ASYMPTOTIC ALMOST AUTOMORPHY FOR ALGEBRAS OF GENERALIZED FUNCTIONS Текст научной статьи по специальности «Математика»

Vladikavkaz Mathematical Journal 2023, Volume 25, Issue 2, P. 38-55

YAK 517.98

DOI 10.46698/j1917-4964-8877-l

ASYMPTOTIC ALMOST AUTOMORPHY FOR ALGEBRAS OF GENERALIZED FUNCTIONS

Ch. Bouzar1 and M. Slimani1

1 Laboratory of Mathematical Analysis and Applications, University Oran 1, Ahmed Ben Bella, 31000 Oran, Algeria E-mail: [email protected], [email protected]

Abstract. The paper aims to to study the concept of asymptotic almost automorphy in the context of generalized functions. We introduce an algebra of asymptotically almost automorphic generalized functions which contains the space of smooth asymptotically almost automorphic functions as a subalgebra. The fundamental importance of this algebra, is related to the impossibility of multiplication of distributions; it also contains the asymptotically almost automorphic Sobolev-Schwartz distributions as a subspace. Moreover, it is shown that the introduced algebra is stable under some nonlinear operations. As a by pass result, the paper gives a Seeley type result on extension of functions in the context of the algebra of bounded generalized functions and the algebra of bounded generalized functions vanishing at infinity, these results are used to prove the fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. As applications, neutral difference-differential systems are considered in the framework of the algebra of generalized functions.

Keywords: asymptotic almost automorphy, generalized functions, neutral difference differential equations.

AMS Subject Classification: 46F10, 34K14, 46F30.

For citation: Bouzar, Ch. and Slimani, M. Asymptotic Almost Automorphy for Algebras of Generalized Functions, Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp. 38-55. DOI: 10.46698/j1917-4964-8877-l.

1. Introduction

Bochner S. defined explicitly almost automorphic functions in the papers [1, 2], where also some of their basic properties are given. In [3] he studied linear difference differential equations in the framework of such functions. It is well known that the concept of almost automorphy is strictly more general than the almost periodicity of H. Bohr [4], however the Stepanoff almost periodicity [5] and the Levitan almost periodicity [6] don't enter into the Bochner concept. Asymptotic almost periodicity of functions as a perturbation of almost periodic functions by functions vanishing at infinity is due to M. Frechet in [7]. Asymptotic almost automorphy of classical functions is considered in [8], see also [9]. The almost periodicity and the asymptotic almost periodicity of Sobolev-Schwartz distributions, [10] and [11], are respectively considered by L. Schwartz in [11] and I. Cioranescu in [12], while almost automorphy and asymptotic almost automorphy in the setting of these distributions are respectively the subject of the recent works [9] and [13].

In view of the result [14] on the impossibility of the multiplication of distributions, algebras of generalized functions containing spaces of Sobolev-Schwartz type distributions have been

© 2023 Bouzar, Ch. and Slimani, M.

studied, see [15-17]. The concepts of almost periodicity and asymptotic almost periodicity as well as almost automorphy in the context of such algebras of generalized functions are introduced, studied and applied in the papers [18-22]. So, the paper first introduces and studies a class of asymptotically almost automorphic generalized functions, denoted by Gaaa. In the sense of multiplication, not only Gaaa is stable under multiplication and it contains the space of asymptotically almost automorphic distributions of [9], but moreover some nonlinear operations are performed within the algebra Gaaa. As a by pass result, we give a Seeley type result on extension of functions in the context of the introduced generalized functions, this is needed in the proof of a fundamental result on the uniqueness of decomposition of an asymptotically almost automorphic generalized function. The papers [20] and [21] can be considered as consequences of this work. The paper aims also, as in [7], to lift a Frechet existence result of asymptotically almost automorphic solutions of differential equations to the level of neutral difference differential systems in the framework of Gaaa.

It is worth noting that the meaning of generalized functions is utilized differently by authors as distributions or ultradistributions, even as hyperfunctions, but in this work by generalized functions we mean in the sense of the works [15-17].

The paper is organized as follows: section two recalls definitions and some properties of asymptotically almost automorphic functions and asymptotically almost automorphic distributions as in [9]. Section three introduces asymptotically almost automorphic generalized functions and gives some of their important properties. The study of a Seeley type result on extensions of generalized functions is given in section four. In section six nonlinear operations on asymptotically almost automorphic generalized function are studied. The last section is dedicated to linear neutral difference differential systems in the framework of asymptotically almost automorphic generalized functions.

2. Asymptotic Almost Automorphy of Functions and Distributions

Let Cb denotes the space of bounded and continuous complex-valued functions defined on R, endowed with the norm || ■ (r) of uniform convergence on R, it is well-known that || ■ (r)) is a Banach algebra.

A complex-valued function g defined and continuous on R is called almost automorphic if for any sequence (sm)meN C R, one can extract a subsequence (smk)k such that

g(x) := lim g(x + smk) (Vx € R),

and

lim g(x - Smk)= g(x) (Vx € R).

The space of almost automorphic functions on R is denoted by Caa.

The space C+;0 is the set of all bounded and continuous complex-valued functions defined on R and vanishing at

Definition 1. We say that a function f € Cb is asymptotically almost automorphic, if there exist g € Caa and h € C+;0 such that f = g + h on J := [0, . The space of asymptotically almost automorphic functions is denoted by Caaa.

For a study of asymptotically almost automorphic functions and asymptotically almost automorphic distributions see [9] and [13] and the references list therein.

Proposition 1. The decomposition of an asymptotically almost automorphic function is unique on J.

Notation 1. If f € Caaa and f = g + h on J, where g € Caa and h € C+,0. Due to the uniqueness of the decomposition of f, the function g is said the principal term of f and the function h the corrective term of f, we denote them respectively by faa and fcor. The notation

f = (faa + fcor) € Caaa means that faa € Caa, fcor € C+,o and f = faa + fcor on J.

Let E(I) be the algebra of smooth functions on I = R or J, and define the space Dlp(I) := {f € E(I) : Vj € Z+, f(j) € Lp(I)}, p € [1, that we endow with the topology defined by the family of semi-norms

k € Z+.

M k,p,I :=E \\f(j)

Kk

LP (I)

So, Dlp(I) is a Frechet subalgebra of E(I). Denote B(I) := DL~(I).

Remark 1. We have lim^0 f(j)(x) exists for every j € Z+ when f € Dlp(J).

>

The space of smooth almost automorphic functions Baa and smooth asymptotically almost automorphic functions Baaa are defined respectively by

Baa := {f € E(R) : V j € Z+, f(j) € Caa}.

Baaa := {f € E(R) : Vj € Z+, f(j) € Caaa}.

We endow Baa and Baaa with the topology induced by B := DL~ (R).

Proposition 2. (1) The space Baaa is a Frechet subalgebra of B stable by translation.

(2) Baaa X Baa C Baaa-

(3) Baaa * L C Baaa-

The space of Lp-distributions, p €]1, denoted by D£p(R), is the topological dual of DLq(R), where 1/p + 1/q = 1. Let B be the closure in B of the space D C E(R) of functions with compact support. The topological dual of B is denoted by D^ (R). The space of bounded distributions (R) is denoted by B'. The following definition follows from the characterizations of almost automorphic distributions, see [9].

Definition 2 (Proposition). The space of almost automorphic distributions, denoted by Baa, is the space of T € B' satisfying one of the following equivalent statements

(1) T * f € Caa, V f € D.

(2) 3 k € Z+ and gj € Caa, 0 < j < k, such that T = gj

The space of bounded distributions vanishing at infinity, denoted by B+ 0, is the space of Q € B' satisfying

lim <twQ,f) := lim <Q,T-wf) = 0 (V f € D),

W^ + TO W^ + TO

where twf (■) := f (■ + w), w € R.

Theorem 1. The space of asymptotically almost automorphic distributions, denoted by Baaa, is the space of T € B' satisfying one of the following equivalent statements

(1) 3 P € Baa, 3 Q € B+ 0 such that T = P + Q on J.

(2) T * f € Caaa, V f € D.

(3) 3 k € Z+ and fj € Caaa, 0 < j < k, such that T = j, j

(4) 3 (0 C Baaa such that 0 m — T in B .

Notation 2. If T € B'aaa and T = P + Q on J, since this decomposition is unique by [9, Proposition 12], the distribution P is called the principal term of T, the distribution Q is called the corrective term of T, we denote them respectively Taa and Tcor. This is summarized by the notation T = (Taa + Tcor) € B'aaa.

3. Asymptotically Almost Automorphic Generalized Functions

We introduce and study an algebra of asymptotically almost automorphic generalized functions.

Let I :=]0,1], (u£)£ € (DLp(I))1, p € [1, +rc>], m € Z and k € Z+, then the notation

KkP,i = 0(em), e ^ 0 ^ (3 c > 0) (3eg € I) (Ve < eg) |ue|fc>p>i < cem.

The space of moderate elements is denoted and defined by

Maaa = {(u£)s € (Baaa) : Vk € Z+, 3m € Z+, |ue|fc>TO>R = 0(e-m), e ^ 0},

and the space of null elements by

Naaa = {(u£)£ € (Baaa)1 : Vk € Z+, Vm € Z+, |ue|fc>TO>R = 0(em), e ^ 0}.

Some properties of Maaa and Naaa are given in the following results. Proposition 3. (1) We have the null characterization of Naaa, i. e.

Naaa = { (ue)e € Maaa : V m € Z+, |u£|q;TO;R = 0(em), e ^ 0}.

(2) The space Maaa is an algebra stable under translation and derivation.

(3) The space Naaa is an ideal of Maaa.

< (1) The proof is based on the following Landau-Kolmogorov inequality

II f(p) II < 27rll f II1 ™ II f(ra) II ™ lf IIl~(R) ^ 2/1 lf IIL~(R) lf IIl°

where 0 < p < n € Z+ and f is of class Cn(R). Let (u£)£ € Maaa, i. e.

(Vk € Z+)(3 m € Z+)(3 c > 0) (3ei € I) (Ve < ei) |ue|fc>TO>R < ce-m, (12) and it satisfies the estimate of order zero, i. e.

(Vm2 € Z+)(3 C2 > 0)(3e2 € I) (Ve < e2) Mq^r < C2em2. (13)

The Landau-Kolmogorov inequality for p = j, n = 2j, (12) and (13) give V k € Z+,

1—i •) -l^eIfc,oo,r ^ / " 27t11u£ 11 ^oo2^ ||u£ II^tcj-r) Kk

< 27t(|u£|o,oo,r)2 IK°III/-(r) ^

Taking m2 € Z+ such that mo = 2( — + € Z+, then we obtain

(Vk € Z+) (Vmo € Z+) (3 c = 2ttc^c| > 0) (3e0 = inf(ei,e2) € I) (Ve<£0) k|fc)CO;R < cem°. Which gives (u£)£ € N^aa-

(2) The stability under translation and derivation of the space Maaa is obvious. Let (u£)£ (v£)£ € Maaa, i. e. they satisfy (12) and for j € Z+,

|(u£v£)(i)|

^ Z^ j)

j!

L~(K) - ^ i!(j - i)!

Iw(i) II IIV(j-i)l

R

V-^ j!

R K

consequently,

(Vk € Z+) (3m = (mi + m2) € Z+) |u£v£|fc;TO;R = O(e-m), e ^ 0.

Which shows that (u£v£)£ € Maaa.

(3) Let (v£)£ € Maaa and (u£)£ € N«aa,, i. e. (v£)£ satisfies (12) and (u£)£ satisfies

(Vk € Z+) (Vm' € Z+) (3 c' > 0) (3 ei € I) (Ve < ei) KU^r < c'em'.

(14)

Since the family of the norms | ■ is compatible with the algebraic structure of B, i. e.

V k € Z+, 3 ck > 0 such that

Take m' € Z+ such that m0 = (—m + m' ) € Z+, so we obtain

(Vk € Z+) (Vmo € Z+) (3C = cfccc' > 0) (3eo =inf (ei,ei) € I) (Ve<eo) Kv£|fcooR

^ Cem0,

which implies (u^^ € N«aa. >

Definition 3. The algebra of asymptotically almost automorphic generalized functions is denoted and defined as the quotient algebra

G •—

^aaa • —

Ma

n«

Example 1. We have Gaap C Gaaa, where Gaap is the algebra of asymptotically almost periodic generalized functions of [20]. Let p € S such that

/p(x) dx — ! and p(x) dx = 0 (V k

Setft(-) :=ip(;), £>0.

Define the following maps

Oaaa

B '

aaa

T

Baaa f

Baaa f

(T * P£)£ + Nao

G

aaa

(f )£ + Naaa

B

aaa

-m2

6

I

Proposition 4. The following diagram of linear embeddings

B laas B'

^aaa * ^aaa

u aaa g ;

aaa aaa

is commutative.

< For f € Baaa, we have Yaaa(f) € Baaa C Caaa• We conclude from ([9, Example 3-(1)]) that Yaaa(f) € Baaa. Let T € Baaa, by the characterization of an asymptotically almost automorphic distribution 3 (fi)i^m C Caaa such that T = ^f(i). If j € Z+,

i^m R i^m R

consequently, there exists cmj > 0 such that hence,

(V k € Z+) (3 mi € Z+) |T * p^^R = 0(e-mi) , e ^ 0,

which gives that (T * p£)£ € Maaa. The linearity of iaaa results from the fact that the convolution is linear. If (T * p£)£ € Naaa, then

(Vm' € Z+) (3 c' > 0) (3ei € I) (Ve < ei) ||T * p£||l~w < c'em'. (15)

Due to regularization we have

(T, = lim / (T * pj (x Wx) dx, ^ € Dli (R).

£^q J

From (15), it holds Vm' € Z+, 3c' > 0, 3ei € I, Ve < ei,

(T * p£)(x)^(x) dx

^ II^!l1(r)c e

consequently, when e ^ 0 we obtain (T, = 0, V ^ € Dli (R), therefore iaaa is injective. Finally,

Iaaa (T(j)) = (T(j) * p£)£ + Naaa = (T * p^j + Naaa = (iaaa(T))(j) (V j € Z+),

i. e. the embedding iaaa commutes with derivatives.

Let f € Baaa, we have to show that (f * p£ — f )£ € N^aa. By Taylor's formula, for 0 €]0,1[ and m € N, we set for j € Z+, that (f(j) * p£ — f (j))(x) equals

. m-i / Ni ^ xm

/ E f{l+J\x)p(y) dy+ f f{m+J\x - 9ey)p(y) dy,

J i=i ' J '

and then

P I

\fu),Pe_fu)\\ ^ sup / K-j/rll/^ix-^Hp^ldj/,

v y m' .xeR J

m

< _ II f(m+i) II 111/"1/ill ^ m! IK IIL°°(R) II" ^IIlh

Hence, (V k € Z+) (V m €

| f * Pe~ /|fc;00;R ^ |/U+fc,oo,R \\ymp||Ll(R),

which means that (f * p£ — f )£ € N«««. >

Remark 2. The application o-aaa is not only a linear embedding but it is also an algebraic embedding.

Recall the algebra of almost automorphic generalized functions, see [22], denoted and defined by

Q2__"^M

•J' aa

where

Maa := {(u£)£ € (Baa)1 : Vk € Z+, 3m € Z+, Klfc^R = O(e-m), e ^ 0}.

N«a := {(u£)£ € (Baa)1 : Vk € Z+, Vm € Z+, |ue|fc>TO>R = O(em), e ^ 0}.

The algebra of Lp-generalized functions on I, 1 ^ p ^ see [23], is defined as the quotient algebra

where

Mlp(I) := {(u£)£ € (DLp(I))1 : Vk € Z+, 3m > 0, |u|fc>p>i = O(e-m), e ^ 0}. Nlp(I) := {(u£)£ € (Dlp(I))1 : Vk € Z+, Vm > 0, Kl^i = O(em), e ^ 0}.

Notation 3. Denote GL~(I) := Gb(I), Gb := Gb(R) and Gli := Gli(R). For w € R, the translate rwu of u = [(u£)£] € Gb is defined by

twu := [(twu^] . For j € Z+, the derivation u(j) of u is defined by

u :=

/ (j) \ K )e

Let v = [(v£)£] € Gb, the product u x v is defined by

V x V := [(u^)^ .

Let V = [(v£)£] € Gli , the convolution V * V is defined by

V * V := [(u£ * v^] .

The following results lift the results of Proposition 2 to Gaaa.

Proposition 5. (1) Gaaa is a subalgebra of Gb stable under translation and derivation.

(2) Gaaa x Gaa C G«aa •

(3) Gaaa * GL1 C Gaaa •

< (1) From Proposition 3 (2), we deduce that Gaaa is an algebra stable under translation and derivation. Let (u£)£ € Maaa, satisfies (12) and as Baaa C B, hence (u£)£ € Mb• If (u£)£ € Naaa, in the same way we prove that (u£)£ € NB•

(2) Let u = [(«£)£j € Gaaa and u = [(v£)£] € Gaa. As (u£)£ € Maaa and (v£)£ € Maa, so (u£)£ and (v£)£ satisfy the estimate (12). In view of Proposition 2 (2) it follows that for all e € I, u£v£ € Baaa, and for every j € Z+,

-m2

||(U£V£)(j)||Lc(R) < 2j |u£|j,^,R |v£< 2jciC2e mie

therefore, (Vk € Z+) (3 m = (m1 + m2) € Z+) |u£v£= O(e-m), e ^ 0, i. e., (u£v£)£ € Maaa. The product U x U is independent on the representatives. Indeed, suppose that (u£)£ € Maaa and (v£)£ € Maa are others representatives of U and U respectively, for j € Z+,

(^£V£ - u£v£)

LC(R)

/„./\(j)

(^£V£ - u£v£ + u£v£ - u£v£)

<

((«£ - u£K)

(j)

LC(

+

m

>£ - v£))

(j)

LC(R)

since («£ - u£)£ € N^aa, (v£ - v')£ € N^a, then

(V k € Z+) (V m € Z+) |u£V£ - u£v£ lfc ^ r = O(em),

as e ^ 0, which implies that (u£v£ - u£v')£ € N^aa.

(3) Let u = [(u£)£ € Gaaa and v = [(v^] € GL, so (u£)£ € Maaa and (v£)£ € MLi, that is (u£)£ satisfies the estimate 12, and (v£)£ satisfies

(Vk € Z+)(3 mi > 0) (3 c> 0)(3 eo € I) (Ve < eo) Mfc.i.R < ce

-mi

Proposition 2 (3) gives for all e € I, u£ * v£ € Baaa. Due to Young inequality we obtain

||(u£ * V£)(j)|L~(R) ^ ||u£j) !ltc(R) ||V£|L1(R).

For every k € Z+,

l U£ ^ V£ l . t™

<

E

Kfc

£ 11LC (

I^Hl^R) ^ I ^Ifc^.Rl ^ l0,1,R'

consequently,

(V k € Z+)(3 m € Z+) ^ * v£

= O(e-m), e ^ 0,

so (u£ * v£)£ € Maaa. The convolution v * v does not depend on the representatives. Indeed, suppose that (u£)£ € Maaa and (v£)£ € MLi are others representatives of v and V respectively, for every k € Z+,

|u£ * v£ — u£ * v£L „ = |u£ * v£ — u£ * v£ + u£ * v£ — u£ * v£|, „

I £ £ £ £lk,^,R I £ £ £ £ £ £ £ £lk,^,R

^ |u£ — u£|k)TO)R Mq^r + |u£|k,^,R K — V£ |Q,1,R ,

as (u£ — u£)£ € Naaa, (V: — v£)£ € N¿1, then

(V k € Z+)(V m € Z+) |u£ * V£ — u£ * v£ as e ^ 0, which means that (u£ * v£ — u£ * v£)£ € N^aa. >

* v' l

* V£ Ifc.oo.R

= O(em),

L

CO

4. A Generalized Seeley Theorem

We give a result on the extension operators in the context of generalized functions. It is needed in the proof of the decomposition of an asymptotically almost automorphic generalized function. Let's first recall a technical Lemma, see [24].

Lemma 1. There are two sequences of real numbers (a^)leZ+ and (bj)leZ+ such that

(1) b < 0, V l € Z+.

(2) |aj||bj|n < Vn € Z+.

(3) Eto alb? = 1, V n € Z+.

(4) b ^ -to, l ^ +to. Define the space

B+o(I):= {^ € B(I) : Vj € Z+, lim <^(j)(x) = 0}.

The algebra of bounded generalized functions vanishing at infinity on I is defined by

G+,o(I) :=

M+,o(I)

N+,o(I) ' where

M+,o(I) := {(u£)£ € (B+,o(I))/ : Vk € Z+, 3m € Z+, |u£|fc;TO;I = O(e-m), e ^ 0}.

N+,o(I) := {(u£)£ € (B+,o(I))/ : Vk € Z+, Vm € Z+, = O(em), e ^ 0}.

Theorem 2. The linear extension operator E : Gb(J) ^ Gb(R), « = [(u£)£] ^ EuZ = [(Eu^], where

{u£(x), if x ^ 0,

E aju£(bix), if x < 0, i=o

is well defined and we have EUj = U. In particular, V U € G+,0(J), EU € G+0(

< Let u = [(u£)£j € Gb (J), and (u£ )£ € Mb (J) be a representative of -u. So V e € I, Eu£ € B(R) and Eu£|j = u£. Indeed, if x < 0, then fyx > 0, Vl € Z+ in view of Lemma 1 (1). Moreover according to Lemma 1 (2) and as u£ € B(J), V e € I, hence V n € Z+, V e € I, V x < 0,

|(E«£ )(n)(x)| < ||u£n)||LTC(J)^ MM" < (16)

i=o

consequently, Vn € Z+, the series

(n)(x) = Y^ ab"u(n)(

(£«e)(n)(x) = ^ a^u^x), e € I, i=0

absolutely converge. Furthermore due to Lemma 1 (3),

lim (Eu£)(n) (x) = V aib" lim u£n)(6|x) = u£n) (0) V a^" = u£n)(0),

x—)- 0 z—/ x—»0 z—/

< i=0 < i=0

so Ve € I, Eu£ € E(R). As Ve € I, u£ € B(J) and by (16), it follows that Vn € Z+, Ve € I, (Eu£)(n) € LTO(R). In order to show that (Eu£)£ € Mb(R), we prove the following estimate

(Ve € I) (Vk € Z+) (3 Ck > 0) |Eu£|fc)TO;R < Cfc(17) Indeed, it is clear that

l(EU )(n)II = yu(ra)n

|(EU£) IIl^(j) = ||u£ ||L^(J)

and the estimate (16), gives

IKEU£)(ra)IURv) < |aiM^

i=o

therefore

II (Eu )(n) II < C IIU(n) II ll(EU£) IIl^(r) < Cnllu£ IIl^(J) ,

where Cra = max (1, N|bj|n). So

(V k € Z+) (3 Cfc := ^ Cn < (Ve € I) (u € B(J))

ra<fc

|Eu£L „ < CfcluJ. T, (18)

this implies (Eu£)£ € Mb(R). The definition of Eu is independent on representatives. Indeed, if (u£)£ and (v£)£ are representatives of u, hence by (18),

(V e € I) (V k € Z+) (3 C > 0) |Eu£ - Ev^,^ < Cfc|u£ - v^,^ .

As (u£ - V£)£ € Nb(J) then

(Vk € Z+)(Vm > 0) |Eu£ - Ev£|fccoR = O(em), e ^ 0,

which shows that (Eu£ - Ev£)£ €

We have EuIj = « in GB(J). Indeed, as Eu = [(Eu£)£] € GB(R) and u = [(u£)£] € GB(J), therefore

Euj - u := [(Eu£|j)^ - u = [(u^] - u = u - u = 0 in Gb(J).

If u = [(u£)£] € G+,o(J) C Gb (J), then Eu = [(Eu£)£] € Gb (R). So V e € I, Eu£ € B(R). The fact that V e € I, Eu£ = u£ on J, and V e € I, u£ € B+,0(J) implies Eu£(x) = 0,

i. e. V e € I, Eu£ € C+,0. By [9, Proposition 5 (5)], we obtain that

(V e € I) Eu£ € C+,o n B(R) = B+,o(R), it follows that Eu € G+,0(R). >

5. The Decomposition

In this section we show that an asymptotically almost automorphic generalized function is uniquely decomposed as in the classical case.

Theorem 3. Let U € Gaaa(R) then there exist U € Gaa(R) and w € G+,o(R) such that U = U + WU on J, and the decomposition is unique on J.

< Let U = [(u£)£] € Gaaa, so Ve € I, Vj € Z+, u£j) € Caaa. Then there exist v£j € Caa, w£,j € such that V j € N, u£j = (v£,j + W£,j ) € Caaa on J, and for j — 0, U — V£ + W£ on J. By [9, Proposition 8], it holds that Vj € N, v£j = (v£)j) on R and w£j = (w£)j) on J, which gives v£ € Baa and w£ € B+,0(J). Let's show that (v£)£ € Maa. As (u£)£ € Maaa, therefore

(Vk € Z+)(3 m € Z+)(3 c > 0) (3eo € I) (Ve < eo) K|fc)^R < ce-m (19)

Due to [9, Proposition 3 (5)], we obtain

(Vj € Z+) ||vj)||Lc(R) < ||u£j)|Lc(J), (20)

it follows that

(Vk € Z+)(3 m € Z+)(3 c > 0) (3eo € I) (Ve < eo) KU^r < ce-m, (21)

this means that (v£)£ € Maa. If (u£)£ € Naaa then

(Vk € Z+)(Vm € Z+)(3 c > 0)(3eo € I) (Ve < eo) K|fc)^R < cem, (22)

and from (20) it holds (v£)£ € N^a. Consequently, U = [(v£)£] € Gaa. On the other hand, we have

(V j € Z+) ¡Wj)|Lc(J) < ¡u(j)|Lc(J) + ¡Vj)|Lc(J) - (23)

The estimates (19), (21) and (23) give

(Vk € Z+) (3 m € Z+) (3 c > 0) (3 eo € I) (Ve < eo) KU^j < 2ce-m,

hence (w£)£ € M+,o(J). If (u£)£ € N^aa, then (w£)£ € N+,o(J) follows from (22) and (23). Thus WU = [(w£)£] € G+,o(J). By Theorem 2 extending WU € G+,o(J) to Ew € G+,o(R) with EUwU = WU on J. Finally, U = U + WU on J. If U € Gaaa has two decompositions, i.e.

U = U + Wj on J, i = 1,2,

where f € Gaa and w € G+,o := G+,o(R). Let (v£;i)£ € Maa and (w £,i)£ € ^+,0 be respeCtively representatives of f and wfj, i = 1,2. So (v£)1 — v£,2)£ + (w£)1 — w£,2)£ € NB(J), i. e.

(V k e Z+) (V m > 0) (3 c > 0) (3 eo e I) (V e < eo)

|v£,i - v£;2 + w£)i - w£;2|fc,TO>J ^ cem. (24)

Due to [9, Proposition 9], as V e € I, v£)i € Baa, i = 1,2, for any real sequence (sm)meN, such that sm ^ there exist (sTO;(£)) a subsequence of (sm)meN and g£)i, i = 1,2, such that V x e R, V j e Z+,

g£ji (x) := , lim (x + sm;(£)) and , lim (x - smi(£)) = (x) •

Furthermore, as V e € I, w£)i € B+,o, i = 1, 2,

liim (x + =0 (V X € R, V j € Z+).

By using (24) we have

(V m > 0) (3 c > 0) (3 eo € I) (V e < eo) (V x ^ -smi(e)),

(X + — (X + sm;(£)) + (x + sm;(£) ) — (x + sm;(£))

so when l ^ we obtain

(Vm > 0) (3 c > 0) (3 eo € I) (V e < eo) (Vx ^ —s^) ^(x) - gj2(x) by taking the translate — sTOl(£) and let l ^ we get

< cem,

^ cem

< cem

(Vm > 0) (3 c > 0) (3 eo € I) (V e < eo) (V x ^ 0) (x) — v^ (x)

By [9, Proposition 3(5)], it follows

(Vk € Z+)(Vm > 0)(3 c > 0)(3eo € I) (Ve < eo) |ve>i — ve,2|k>^,R < cem, (25)

which shows that (v£)i — v^) e € NB(R), so Vi — V2 in Gb(R). From (24) and (25) it holds that (we>1 — w£;2)e € NB(J), i. e. WV1 — WV2 on J. >

Notation 4. Let V € Gaaa, and V — V + WV on J, where V € Gaa and WV € G+,o, then V and WV are called respectively the principal term and the corrective term of V and we denote them respectively Vaa and Vcor. Also V — (Vaa + Vcor) € Ga(ia means that Vaa € Ga(l, Vcor € G+,o and V — Vaa + Vcor on J.

6. Nonlinear Operations

The algebra of tempered generalized functions on C denoted by GT (C), see [23] for more details, is the quotient algebra

G(C) - C)

where

Mt(C):— / € ^ ^V j ^, 3 m € Z+, 1.

\ supxeR2 (1 + |x|)-m |/e(j)(x)| — O (e-m), e ^ 0 J

Nt( C) \ (/e)e € (E (R2))7 : V j € Z+, 3 n € Z+, Vm € Z+, 1 T I supxgR2 (1 + |x|)-n |/F(x)| — O(em) ,e ^ 0 j

Example 2. Any polynomial function is a tempered generalized function. Theorem 4. Let V — [(ue)e] € Gaaa and F — [(/e)ej € Gt(C), then

F O V :— [(/e O U£)£]

is a well-defined element of Gaaa. The principal term and the corrective term of F o V are respectively F(Vaa) and F(Vaa + Vcor) — F(Vaa), where V — Vaa + Vcor on J.

< Let (u£)£ € Maaa and (/£)£ € MT(C), by the classical Faa di Bruno formula, we have Vj € Z+,

(/ O u£)(j)(x) ^ /ir)(u£(x)) j /u£°(x)

it

/11 LI ll ~T" * <26)

j! ^ li! ...j! 11 \ i!

J 1i+2l2+-----hjij =j, 1 j i=1 \

r=ii+-+j

As Ve € I, Vj € Z+, u(j) € Caaa and / € E(R2), it follows by [9, Proposition 3(4)], that /ir)(u£) € Caaa, and since Caaa is an algebra, then V£ € I, / O U£ € Baaa. As (u£)£ € Maaa, then

(Vk € Z+) (3nfc € Z+) (3 Ck > 0) (3£fc € I) (V£ < £fc) K^r < cfce-nk. The fact that (/£)£ € MT(C) gives

(V j € Z+) (3 Nj € Z+) (3 Cj > 0) (3 ej € I) (V £ < ej)

II/^MUr) < Cje-Nj II1 + U£llN'

Consequently, by (26) we obtain

-Nr in , „, ||Nr j { l,(i)

||(/£ O U£)(j)|LTC(R) Cr£-Nr ||1 + u

L°°( R) ^ 11 "-e 11 ¿00 (]R) -p-j- j N"6 NL°°(R)

77i u 11

U£

j! ^ li! ...j! 11 i!

J 1i+2l2+-----hjij=j, 1 j i=1 V

r=ii+-+j

hence there exists c > 0,

\\(feOU£)^\\Loo{R) C£-Nr{ 1+np) i/^-n^I

j! ^ /1! •••/,! 11

J 1i+2l2+-----hjij =j, 1 j i=1

where

-(Nr(1+no)+ E nth) j

< e - ,,,...,;■ msr«*-.

1i+2l2+-----hjij =j, 1 j i=1

r=1i+-----hi,'

J | j ,

c -r-r /CA1t

r^ + ^+j, ^ i=1 J ii+^+^+jj-j, i=1

1<r<j r = ii + "+j

Finally, with C = Cj!, it holds

(Vk € Z+) (3m € Z+) (3C > 0) (3= ^inf (ei,ej-)) (Ve < £') |/£ ou^^r < Ce-m,

which means that (/£ ou£)£ € Maaa. This composition does not depend on the representatives. Indeed, suppose that (v£)£ € Maaa and (g£)£ € MT(C) are others representatives of « and F respectively. Set (n^ := ((v£)£— (u^) € N^aa and (m£)£ := ((/)£— (g^) € NT(C). To show that (/£ou£-g£ov£)£ € N^aa, since (/£ou-g£o€ Maaa, according to Proposition 3(1), it is enough to prove that (/£ o u£ — g£ o v£)£ satisfies (13). Indeed, we have

(/£ 0 U£ — g£ 0 V4,c,R < (/£ 0 U£ — ^ 0 V4,co,R + (/£ 0 V£ — & O ^G.c.R (/£(u £) — (u £ + n£)|g^,R + |m£(v £)|G^,R < |n£ |G^,R |/£(u £)|G^,R + |m£(v£)|G^,R"

It is clear that

(Vk € Z+) |ne|o>M>R |/e(«e)|o,«,R — °(ek), e ^ 0, and also | | ( )

(Vl € Z+) |me(ve)|o>TO>R — °(el), e ^ 0. Therefore, | | ( )

(Vq € Z+) |/e O Ue — ge ◦ ve^^ — °(eq), e ^ 0. Let V — (Vaa, + Vcor) € Gaaa. As F o V — iV(Vaa) + (FV(V) — iV(Vaa)), then F o V — F(Vaa) + (-F(Vaa + Vcor) — iV(Vaa)) on J.

In view of [22, Proposition 9], we obtain .F(Vaa) € Gaa. It remains to prove that^F(Vaa+Vcor) — FV(Vaa) € G+,0. Since Gaaa and Gaa are contained in Gb then iV(Vaa + Vcor) — F(Vaa) € Gb. It suffices to show that

(V e € I) /e (M«a,e + Ucor,e) y*e (uaa,e) € B+,o,

where (/e)e, (uaa,e)e and (ucor,e )e are respective representatives of FV, Vaa and Vcor. The classical result on composition of asymptotically almost automorphic function with continuous function shows that the correctiveJerm of /e(Uaa,e + UCOr,e) is /e(Uaa,e + Ucor,e) — /e(«aa,e) and the fact that -F(Vaa + Vcor) — -F(Vaa) € Gb gives

(Ve € I) (/e(Uaa,e + UCOr,e) — /e(Uaa,e)) € B.

By [9, Proposition 5 (5)], we have

(V e € I) /e(Uaa,e ) € C+,o n B — B+,o. >

7. Linear Neutral Difference Differential Systems

We consider linear neutral difference differential systems for the unknown vector function V — (Vi,..., Vn),

LwV :— ^ ^ AVij (rWj. V)(i) + KV * V — /, (27)

i=o j=o

where^/ — (/1,..., /„) € (Gaaa)n, w — (w)o<j<q C R+ and for i < p, j < q, Ay — (Aj)1<r>l<n and KK — (KKrl) 1<rl<n are square matrices of almost automorphic generalized functions and L1 — generalized functions respectively.

Lemma 2. If V € (Gaaa)n then V € (Gaaa)n < If V € (Gaaa)n then due to results of Proposition 5 we obtain that

(V i < P, j < q) (V W — (W )o<j<q C R+) (rWj. V)W) € (Gaaa)" ,

and also

KK * V € (Gaaa)" ,

so

(EE Aj K V (i) + KK * « ) € ,

V i=0 j=0 /

i. e. S € (Gaaa)n >

Definition 4. A generalized function u € (Gb)n is said a generalized solution of (27) on J if it satisfies

(n p q \

EEE^j (ji^)(i) + Kri * ui>e - frA € Nb (J), r = i=i i=i j=i J£

where (ui>£)£, (A.rj£)£, (Krz)£ and (fr,£)£, r, l = 1,..., n, are respective representatives of ur, Aj, KKri and fr.

We give the main result of this section.

Theorem 5. Let f = (faa, + fcor) € (Gaaa,)n, the equation (27) admits a generalized solution u € (Gaaa)n on J if and only if there exist v € (Gaa)n and £y € (G+,0)n such that

Lw U = fa« on R, (28)

and

wv = fcor on J. (29)

< If u = (uaa + uCor) € (Gaaa,)n is a generalized solution of (27 ), then the equations (27) explicitly are written as follows

f n / p q __N

E E E AVlj (rWj. SO(i) + kv 11 * uA = fi,

i=1 V i=0 j=0 /

n / p q _ \ —

E E E Aji (rWj.Si)(i) + Kun1 * Si = fn. ^ i=1 \ i=0 j=0 /

Let (uaa)i)£)£,

(^j)^ (faa,l,£)£ and (fcor,l,£)£ be respective representatives

of u«a,i, Scor,i, A^j, fa«,i and f^i, 1 ^ l ^ n. For every r = 1,..., n and e € I, denoting

n / p q \

S«a,r,£ := ^ ^ I ^ ^ ^ ^ Arj,£ Uaa,i,^ + K£ * Uaa,i,£ I j

i=1 V i=1 j=1 /

n / p q

SCOr,r,£ :- ^ ^ ( ^ ^ Arj,£ UCOr,i,^ + K£ * UCOr,i,£ I ,

i=1 Vi=ij=i /

we have V e € I, Saa,r,£ € Baa and £c0r,r,£ € B+,0, and also [(S^r,^] € Gaa and )£] € G+,0. Since u is a generalized solution of (27) on J, it means that

(V k € Z+) (V m > 0) (3 cfc > 0) (3 efc € I) (V e < efc) (V x € J)

^ ^ |SOQ,),r,£(x) — f(ltt,r,£(x) + SCOr,r,£(x) — fCOr,r,£(x) | ^ e , r 1, . . . , n. (30)

j^fc

For any real sequence (sm)meN, such that sm ^ there exist (sTOp(£) )p a subsequence of (s m)meN, such that taking the translate at sTOp(£) — sTOq(£) in the estimate (30) and let p, q ^ we obtain due to [9, Proposition 9], that

(Vk € Z+) (Vm > 0)(3 cfc > 0)(3efc € I) (Ve < efc) (Vx € J)

E |Sj),r,£ (x) — /j)r>£ (x)| < cfcem (V r = 1,...,n), j ^fc

consequently, by [9, Proposition 3(5)], it holds

(Vk € Z+) (Vm > 0)(3 cfc > 0)(3efc € I) (Ve < efc) |Saa,r, £ — /aa,r,£ |fc;^;R ^ cfc£™ (V r = 1, . . . ,n),

i. e.

(Saa,r,£ — /aa,r,£)£€ NB(R) (Vr = 1,...,n), (31)

which means that Uaa = (Uaa>1,..., Uaa>ra) is a generalized solution of (28) on R. By (30) and (31), we deduce

(Scow — /cor,r,£)£ € NB (J) (Vr = 1, . . . , n),

i.e. Ucor = (Ucor>1,... ,Ucor>ra) is a generalized solution of (29) on J.

Conversely if there exist U € (Gaa)n and w € (G+;o)n such that (28) and (29) hold, then we have U := (U + W) € (Gaaa)n is a generalized solution of (27) on J. >

Remark 3. Theorem 5 generalizes Theorem 6 of [21] and Theorem 3 of [9]. As a particular case we consider linear systems of ordinary differential equations

U + AU = /, (32)

where A is a square matrix of almost automorphic generalized functions.

Corollary 1. Let / = (/aa + /cor) € (Gaaa)n, the system (32) admits a generalized solution U € (Gaaa)n on J if and only if there exist U € (Gaa)n and w € (G+;o)n such that

U + AU = /"aa on R,

and

W + AWU = /cor on J. Let U = [(u£)£] € GB, xo € R and define the primitive of U by U = [(U£)£], where

x

U£(x) := J u£(t) dt, e € I.

xo

Corollary 2. A generalized function U = (Uaa + Ucor) € Gb is a primitive of U = (Uaa + Ucor) € Gwa on J if and only if

Uaa is a primtive of Uaa on R,

and

Ucor is a primitive of Ucor on J.

References

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3. Bochner, S. A New Approach to Almost Periodicity, Proceedings of the National Academy of Sciences USA, 1962, vol. 48, no. 12, p. 2039-2043. DOI: 10.1073/pnas.48.12.2039.

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5. Stepanoff, V. V. Sur Quelques Generalisations des Fonctions Presque Periodiques, Comptes Rendus de l'Académie des Sciences, Paris, 1925, vol. 181, pp. 90-92.

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8. N'Guerekata, G. M. Some Remarks on Asymptotically Almost Automorphic Functions, The Mathematical Revue of the University of Parma (4), 1987, vol. 13, pp. 301-303.

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Received February 7, 2022 Chikh Bouzar

Laboratory of Mathematical Analysis and Applications, Université Oran 1,

Ahmed Ben Bella, 31000, Oran, Algeria,

Professor

E-mail: ch. bouzar@gmail. com

https://orcid.org/0000-0002-1081-0993

Meryem Slimani

Laboratory of Mathematical Analysis and Applications, Universite Oran 1,

Ahmed Ben Bella, 31000, Oran, Algeria,

Researcher

E-mail: [email protected] https://orcid.org/0000-0003-4315-431X

Владикавказский математический журнал 2023, Том 25, Выпуск 2, С. 38-55

АСИМПТОТИЧЕСКАЯ ПОЧТИ АВТОМОРФНОСТЬ ДЛЯ АЛГЕБР ОБОБЩЕННЫХ ФУНКЦИЙ

Бузар Ш.1, Слимани М.1

1 Лаборатория математического анализа и приложений, Университет Оран 1, Ахмед Бен Белла, 31000, Оран, Алжир E-mail: [email protected], [email protected]

Аннотация. Цель статьи — изучение понятие асимптотической почти автоморфности в контексте обобщенных функций. Вводится алгебра асимптотически почти автоморфных обобщенных функций, содержащих пространство гладких асимптотически почти автоморфных функции как подалгебру. Фундаментальное значение этой алгебры связана с невозможностью умножения распределений; оно также содержит асимптотически почти автоморфные распределения Соболева — Шварца как подпространство. Более того, показано, что введенная алгебра устойчива относительно некоторых нелинейных операций. Как побочного результата приводится результат типа Сили о продолжении функций в контекст алгебры ограниченных обобщенных функций и алгебры ограниченных обобщенных функций, обращающихся в нуль на бесконечности; эти результаты используется для доказательства фундаментального результата о единственности разложения асимптотически почти автоморфной обобщенной функции. В качестве приложений рассмотрены разностно-дифференциальные системы нейтрального типа в рамках изучаемой алгебра обобщенных функций.

Ключевые слова: асимптотическая почти автоморфность, обобщенные функции, нейтральные дифференциально-разностные уравнения.

AMS Subject Classification: 46F10, 34K14, 46F30.

Образец цитирования: Bouzar, Ch. and Slimani, M. Asymptotic almost automorphy for algebras of generalized functions // Владикавк. мат. журн.—2023.—Т. 25, № 2.—C. 38-55 (in English). DOI: 10.46698/j1917-4964-8877-l.