are named almost periodic.
2. \scrE (\BbbR , X, \mu , \nu ) is the space of f \in X\BbbR such that
\mathrm{l}\mathrm{i}\mathrm{m}
r - \rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\| f(s)\| d\mu (s) = 0,
where \mu and \nu are two positive measures. Such functions are named ergodic.
3. \scrE U(\BbbR \times Y,X, \mu , \nu ) is the set of all continuous functions f \in X\BbbR \times Y such that f(., y) \in
\in \scrE (\BbbR , X, \mu , \nu ) \forall y \in Y and f : y \mapsto - \rightarrow f(., y) is uniformly continuous on each compact K \subset Y,
where \mu and \nu are two positive measures. Such functions are named ergodic uniformly.
4. PAP (\BbbR , X, \mu , \nu ) = AP (\BbbR , X)+\scrE (\BbbR , X, \mu , \nu ) is the set of pseudo almost periodic functions.
5. Let f \in X\BbbR be a function. The function given by
f b : \BbbR \rightarrow Lp((0; 1);X),
t \mapsto \rightarrow f b(t) = f(t+ .)
is said to be a Bochner transform of f.
6. The boundedness in Stepanov’s sense of a function f \in Lp
\mathrm{l}\mathrm{o}\mathrm{c}(\BbbR , X), p \geq 1, is characterized
by
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1403
\mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
= \mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\left( 1\int
0
\| f(t+ s)\| pds
\right)
1
p
< \infty ,
which define a norm on the set BSp(\BbbR , X) of such function, i.e.,
\| f\| BSp = \mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
= \mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\| f(t+ .)\| p.
7. For f \in BSp(\BbbR , X), p \geq 1, we say that f is almost periodic in Stepanov’s sense (or Sp-
almost periodic), if f b \in AP (\BbbR , Lp((0, 1), X)). The set of such function is denoted by SpAP (\BbbR , X).
8. A function f \in X\BbbR \times Y , p \geq 1, where f(., u) \in BSp(\BbbR , X) \forall u \in Y, is said to be Sp-almost
periodic in t \in \BbbR uniformly for u \in Y if for all \varepsilon > 0 and K \subset Y, compact, there exists a relatively
dense set P = P (\varepsilon , f,K) satisfying
\mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\left( 1\int
0
\| f(t+ s+ \tau , u) - f(t+ s, u)\| pds
\right)
1
p
< \varepsilon for all t \in \BbbR , \tau \in P, u \in K.
The set of such functions is denoted by SpAPU(\BbbR \times Y,X).
The following interest results are useful in the sequel:
1. BSp(\BbbR ;X) is a Banach space under the norm \| .\| BSp (see [7, 8]).
2. (SpAP (\BbbR , X), \| .\| BSp) is a Banach space under the norm \| .\| BSp (see [8]).
2. Main results. 2.1. (\bfitmu , \bfitnu )-Ergodic functions in Stepanov’s sense. Motivited by Diagana et
al. in [6], we define the (\mu , \nu )-ergodic functions in Stepanov’s sense and give some properties. The
most important theorems and properties are contained in this subsection.
Definition 1. For f \in BSp(\BbbR ;X), we say that f is (\mu , \nu )-ergodic in Stepanov’s sense (or
Sp - (\mu , \nu )-ergodic) if f b \in \scrE (\BbbR , Lp((0; 1);X), \mu , \nu ), i.e.,
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) = 0,
where \mu , \nu \in \scrM and p \geq 1. The set of such functions is denoted \scrE p(\BbbR , X, \mu , \nu ).
Theorem 1. For p \geq 1 and \mu , \nu \in \scrM satisfy (M0), the space (\scrE p(\BbbR , X, \mu , \nu ), \| .\| BSp) is a
Banach space.
Proof. Let f, g \in \scrE p(\BbbR , X, \mu , \nu ) and \lambda \in \BbbC . It is obvious to see that \lambda f +g \in \scrE p(\BbbR , X, \mu , \nu ) \subset
\subset BSp(\BbbR , X). It is sufficient to show that (\scrE p(\BbbR , X, \mu , \nu ) is closed in BSp(\BbbR , X).
For (fn)n \subset \scrE p(\BbbR , X, \mu , \nu ) satisfying \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| fn - f\| BSp = 0. It follows that
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \leq
\int
[ - r,r]
\left( t+1\int
t
\| f(s) - fn(s)\| pds
\right)
1
p
d\mu (t) +
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1404 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
+
\int
[ - r,r]
\left( t+1\int
t
\| fn(s)\| pds
\right)
1
p
d\mu (t) \leq
\leq
\int
[ - r,r]
\| f - fn\| BSpd\mu (t) +
\int
[ - r,r]
\left( t+1\int
t
\| fn(s)\| pds
\right)
1
p
d\mu (t).
Then
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \leq \mu ([ - r, r])
\nu ([ - r, r])
\| f - fn\| BSp +
+
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| fn(s)\| pds
\right)
1
p
d\mu (t).
Therefore,
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \leq M\| f - fn\| BSp \forall n \in \BbbN .
Since \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow +\infty \| f - fn\| BSp = 0, one can see that
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) = 0.
Theorem 1 is proved.
Proposition 1. The space \scrE p(\BbbR , X, \mu , \nu ) is translation invariant, where \mu , \nu \in \scrM satisfy (M2)
and p \geq 1.
Proof. Suppose that f \in \scrE p(\BbbR , X, \mu , \nu ), we define F by F (t) =
\biggl( \int t+1
t
\| f(s)\| pds
\biggr) 1
p
. Then
F belongs to \scrE (\BbbR ,\BbbR , \mu , \nu ) and, from the translation invariance of \scrE (\BbbR ,\BbbR , \mu , \nu ), we have
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s+ a)\| pds
\right)
1
p
d\mu (t) =
=
1
\nu ([ - r, r])
\int
[ - r,r]
F (t+ a)d\mu (t) \rightarrow 0 as r \rightarrow +\infty .
Therefore, f(.+ a) \in \scrE p(\BbbR , X, \mu , \nu ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1405
Theorem 2. \scrE (\BbbR , X, \mu , \nu ) \subseteq \scrE p(\BbbR , X, \mu , \nu ), where \mu , \nu \in \scrM satisfy (M0) and (M2), p \geq 1.
Proof. Suppose that f \in \scrE (\BbbR , X, \mu , \nu ). By Hölder’s inequality, we get
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) =
\int
[ - r,r]
\left( 1\int
0
\| f(s+ t)\| pds
\right)
1
p
d\mu (t) \leq
\leq
\left( \int
[ - r,r]
d\mu (t)
\right)
1 - 1
p
\left( \int
[ - r,r]
\left( 1\int
0
\| f(s+ t)\| pds
\right) d\mu (t)
\right)
1
p
=
= (\mu ([ - r, r]))
1 - 1
p
\left( \int
[ - r,r]
\left( 1\int
0
\| f(s+ t)\| p - 1\| f(s+ t)\| ds
\right) d\mu (t)
\right)
1
p
\leq
\leq (\mu ([ - r, r]))
1 - 1
p \| f\|
p - 1
p
\infty
\left( \int
[ - r,r]
\left( 1\int
0
\| f(s+ t)\| ds
\right) d\mu (t)
\right)
1
p
.
Using the Fubini theorem, we have
\int
[ - r,r]
\left( 1\int
0
\| f(s+ t)\| ds
\right) d\mu (t) =
1\int
0
\left( \int
[ - r,r]
\| f(s+ t)\| d\mu (t)
\right) ds.
It follows that
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \leq
\leq 1
\nu ([ - r, r])
(\mu ([ - r, r]))
1 - 1
p \| f\|
p - 1
p
\infty
\left( 1\int
0
\left( \int
[ - r,r]
\| f(s+ t)\| d\mu (t)
\right) ds
\right)
1
p
=
=
\biggl(
\mu ([ - r, r])
\nu ([ - r, r])
\biggr) 1 - 1
p
\| f\|
p - 1
p
\infty
\left( 1\int
0
\left( 1
\nu ([ - r, r])
\int
[ - r,r]
\| f(s+ t)\| d\mu (t)
\right) ds
\right)
1
p
.
From translation invariance of \scrE (\BbbR , X, \mu , \nu ), we have
\mathrm{l}\mathrm{i}\mathrm{m}
r - \rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\| f(s+ t)\| d\mu (t) = 0 \forall s \in [0, 1].
Apply the Lebesgue dominated convergence, we get
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1406 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
\mathrm{l}\mathrm{i}\mathrm{m}
r - \rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) = 0.
Theorem 2 is proved.
Example 1. Let p \geq 1 and the function f given on \BbbR by
f(t) =
1
[t]
\chi [[t],[t]+ 1
2p[t]p
]\cap [1,+\infty [(t).
Obviously, f is not continuous on \BbbR , then f is not ergodic. But f is ergodic in Stepanov’s sense.
Indeed, if we take r > 1 and p \geq 1, we have
1
2r
r\int
- r
\left( t+1\int
t
(f(s))pds
\right)
1
p
dt \leq 1
2r
+\infty \int
1
\left( t+1\int
t
(f(s))pds
\right)
1
p
dt \leq
\leq 1
2r
+\infty \int
1
\left( [t]+2\int
[t]
(f(s))pds
\right)
1
p
dt \leq
\leq 2
2r
+\infty \sum
k=1
k+1\int
k
\left( k+ 1
2pkp\int
k
(f(s))pds
\right)
1
p
dt \leq
\leq 1
2r
+\infty \sum
k=1
1
k2
=
\pi 2
12r
- \rightarrow 0 as r - \rightarrow +\infty .
A characterization of (\mu , \nu )-ergodic functions in Stepanov’s sense is given by the following proposi-
tion.
Proposition 2. Let \mu , \nu \in \scrM and I be a bounded interval (eventually I = \phi ). Let f \in
\in BSp(\BbbR , X). Then the following statements are equivalent:
1) f \in \scrE p(\BbbR , X, \mu , \nu ),
2) \mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow \infty
1
\nu ([ - r, r] \setminus I)
\int
[ - r,r]\setminus I
\biggl( \int t+1
t
\| f(s)\| pds
\biggr) 1
p
d\mu (t) = 0,
3) for any \varepsilon > 0, \mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow \infty
\mu
\Biggl(
\{ t \in [ - r, r] \setminus I :
\biggl( \int t+1
t
\| f(t)\| pds
\biggr) 1
p
> \varepsilon \}
\Biggr)
\nu ([ - r, r] \setminus I)
= 0.
Proof. Let A = \nu (I), B =
\int
I
\biggl( \int t+1
t
\| f(s)\| pds
\biggr) 1
p
d\mu (t), C = \mu (I).
(i)\Rightarrow (ii). We have
1
\nu ([ - r, r] \setminus I)
\int
[ - r,r]\setminus I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) =
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1407
=
1
\nu ([ - r, r] - A)
\left( \int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) - B
\right) =
=
\nu ([ - r, r])
\nu ([ - r, r] - A)
\left( 1
\nu ([ - r, r])
\int
[ - r,r]
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) - B
\nu ([ - r, r])
\right) .
(iii)\Rightarrow (ii). Let
A\varepsilon
r =
\left\{ t \in [ - r, r] \setminus I :
\int
I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
> \varepsilon
\right\}
and
B\varepsilon
r =
\left\{ t \in [ - r, r] \setminus I :
\int
I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
\leq \varepsilon
\right\} .
Suppose that (iii) holds, that is,
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow \infty
\mu (A\varepsilon
r)
\nu ([ - r, r] \setminus I)
= 0.
From the equality
\int
[ - r,r]\setminus I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) =
\int
A\varepsilon
r
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) +
\int
B\varepsilon
r
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t)
we deduce that for r sufficiently large
1
\nu ([ - r, r] \setminus I)
\int
[ - r,r]\setminus I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \leq
\leq \| f\| BSp
\mu (A\varepsilon
r)
\nu ([ - r, r] \setminus I)
+ \varepsilon
\mu (B\varepsilon
r)
\nu ([ - r, r] \setminus I)
\leq
\leq \| f\| BSp
\mu (A\varepsilon
r)
\nu ([ - r, r] \setminus I)
+ \varepsilon
\mu ([ - r, r] \setminus I)
\nu ([ - r, r] \setminus I)
\leq
\leq \| f\| BSp
\mu (A\varepsilon
r)
\nu ([ - r, r] \setminus I)
+ \varepsilon
\mu ([ - r, r] - C)
\nu ([ - r, r] - A)
\leq
\leq \| f\| BSp
\mu (A\varepsilon
r)
\nu ([ - r, r] \setminus I)
+ \varepsilon
\mu ([ - r, r])
\nu ([ - r, r])
\left( 1 - C
\mu ([ - r, r])
1 - A
\mu ([ - r, r])
\right) .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1408 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
Then, for all \varepsilon > 0,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow \infty
1
\nu ([ - r, r] \setminus I)
\int
[ - r,r]\setminus I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) < \varepsilon .
Therefore, (ii) holds.
(ii)\Rightarrow (iii). We have
1
\nu ([ - r, r] \setminus I)
\int
[ - r,r]\setminus I
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \geq
\geq 1
\nu ([ - r, r] \setminus I)
\int
A\varepsilon
r
\left( t+1\int
t
\| f(s)\| pds
\right)
1
p
d\mu (t) \geq
\geq \varepsilon
\nu ([ - r, r] \setminus I)
\mu (A\varepsilon
r)
.
Proposition 2 is proved.
2.2. (\bfitmu , \bfitnu )-Pseudo almost periodic functions in Stepanov’s sense.
Definition 2. A function f \in X\BbbR is said to be (\mu , \nu )-pseudo almost periodic in Stepanov’s sense
if it can be written in the form
f = g + h,
where g \in SpAP (\BbbR , X) and h \in \scrE p(\BbbR , X, \mu , \nu ). The set of such functions is denoted by
SpPAP (\BbbR , X, \mu , \nu ).
We have the following diagram:\left[ AP (\BbbR , X) \subset PAP (\BbbR , X, \mu , \nu ) \subset BC(\BbbR , X) \subset C(\BbbR , X)
\cap \cap \cap \cap
SpAP (\BbbR , X) \subset SpPAP (\BbbR , X, \mu , \nu ) \subset BSp(\BbbR , X) \subset Lp
\mathrm{l}\mathrm{o}\mathrm{c}(\BbbR , X)
\right] .
Theorem 3. Let \mu , \nu \in \scrM satisfy (M2). Then the decomposition of a (\mu , \nu )-pseudo almost
periodic function in the form f = g+h, where g \in SpAP (\BbbR , X) and h \in \scrE p(\BbbR , X, \mu , \nu ), is unique
and so
SpPAP (\BbbR , X, \mu , \nu ) = SpAP (\BbbR , X)\oplus \scrE p(\BbbR , X, \mu , \nu ).
Proof. Let the operator
B : BSp(\BbbR , X) \rightarrow L\infty (\BbbR , Lp([0; 1];X),
f \mapsto \rightarrow f b.
Clearly, this operator is isometric linear. We have SpAP (\BbbR , X) = B - 1(AP (\BbbR , Lp((0; 1);X))) and
\scrE p(\BbbR , X, \mu , \nu ) = B - 1(\scrE (\BbbR , Lp((0; 1);X), \mu , \nu )). Using the fact that
AP (\BbbR , Lp((0; 1);X))
\bigcap
\scrE (\BbbR , Lp((0; 1);X), \mu , \nu ) = \{ 0\}
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1409
and B is isometric linear, we obtain
B - 1(AP (\BbbR , Lp((0; 1);X)))
\bigcap
B - 1(\scrE (\BbbR , Lp((0; 1);X), \mu , \nu )) = \{ 0\}
and so
SpAP (\BbbR , X)
\bigcap
\scrE p(\BbbR , X, \mu , \nu ) = \{ 0\} .
Theorem 3 is proved.
Theorem 4. Let \mu \in \scrM satisfies (M1). Then SpPAP (\BbbR , X, \mu , \nu ) equipped with the norm
\| f\| SpPAP (\BbbR ,X,\mu ,\nu ) = \| g\| SpAP (\BbbR ,X) + \| h\| \scrE p(\BbbR ,X,\mu ,\nu ),
where f = g + h, g \in SpAP (\BbbR , X) and h \in \scrE p(\BbbR , X, \mu , \nu ) is a Banach space.
Proof. Let (fn)n be a Cauchy sequence in SpPAP (\BbbR , X, \mu , \nu ), then, for all n \in \BbbN , there exists
(gn, hn) \in SpAP (\BbbR , X)\times \scrE p(\BbbR , X, \mu , \nu ) such that fn = gn + hn.
For \varepsilon > 0, there exists n0 \in \BbbN , n \geq n0, such that, for all m, we have
\| fn - fm\| BSp = \| gn - gm\| BSp + \| hn - hm\| BSp < \varepsilon .
Then, for all m,n \geq n0, we get
\| gn - gm\| BSp < \varepsilon and \| hn - hm\| BSp < \varepsilon .
Therefore, (gn)n and (hn)n are Cauchy sequences in the Banach spaces SpAP (\BbbR , X) and
\scrE p(\BbbR , X, \mu , \nu ), respectively. Then there exists (g, h) \in SpAP (\BbbR , X)\times \scrE p(\BbbR , X, \mu , \nu ) such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\| gn - g\| BSp = 0 and \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\| hn - h\| BSp = 0.
Let f = g + h \in SpAP (\BbbR , X)\oplus \scrE p(\BbbR , X, \mu , \nu ) = SpPAP (\BbbR , X, \mu , \nu ), then
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\| fn - f\| BSp = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\| gn - g\| BSp + \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow +\infty
\| hn - h\| BSp = 0.
So, (SpPAP (\BbbR , X, \mu , \nu ) is a Banach space.
Theorem 4 is proved.
Definition 3. For \mu , \nu \in \scrM and f \in X\BbbR \times Y , continuous, we say that f is uniformly (\mu , \nu )-
pseudo almost periodic in Stepanov’s sense if f = g+h, g \in SpAPU(\BbbR \times Y,X) and h \in \scrE pU(\BbbR \times
\times Y,X, \mu , \nu ). The set of such functions is designated by SpPAPU(\BbbR \times Y,X, \mu , \nu ).
The following hypothesis is useful in the rest.
(H1) For all 1 \leq p < \infty , there exists L > 0 such that
\| f(t, x) - f(t, y)\| \leq L\| x - y\| \forall t \in \BbbR \forall x, y \in Y.
Theorem 5 [8]. Let 1 \leq p < \infty and f \in X\BbbR \times Y with f(., x) \in SpAP (\BbbR , X) \forall x \in Y. Assume
that f satisfies (H1) and x \in AP (\BbbR , Y ). Then f(., x(.)) \in SpAP (\BbbR , X).
Theorem 6. Let \mu , \nu \in \scrM satisfy (M0). Assume that x \in SpAP (\BbbR , Y ), K = \{ x(t); t \in \BbbR \} is
a compact subset of Y and f \in \scrE pU(\BbbR \times Y,X, \mu , \nu ). Then f(., x(.)) \in \scrE p(\BbbR , X, \mu , \nu ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1410 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
Proof. Let f \in \scrE pU(\BbbR \times Y,X, \mu , \nu ) and K = \{ x(t); t \in \BbbR \} be fixed. Then, for all \varepsilon > 0, there
exists \delta \varepsilon ,K such that, for all x1, x2 \in K, one has
\| x1 - x2\| \leq \delta \varepsilon ,K \Rightarrow
\left( t+1\int
t
\| f(s, x1) - f(s, x2)\| p
\right)
1
p
\leq \varepsilon
M
\forall t \in \BbbR .
Since K is a compact, then there exists a finite subset \{ x1, x2, . . . , xn\} \subset K, n \in \BbbN \ast , satisfying
K \subset
\bigcup n
i=1
B(xi, \delta \varepsilon ,K). Therefore,
\forall t \in \BbbR \exists i(t) = 1, . . . , n : \| x(t) - xi(t)\| \leq \delta ,
\left( t+1\int
t
\| f(s, x(s))\| pds
\right)
1
p
\leq
\left( t+1\int
t
\| f(s, x(s)) - f(s, xi(t))\| pds
\right)
1
p
+
\left( t+1\int
t
\| f(s, xi(t))\| pds
\right)
1
p
\leq
\leq \varepsilon
M
+
n\sum
i=1
\left( t+1\int
t
\| f(s, xi)\| pds
\right)
1
p
.
Note that, for all i = 1, . . . , n, f(., xi) \in \scrE p(\BbbR , X, \mu , \nu ). Hence, for r large enough,
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| f(s, x(s))\| pds
\right)
1
p
d\mu (t) \leq \varepsilon
M
\mu ([ - r, r])
\nu ([ - r, r])
+
+
1
\nu ([ - r, r])
n\sum
i=1
r\int
- r
\left( t+1\int
t
\| f(s, xi)\| pds
\right)
1
p
d\mu (t).
Consequently,
\mathrm{l}\mathrm{i}\mathrm{m} \mathrm{s}\mathrm{u}\mathrm{p}
r\rightarrow +\infty
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| f(s, x(s))\| pds
\right)
1
p
d\mu (t) \leq M
\varepsilon
M
= \varepsilon .
Finally, we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow +\infty
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| f(s, x(s))\| pds
\right)
1
p
d\mu (t) = 0.
Theorem 6 is proved.
Corollary 1. Let \mu , \nu \in \scrM . For x \in AP (\BbbR , Y ) and f \in \scrE pU(\BbbR \times Y,X, \mu , \nu ), we have that
f(., x(.)) \in \scrE p(\BbbR , X, \mu , \nu ).
Proof. Since x \in AP (\BbbR , Y ), we obtain that x \in SpAP (\BbbR , Y ) and K = \{ x(t); t \in \BbbR \} is a
compact subset of Y. Hence, Theorem 6 is satisfied.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1411
Theorem 7. Let \mu , \nu \in \scrM satisfy (M2), f : \BbbR \times Y \rightarrow X be a function such that f = g + h \in
\in SpPAPU(\BbbR \times Y,X, \mu , \nu ) satisfying:
(i) \forall x \in Y, g(., x) \in SpAP (\BbbR , Y ) and h(., x) \in \scrE p(\BbbR , Y, \mu , \nu ),
(ii) x = x1 + x2 \in PAP (\BbbR , Y, \mu , \nu ), where x1 \in AP (\BbbR , Y ) and x2 \in \scrE (\BbbR , Y, \mu , \nu ),
(iii) f satisfies (H1).
Then f(., x(.)) \in SpPAP (\BbbR , X, \mu , \nu ).
Proof. Let f = g + h, where g \in SpAPU(\BbbR \times Y,X) and h \in \scrE pU(\BbbR \times Y,X, \mu , \nu ). Then we
have
f(t, x(t)) = g(t, x1(t)) + [f(t, x(t)) - f(t, x1(t))] + h(t, x1(t)) =
= G(t) + F (t) +H(t).
From Theorem 5, it follows that G(.) \in SpAP (\BbbR , X) and, by Corollary 1, we have that H(.) \in
\in \scrE p(\BbbR , X, \mu , \nu ). Now, it is sufficient to show that F (.) \in \scrE p(\BbbR , Y, \mu , \nu ). Indeed, for r > 0 large
enough, we obtain
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| F (s)\| pds
\right)
1
p
d\mu (t) \leq 1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
Lp\| x(s) - x1(s)\| pds
\right)
1
p
d\mu (t) =
=
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
Lp\| x2(s)\| pds
\right)
1
p
d\mu (t) =
=
L
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| x2(s)\| pds
\right)
1
p
d\mu (t).
Therefore, using the fact that x2 \in \scrE p(\BbbR , X, \mu , \nu ), we get
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow +\infty
1
\nu ([ - r, r])
r\int
- r
\left( t+1\int
t
\| F (s)\| pds
\right)
1
p
d\mu (t) = 0.
Theorem 8 [1]. If u \in SpAP (\BbbR , X) and v defined by
v(t) :=
t\int
- \infty
T (t - s)u(s)ds for all t \in \BbbR ,
then v \in AP (\BbbR , X).
Theorem 9. Let \mu , \nu \in \scrM satisfy (M0) and (M2). If u \in \scrE p(\BbbR , X, \mu , \nu ) and v given by
v(t) :=
t\int
- \infty
T (t - s)u(s)ds for all t \in \BbbR ,
then v \in \scrE (\BbbR , X, \mu , \nu ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1412 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
Proof. Let vk(t) =
\int t - k+1
t - k
T (t - s)u(s)ds, k = 1, 2, . . . , vk is a continuous function and
satisfies
\| vk(t)\| \leq
t - k+1\int
t - k
\| T (t - s)u(s)\| ds \leq
\leq
t - k+1\int
t - k
\| T (t - s)A\alpha \| \| A - \alpha u(s)\| ds =
=
k\int
k - 1
\| A\alpha T (\tau )\| \| A - \alpha u(t - \tau )\| d\tau =
= M\alpha \| A - \alpha \|
k\int
k - 1
e - \omega \tau
\tau \alpha
\| u(t - \tau )\| d\tau \leq
\leq M\alpha \| A - \alpha \| e
- \omega (k - 1)
(k - 1)\alpha
k\int
k - 1
\| u(t - \tau )\| d\tau , k > 1.
Hence, by using the Hölder inequality, we get
\| vk(t)\| \leq M\alpha \| A - \alpha \| e
- \omega (k - 1)
(k - 1)\alpha
\left( k\int
k - 1
\| u(t - \tau )\| pd\tau
\right)
1
p
, k > 1.
Then, for r > 0 and k > 1, we have
1
\nu ([ - r, r])
\int
[ - r,r]
\| vk(t)\| d\mu (t) \leq
\leq M\alpha \| A - \alpha \| e
- \omega (k - 1)
(k - 1)\alpha
1
\nu ([ - r, r])
\int
[ - r,r]
\left( k\int
k - 1
\| u(t - \tau )\| pd\tau
\right)
1
p
d\mu (t).
Since h \in \scrE p(\BbbR , X, \mu , \nu ) which is translation invariant (Proposition 1), we deduce that the function
t \mapsto \rightarrow u(t - \tau ) also ergodic in Stepanov’s sense. Then
\mathrm{l}\mathrm{i}\mathrm{m}
r - \rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\| vk(t)\| d\mu (t) = 0,
which gives vk \in \scrE (\BbbR , X, \mu , \nu ) for each k > 1. Moreover,
\| vk(t)\| \leq M\alpha \| A - \alpha \| e
- \omega (k - 1)
(k - 1)\alpha
\| u\| BSp .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1413
Hence
+\infty \sum
k=2
\| vk(t)\| \leq M\alpha \| A - \alpha \|
+\infty \sum
k=2
\Biggl(
e - \omega (k - 1)
(k - 1)\alpha
\Biggr)
\| u\| BSp < \infty .
By using this, we conclude that the series
\sum
k
vk(t) converges uniformly on \BbbR . Moreover, v(t) :=
:=
\int t
- \infty
T (t - s)u(s)ds =
\sum +\infty
k=1
vk(t) is continuous on \BbbR and
\| v(t)\| \leq
+\infty \sum
k=1
\| vk(t)\| \leq M\alpha \| A - \alpha \|
+\infty \sum
k=2
\Biggl(
e - \omega (k - 1)
(k - 1)\alpha
\Biggr)
\| u\| BSp ,
1
\nu ([ - r, r])
\int
[ - r,r]
\| v(t)\| d\mu (t) \leq 1
\nu ([ - r, r])
\int
[ - r,r]
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| v(t) -
n\sum
k=1
vk(t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| d\mu (t)+
+
n\sum
k=1
1
\nu ([ - r, r])
\int
[ - r,r]
\| vk(t)\| d\mu (t).
Let \varepsilon > 0, there exists N \in \BbbN such that, for all n \geq N, we have
\mathrm{s}\mathrm{u}\mathrm{p}
t\in \BbbR
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| v(t) -
n\sum
k=1
vk(t)
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| < \varepsilon .
From (M0), we obtain
1
\nu ([ - r, r])
\int
[ - r,r]
\| v(t)\| d\mu (t) \leq M\varepsilon +
n\sum
k=1
1
\nu ([ - r, r])
\int
[ - r,r]
\| vk(t)\| d\mu (t).
Since vk \in \scrE (\BbbR , X, \mu , \nu ) for all k \geq 1, we get
\mathrm{l}\mathrm{i}\mathrm{m}
r - \rightarrow +\infty
1
\nu ([ - r, r])
\int
[ - r,r]
\| v(t)\| d\mu (t) = 0.
Consequently,
t \mapsto \rightarrow v(t) =
+\infty \sum
k=1
vk(t) \in \scrE (\BbbR , X, \mu , \nu ).
Thus, t \mapsto \rightarrow v(t) :=
\int t
- \infty
T (t - s)u(s)ds is ergodic.
Theorem 9 is proved.
From Theorems 8 and 9, we get the following result.
Theorem 10. Let \mu , \nu \in \scrM satisfy (M0) and (M2). If u \in SpPAP (\BbbR , X, \mu , \nu ) and v defined
by
v(t) :=
t\int
- \infty
T (t - s)u(s)ds for all t \in \BbbR ,
then v \in PAP (\BbbR , X, \mu , \nu ).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
1414 A. REBEY, H. BEN-ELMONSER, M. ELJERI, M. MIRAOUI
2.3. (\bfitmu , \bfitnu )-Pseudo almost periodic solutions. In this subsection, we treat the existence and
uniqueness of pseudo almost periodic solutions to the neutral equation (1). In the following, we
assume that
(H1) f \in SpPAPU(\BbbR \times X\alpha , X, \mu , \nu ) and there exists Lf > 0 such that
\| f(t, x) - f(t, y)\| \leq Lf\| x - y\| \alpha
for every x, y \in X\alpha and t \in \BbbR ;
(H2) g \in PAPU(\BbbR \times X\alpha , X\alpha , \mu , \nu ) such that, for all bounded subset C of X\alpha , the function g
is bounded on \BbbR \times C and there exists Lg > 0 such that
\| g(t, x) - g(t, y)\| \alpha \leq Lg\| x - y\| \alpha
for every x, y \in X\alpha and t \in \BbbR .
Definition 4. A continuous function v : ( - \infty ,+\infty ) - \rightarrow X\alpha is said to be a mild solution of
equation (1) on \BbbR , if
v(t) = T (t - \sigma )
\Bigl[
v(\sigma ) - g(\sigma , v(\sigma ))
\Bigr]
+ g(t, v(t)) +
t\int
\sigma
T (t - s)f(s, v(s))ds for any t \geq \sigma .
Theorem 11. Let \mu , \nu \in \scrM satisfy (M2). Under conditions (H0), (H1) and (H2), suppose that
Lg +M\alpha Lf
\Gamma (1 - \alpha )
\omega 1 - \alpha
< 1.
Then equation (1) has a unique (\mu , \nu )-pseudo almost periodic mild solution, and we have
v(t) = g(t, v(t)) +
t\int
- \infty
T (t - s)f(s, v(s))ds for t \in \BbbR .
Proof. First, suppose that \Lambda : PAP (\BbbR , X\alpha , \mu , \nu ) \rightarrow C(\BbbR , X\alpha ) given by
\Lambda v(t) := g(t, v(t)) +
t\int
- \infty
T (t - s)f(s, v(s))ds for t \in \BbbR .
We can see that \Lambda v \in C(\BbbR , X\alpha ) is well-defined and continuous. Also, from Theorems 7, 10 and 2.26
in [6], \Lambda v \in PAP (\BbbR , X\alpha , \mu , \nu ), that is, \Lambda : PAP (\BbbR , X\alpha , \mu , \nu ) \mapsto \rightarrow PAP (\BbbR , X\alpha , \mu , \nu ). It remains to
prove that \Lambda is a strict contraction on PAP (\BbbR , X\alpha , \mu , \nu ).
For u, v \in PAP (\BbbR , X\alpha , \mu , \nu ) and t \in \BbbR , we obtain
\| \Lambda u(t) - \Lambda v(t)\| \alpha \leq Lg\| u(t) - v(t)\| \alpha +M\alpha
t\int
- \infty
e - \omega (t - s)
(t - s)\alpha
Lf\| u(s) - v(s)\| \alpha ds \leq
\leq
\left( Lg +M\alpha Lf
+\infty \int
0
e - \omega s
s\alpha
ds
\right) \| u - v\| \infty .
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
PSEUDO ALMOST PERIODIC SOLUTIONS IN THE ALPHA-NORM AND STEPANOV’S SENSE . . . 1415
Then
\| \Lambda u - \Lambda v\| \infty \leq
\left( Lg +M\alpha Lf
+\infty \int
0
e - \omega s
s\alpha
ds
\right) \| u - v\| \infty <
<
\biggl(
Lg +M\alpha Lf
\Gamma (1 - \alpha )
\omega 1 - \alpha
\biggr)
\| u - v\| \infty .
From the Banach fixed-point theorem, we deduce that the operator \Lambda has a unique fixed-point, which
is clearly belong to PAP (\BbbR , X\alpha , \mu , \nu ).
Theorem 11 is proved.
References
1. E. Alvarez, C. Lizama, Weighted pseudo almost periodic solutions to a class of semilinear integro-differential
equations in Banach spaces, Adv. Difference Equat., 1 – 18 (2015).
2. J. Blot, P. Cieutat, K. Ezzinbi, New approach for weighted pseudo-almost periodic functions under the light of
measure theory, basic results and applications, Appl. Anal., 1 – 34 (2011).
3. H. Bohr, Zur Theorie der fastperiodischen Funktionen I, Acta Math., 45, 29 – 127 (1925).
4. T. Chtioui, K. Ezzinbi and A. Rebey, Existence and regularity in the \alpha -norm for neutral partial differential equations
with finite delay, CUBO, 15, № 1, 49 – 75 (2013).
5. T. Diagana, Stepanov-like pseudo-almost periodicity and its applications to some nonau- tonomous differential
equations, Nonlinear Anal.: Theory, Methods and Appl., 69, № 12, 4277 – 4285 (2008).
6. T. Diagana, K. Ezzinbi, M. Miraoui, Pseudo-almost periodic and pseudo-almost automorphic solutions to some
evolution equations involving theoretical measure theory, Cubo, 16, № 2, 1 – 31 (2014).
7. T. Diagana, Gisèle M Mophou, Gaston N’Guérékata, Existence of weighted pseudo-almost periodic solutions to some
classes of differential equations with Sp -weighted pseudo-almost periodic coefficients, Nonlinear Anal., 72, № 1,
430 – 438 (2006).
8. Baroun Mahmoud, Khalil Ezzinbi, Khalil Kamal, Maniar Lahcen, Pseudo almost periodic solutions for some parabolic
evolution equations with Stepanov-like pseudo almost periodic forcing terms, J. Math. Anal. and Appl., 462, № 1,
233 – 262 (2018).
9. A. Pazy, Semigroups of linear operators and application to partial differental equation, Appl. Math. Sci., 44 (1983).
10. C.Y. Zhang, Pseudo almost periodic solutions of some differential equations, J. Math. Anal. and Appl., 151, 62 – 76
(1994).
11. C. Zhang, Pseudo almost periodic type functions and ergodicity, Sci. Press, Kluwer Acad. Publ., Dordrecht (2003).
Received 27.09.20
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 10
|
| id | umjimathkievua-article-6315 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:27:02Z |
| publishDate | 2022 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b7/0a231aa9a15578e6a190ec2b41aad3b7.pdf |
| spelling | umjimathkievua-article-63152023-01-07T13:45:39Z Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. STEPANOV’S SENSE UDC 517.9 Our aim is to present the concept of double-measure ergodic and double-measure pseudo almost periodic functions&nbsp; in Stepanov's sense.&nbsp;&nbsp;In addition, we present numerous interesting results, such as the composition theorems and completeness properties for these two&nbsp;&nbsp;spaces of the considered functions.&nbsp;&nbsp;We also establish the&nbsp; existence and&nbsp; uniqueness for the&nbsp; double-measure pseudo almost periodic mild solutions&nbsp; in Stepanov's sense for some evolution equations. УДК 517.9 Псевдо-майже періодичні розв’язки в альфа-нормі та в розумінні Степанова для деяких еволюційних рівнянь Введено поняття ергодичних функцій подвійної міри та псевдо-майже періодичних функцій подвійної міри в розумінні Степанова. Крім того, наведено багато цікавих результатів, що включають як теореми про композицію, так і властивості повноти для цих двох просторів розглянутих функцій.&nbsp;&nbsp;Встановлено також існування та єдиність псевдо-майже періодичних слабких розв’язків подвійної міри в розумінні Степанова для деяких еволюційних рівнянь.&nbsp; Institute of Mathematics, NAS of Ukraine 2022-11-27 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6315 10.37863/umzh.v74i10.6315 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 10 (2022); 1401 - 1415 Український математичний журнал; Том 74 № 10 (2022); 1401 - 1415 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6315/9315 Copyright (c) 2022 Mohsen Miraoui, Hedi Ben-Elmonser, Mosbah Eljeri, Amor Rebey |
| spellingShingle | Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. Rebey, A. Ben-Elmonser, H. Eljeri, M. Miraoui, M. Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_alt | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_full | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_fullStr | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_full_unstemmed | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_short | Pseudo almost periodic solutions in the alpha-norm and Stepanov's sense for some evolution equations |
| title_sort | pseudo almost periodic solutions in the alpha-norm and stepanov's sense for some evolution equations |
| topic_facet | STEPANOV’S SENSE |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6315 |
| work_keys_str_mv | AT rebeya pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT benelmonserh pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT eljerim pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT miraouim pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT rebeya pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT benelmonserh pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT eljerim pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations AT miraouim pseudoalmostperiodicsolutionsinthealphanormandstepanov039ssenseforsomeevolutionequations |