ASYMPTOTIC ALMOST AUTOMORPHY OF FUNCTIONS AND DISTRIBUTIONS

URAL MATHEMATICAL JOURNAL, Vol. 6, No. 1, 2020, pp. 54-70

DOI: 10.15826/umj.2020.1.005

ASYMPTOTIC ALMOST AUTOMORPHY OF FUNCTIONS AND DISTRIBUTIONS

Chikh Bouzar

Laboratory of Mathematical Analysis and Applications, University of Oran 1, Ahmed Ben Bella, 31000, Oran, Algeria ch.bouzar@gmail.com

Fatima Zahra Tchouar

University Center of Ain Temouchent, 46000 Ain Temouchent, Algeria fatima.tchouar@gmail.com

Abstract: This work aims to introduce and to study asymptotic almost automorphy in the context of Sobolev—Schwartz distributions. Applications to linear ordinary differential equation and neutral difference differential equations are also given.

The paper aims to study asymptotic almost automorphy in the context of functions and Sobolev-Schwartz distributions, it is well known that the concept of almost automorphy is strictly more general than the almost periodicity studied in a full generality by H. Bohr, see [4] and [8]. The concept of asymptotic almost periodicity as a perturbation of almost periodic functions by functions that vanish at infinity belongs to M. Frechet in [9], one of the main motives of which is the introduction of this concept in obtaining the existence of an almost periodic solution to differential equations if they admit an asymptotic almost periodic solution. In the same vein as Frechet motivation, we study the existence of solutions of linear neutral difference differential equations with variable coefficients in the framework of asymptotically almost automorphic distributions. Almost periodicity in the framework of distributions extending the classical Bohr and Stepanov almost periodicity [16] is considered by L. Schwartz [13]. The paper [7] deals with asymptotic almost periodicity of distributions.

In [1] and [3], S. Bochner defined explicitly almost automorphic functions, where some basic properties have been established. He studied linear difference differential equations in the framework of almost automorphic functions in [2]. Almost automorphy of primitives and asymptotically almost automorphic functions are also considered, see [12, 18].

We first investigated the almost automorphy in the settings of distributions and generalized functions respectively in [6] and [5], then we addressed the issue of asymptotic almost automorphy in these contexts, see the communication [17].

The paper is organized as follows: the second section studies asymptotically almost automorphic functions following an appropriate definition, essential properties of these functions are proved; the third section deals with smooth asymptotically almost automorphic functions. The fourth section is dedicated to asymptotically almost automorphic distributions; we give their definition,

characterizations and some of their properties. The last section is an application to linear neutral difference differential equations of asymptotically almost automorphic distributions.

It is worth noting that the definition of an asymptotically almost automorphic function depends on the choice of authors, but in general the essential idea of the decomposition in the definition of an asymptotically almost automorphic function is preserved. The differences in their definitions lie in the domain of definition of the considered functions, their regularity and finally in the choice of the interval of decomposition. We consider functions defined, continuous and bounded on the whole space of real numbers r and the decomposition on the closed interval [0, +to[. So, we have to precise some results on asymptotically almost automorphic functions. Let Cb denotes the space of bounded and continuous complex-valued functions defined on r, endowed with the norm of uniform convergence on r, it is well-known that (Cb, ||-||^) is a Banach algebra. Let w € r and f, p functions, we recall that the translation operator rw is defined by f (■) = f (■ + w), and p by p(x) = p(-x). Denote j := [0, +to[.

Definition 1. The space C+,o is the set of all bounded and continuous complex-valued functions defined on r and vanishing at +to.

We give some properties of the space C+,o which are proved in a straight way.

Proposition 1. The following is true:

(1) The space C+,o is a Banach subalgebra of Cb.

(2) twC+,o C C+,o, Vw € r.

(3) C+,o x Cb C C+,o.

(4) C+,o * L1 C C+,o.

(5) Let h € C+,o, if h& exists and is uniformly continuous on j, then there exists a function H € C+,o such that H = h& on j.

+(» x

(6) There exists H € C+,o a primitive of h on j if and only if f h (t) dt < to and f h (t) dt is

bounded on j.

Remark 1. In (5) if h& exists and is uniformly continuous on r, then H = h& on r.

+c»

Remark 2. If h is a locally integrable function, we denote by f h (t) dt the improper integral,

+c» +c»

and f h (t) dt < to means f h (t) dt is finite. oo

Recall some properties of almost automorphic functions, see [1, 3, 12, 18].

Definition 2. 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) exists for every x € r,

lim g(x — smk)= g(x) for every x € R.

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

Remark 3. The function g is not necessary continuous but g € Lc

Proposition 2. The following is true:

(1) The space Caa is a Banach subalgebra of Cb.

(2) Caa C Caa, Vw € r.

(3) C aa * L C Caa(4) C aa n C+,0 = {0} .

(5) A primitive of an almost automorphic function is almost automorphic if and only if it is bounded.

We give now the definition of an asymptotically almost automorphic function.

Definition 3. 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. The space of asymptotically almost automorphic functions is denoted by Caaa.

Example 1. Caa C Caaa and C+,0 C Caaa.

It can be seen easly that the decomposition of an asymptotically almost automorphic function is unique on j, so if f € Caaa and f = g + h on j, where g € Caa and h € C+,0, the function g is said the principal term of f and the function h is the corrective term of f, we denote them respectively by faa and fcor. Then the notation f = (faa + fcor) € Caaa means that faa € Caa, fcor € C+,0 and f = faa + fcor on j.

Proposition 3. The following is true:

(1) TuCaaa C Caaa, Vw € r+.

(2) Caaa X Caa C Caaa.

(3) Caaa * C Caaa.

(4) Let f € Caaa and 0 is a continuous function on c, then 0 o f

€ Caaa.

(5) If f = (faa + fcor) € Caaa> then \\\\faa ^ supxgJ |f (x)|. In particular, for f € Caa and

w € r, \\\\f = supx>w If(x)|.

(6) Let (fm)meN = (fm ,aa + fm,cor )m C Caaa converges uniformly on j to a function f, then there exists 0 = (g + h) € Caaa, such that 0 = f on j, g € Caa is the uniform limit on r of (fm,aa )m and h € C+,0 is the uniform limit on j of (fm ,cor) m.

Proof. The proofs of (1) and (2) are easy.

(3) Let ^ € L1 and f = (/„„ + fcor) € Caaa. Since f = faa + (f - faa), where (f - /„„) € C+,o, it follows from Proposition 2-(3) and Proposition 1-(4) that f * ^ € Caaa. Now we show explicitly the principal part and the corrective part of f * For x € j, we have

(f * (x) = J f (y) ^ (x — y) = J f (y) ^ (x — y) + J (faa (y) + fcor (y)) ^ (x — y) dy,

= (faa * (x) + (fcor * (x) + J (f — faa — fcor) (y) ^ (x — y) dy.

By Proposition 2-(3), (faa * € Caa and by Proposition 1—(4), (fcor * € C+,0. On the other hand, for x € r,

/ (f — faa — fcor) (y) ^ (x — y) dy = J (f — faa — fcor) (x — y) X]x,+x[ (y) ^ (y) dy.

It is easy to see that the latter function is continuous and bounded on r and by the dominated

convergence theorem it vanishes at infinity. Then f *^ = (^aa + ^cor) € Caaa, where ^aa := faa *^

and ^cor := fcor * ^ + J (f — faa — fcor) (y) ^ (. — y) dy. -•

(4) Let f = (faa + fcor) € Caaa and 0 be a continuous function on c, then it is well-known that 0 (f) € Cb and also 0 (faa) € Caa. On the other hand, it is easy to see that the function 0 (f) — 0 (faa) defined on r belongs to C+0. Consequently we have 0 (f) = (0 (f)aa + 0 (f)cor) € Caaa, where

(5) Let f = (faa + fcor) €

Caaa and (smk)k a subsequence of (sm)mgN C j which tends to

infinity. Let x € r and k0 € z+ such that the sequence (x + smk)k>ko C j tends to infinity, then for k > k0, we have

| faa (X + Smk )| < |f (x + Smk )| + |/cor (x + Smk )| < SUp |f (x) | + |/cor (x + STOfc )|

so Vx € r,

It follows then

Ifaa(x)| = lim |faa(x + smk)I < sup |f (x)|.

I faa(x) | = lim | faa(x — smk )|< sup |f (x)|, Vx € r.

Consequently, we obtain the results.

(6) Let (fm)m = (fm,aa + fm,cor)m C Caaa converges uniformly to f on j, by (5) we have

\\\\fn ,aa fm,aa\\\\x < sup |fn(x) — fm(x) |,

hence (fm,aa)meN is a Cauchy sequence in the Banach space Caa, i.e. (fm,aa) converges uniformly on r to a function g € Caa. Let&s define the function h by

h( ) = / (f — g)(x), x > 0,

h(x) I (f — g)(0), x < 0.

Then h € and (fm cor)m converges uniformly on j to h, i.e. lim h (x) = 0 hence h e C+ 0. Define 0 = g + h on r, then 0 € Caaa and 0 = f on j. □

The space (Caaa, ||-||TO) is complete and it is a consequence of point (6). Corollary 1. The space (Caaa, ||-||TO) is a Banach subalgebra of We have the following results on the derivative and the primitive.

Proposition 4. The following is true:

(1) Let f = (faa + fcor) € Caaa be such that f& exists and is uniformly continuous on j, then there exists 0 = (g + h) € Caaa, such that 0 = f& on j, (fa«)& = g on r and (fcor)& = h on j.

(2) Let f = (faa + fcor) € Caaa be such that f is uniformly continuous on j, then there exists F €

Caaa being a primitive of f on j if and only if J faa (t) dt is bounded on r, / fcor (t) dt is

bounded on j, and J fcor (t) dt < to. o

Proof. (1) Let (cm)meN C j converging to zero and define the sequence (0m)meN C Caaa by

<f>m{x)=f{X + am)-f{X\\ X<=R Cm

= J f &(x + 6»Cm)d6», x € j.

then the sequence (0m)m converges uniformly to f& on j and for x € j,

, / \\__/q.q. (x "b Cm) /q.q. (x) , . .__/cor (x "b CTO) /cor (x)

<Pm,aa (,XJ .— , Ç>m,cor (,XJ .—

By (1), there exists 0 = (g + h) € C«««, such that 0 = f& on j, g € C«« is the uniform limit of (0m,«a)m on r and h € C+,o is the uniform limit of (0m,cor)m on j. Hence (faa)& := lim 0m,«« = g

on r and (fcor)& := lim 0mcor = h on j.

(2) If F = (Faa + Fcor) € C««« is a primitive of f on j, then F& = f is uniformly continuous on j. By (2), (Faa,)& € C««, there exists h € C+,0 such that (Fcor)& = h on j and F& = (Faa)& + (Fcor)& on j. Consequently, by Proposition 1-(6) and Proposition 2-(5), we obtain the result. Conversely, as

+œ x

f fcor (t) dt < œ and J fcor (t) dt is bounded on j, by Proposition 1-(6) , there exits H € C+,0 00

which is a primitive on j of fcor and as J" f«« (t) dt is bounded on r, by Proposition 2-(5), there

exits G € which is a primitive on r of faa, so F := G + H is a primitive on j of f. □

Corollary 2. Let f = (faa + fcor) € Caaa such that f& exists and is uniformly continuous on r, then f& = (g + h) € Caaa, where (faa)& = g on r and (fcor)& = h on j.

Let E (i) be the space of infinitely derivable functions on i = r or j , and p € [1, , the space

vlp (i) := € E (i) : Vj € z+, <^(j) € Lp (i) } endowed with the topology defined by the family of seminorms

Mk,p,I :=Y1 Nlp(I)> k € ^ j<k

is a Frechet subalgebra of E (i). The spaces Dlp (i) studied in [13] are connected with the classical Sobolev spaces Wm&p (i), see [15]. We denote B (i) := (i). Let B be the closure in B := B (r) of the space D of smooth functions with compact support.

Remark 4- By the definition ^ € B (j) requires that lim ^>(j)(x) exists Vj € z+.

Let B+,0 be the space of smooth functions vanishing at infinity, i.e.

B+,0 := {^ € E (r) : Vj € z+, <^(j) € C+,0}. We endow B+,0 with the topology induced by B.

Proposition 5. The following is true:

(1) The space B+,0 is a Frechet subalgebra of B.

(2) twB+,0 C B+,0, Vw € r.

(3) B+,0 X B C B+,0.

(4) B+,0 * L1 C B+,0.

(5) B+,0 = C+,0 n B.

(6) There exists H € B+,0 which is a primitive on j of h € B+,0 if and only if f h (t) dt is

bounded on j and J h (t) dt < to.

Proof. (1) It is easy to see that B+,0 is an algebra and since B is complete, it suffices to show that B+,0 is closed. Let (hm)meN be a sequence of B+,0 that converges to h € B, i.e. Vi € z+, (h^ )m converges uniformly on r to h(i). By Proposition 1-(1), h(i) € C+,0, Vi € z+, i.e. h € B+,0.

(2) This inclusion is obvious.

(3) If € B and h € B+,0, then by Leibniz&s formula and Proposition 1-(3), Vi € z+, (h^)(i) € C+,0.

(4) Let ^ € L1 and h € B+,0, then by Proposition 1-(4), Vi € z+, (h * ^)(i) = h(i) * ^ € C+,0.

(5) It is clear that B+,0 C C+,0 n B. Conversely, if h € C+,0 n B, then h& is uniformly continuous on r, so by Remark 2, h& € C+,0. By repeating this to all derivatives, we obtain that h € B+,0.

(6) The necessity is a consequence of Proposition 1-(6). To prove the sufficiency we need the following preliminary result on extension operators, it can be obtained from [14]: there exist

two sequences of real numbers (al)leZ+ and (bl)leZ+ such that b < 0, Vl € z+, and the operator E : B (j) — B (r) defined by

f f (x) if x > 0,

Ef(x) := <

af (bix) if x < 0

x +c»

is linear and continuous. Suppose that J h (t) dt is bounded on j and f h (t) dt < to. By

Recall the definition and some properties of the space of smooth almost automorphic functions, see [6] for details.

Baa := € E : Vj € z+, ) € C«*}.

Proposition 6. The following is true:

(1) Baa is a Frechet subalgebra of B.

(2) TWBaa C Baa, Vw € r.

(3) Baa * L1 C Baa.

(4) Baa = Caa n B.

(5) Let f € Baa and F is its primitive on r, then F € Baa if and only if F is bounded.

We now introduce smooth asymptotically almost automorphic functions.

Definition 4. The space of smooth asymptotically almost automorphic functions is denoted and defined by

Baaa := € E : Vj € z+, <^(j) € Caa4.

Example 2. Baa C Baaa and B+,0 C Baaa.

We endow Baaa with the topology induced by B. The following proposition is proved in the same way as Proposition 5 by using results of Propositions 3 and 4.

Proposition 7. The following is true:

(1) The space Baaa is a Frechet subalgebra of B.

(2) TWBaaa C Baaa, Vw € r.

(3) Baaa X Baa C Baaa.

(4) Baaa * L1 C Baaa.

(5) Baaa = C aaa nB.

(6) There exists F € Baaa being a primitive on j of f € Baaa if and only if J faa (t) dt is bounded,

x + x

on r, / fcor (t) dt is bounded on J, J fcor (t) dt < to.

Remark 5. Baaa c Caaa n E.

We have the following result needed in the sequel.

Proposition 8. Let f € Baaa, i.e. f = faa + fcor and for i € n, f(i) = faa,i + fcor,i on j. Then faa,i = (faa)W on r and fcor,i = (fcor)w on j.

Proof. If f € Baaa, then f& is uniformly continuous on r and by Proposition 4-(2), we have f& = (faa)& + h on j, where (faa)& € Caa, h € C+,0 and (fcor)& = h on j. By hypothesis, f& = faa,1 + fcor,1 on j and since the decomposition of an asymptotically almost automorphic function is unique, then (faa)& = faa,1 on r and (fcor)& = fcor,1 on j. By repeating this to all derivative, we obtain the desired result. □

In order to prove the main result on linear neutral difference differential equations in the framework of asymptotically almost automorphic distributions, we need the following characterization of the space Baa.

Proposition 9. Let g € E, the following statements are equivalent:

(1) g € Baa.

(2) For every sequence (pm)meN C r there exist a subsequence (pmk )k and g € B such that for all x € r and i € z+, we have

g(i) (x) = lim g(i) (x + pmk) and lim g(i) (x — pmk)= g(i) (x). (3.1)

k^+x k^+x

Proof. (1) ^ (2) Let g € Baa, so Vi € z+, V(pm)m€N C r, 3(p mi k )k C (Pm)m, 3(gi)i C Lx such that Vx € r,

lim g(i)(x + Pmi k) =: gi(x) and lim gi(x — pmi k)= g(i)(x).

k^+x & k^+x &

There exist subsequences (pmn k )k, n € z+, of the sequence (pm)m such that

Vi < n, lim g(i) (x + pmn k)= gi(x), Vx € r. (3.2)

k^+x &

Indeed, the proof is done by induction, if g € Caa it is clear that (3.2) holds for n = 0. Now, let n € n such that (3.2) holds. As g and gn+1 € Lx such that Vx € r,

n € n such that (3.2) holds. As g(n+1) € Caa, there exists a subsequence (pm(n+1) k)k of (pmn k)k

gn+i(x) := lim g(n+1)(x + Pm(n+1) fc)•

Furthermore, as Vi < n, Vx € r, the subsequence (g(i) (x + pm(n+1) k))k is extracted from

(g(i)(x + pmri, k))k then

fclimxg(i)(x + Pm(n+1), k) = gi(x), Vx € R.

By construction, Vk,i, € z+, mi;k < m(i+1),k and since k -—> is strictly increasing

from n to n, then in particular we have mi;i < m^+^j < m(i+1) (i+1), Vi € z+. This gives that the map k i—> mk;k is strictly increasing from n to n. The sequence (pTOfc k)k, which we denote by (pmfc)k, is extracted from the subsequences (pmik)k, i € z+, which is in fact extracted from the sequence (pm)m. Consequently,

lim g(i)(x + pTOfc)= gj(x) exists Vx € r, Vi € z+.

With the same steps we have that

lim gi(x - Pmk) = g(i)(x), Vx € r, Vi € z+.

Let (cn)neN C j converging to zero and consider the sequence of functions (0ra;k)ra,fcgN defined on r by the equality

M) = + = f1 gl{. + ^ + dan)de_

Since g € Baa C B, then g& is bounded and uniformly continuous on r, so lim lim / g&( + pm, + = lim lim / g&( + p,

fc^+TOra^+TO Jo «,^+toJo

Consequently, Vx € r, lim lim 0nk(x) = lim lim k (x) which gives that Vx € r,

k^+œra^+œ & ra^+œk^+œ &

gi(x) = lim lim k(x) = lim lim k(x) := g0(x).

k^+œra^+œ ra^+œk^+œ

By iterating to all derivatives, we obtain that go € E and go = gi € Lœ, Vi € z+, i.e. go € B such that relations (3.1) hold.

(2) ^ (1) is obvious. □

The space of Lp—distributions, denoted by VLP, is the topological dual of DLq, where 1/p + 1/q = 1. The topological dual of B is denoted by DLi • The space of bounded distributions DL- is denoted by B&. The translate twT, w € r, of a distribution T € D is defined by <rwT,p) = <T,T_wp), V^ €D.

Definition 5. By B+0 we denote the space of distributions Q € B& vanishing at infinity, i.e. satisfying

lim <twQ, <p) =0, Vp € D.

We have the following characterizations of B+, 0 , see [7].

Theorem 1. Let Q € B&, the following assertions are equivalent:

(1) Q € B+ ,0.

(2) Q * € C+,o, Vp € D.

(3) 3fc € z+ and hj eC+ 0, 0 < j < k, such that Q = £ hj).

We study some properties of the space B+, 0. Proposition 10. The following is true:

(1) If Q € B+0, then Q(i) € B+0, Vi € z+.

(2) twB+0 C B+0, Vw € r.

(3) B+o xBC B+o(4) B+ o C B+ o.

(5) Let Q € B&, then Q € B+0 if and only if there exists a sequence (^>m)meN C B+,0 converging to Q in B&.

Proof. (1) and (2) are obvious.

(3) Let ^ € B and Q € B+ 0, then by Theorem 1—(3), there exist (hi)i<k C C+,o, such that

Q = E h(i). So

k k i / & \\ ^Q — £ ^h« — ££(-1)j( M (^(j) hi )(i-j ). i=0 i=0 j=0

By Proposition 1-(3), ^>(j)hi € C+,0, hence ^>Q € B+0.

(4) Let Q € B+0, then there exists (hi)i<k C C+,0 such that Q — ^ h(i , and let S € D^i,

by [13, Theorem XXV, Section 8, Chapter VI], there exist (^)j<m C L1 such that S — ^ ^(j).

j<m j=0

(Q * S) —* ^ )(i+j). i=0 j=0

By Proposition 1-(4), h& * ^j € C+,0, hence Q * S € B+,0.

lim <fim — Q in

m^+x

U : — {r_x<p : x €

(5) Let <m)mgN C B+ 0 such that lim <m = Q in B&. For a fixed ^ € D, the set

is bounded in DLi, so

sup |(0m * ^)(x) - (Q * ^)(x)| — sup |(0m - Q,T_x<p)| ,

— sup |(0m - Q,^)| -► 0,

i.e. (0 m * ^)meN C C+,0 is uniformly convergent to (Q * . By Proposition 1 (1), Q * € C+,0, V^> € D, and by Theorem 1, we obtain Q € B+,0.

Conversely, let Q € B+ 0 and take a sequence of positive test functions (0m)meN such that

supp 6m C

and y dm (x) dx = 1.

Define 0m := 0m * Q € B+,0 , we have

(0m - Q, f ) = (Q, 0m * f - f > ,

and there exist l € z+, C > 0 such that

KQ,0m * f - f> I < C 10m * f - f ^ , Vf € DL1. By Minkowski&s inequality and the mean value theorem we obtain for t € ]0,1[,

(0m * f)W - fW||Li < J Uy^ |y||^(i+1) (X + ty)|dx)dy

< — Mi+1,1»

Let U be a bounded set of Dli and f € U, then 3M > 0 such that

sup \\em * if - if , <— —► o,

which gives 0m —^ Q in B&. D

We recall the definition, characterizations and some properties of almost automorphic distributions, see [6].

Definition 6. A distribution T € B& is said almost automorphic if it satisfies one of the following equivalent conditions:

(1) T * f € , Vf € D.

(2) 3k € z+ and gj € Caa, 0 < j < k, such that T = £ g(i).

(3) For every sequence (sm)meN C r, there is a subsequence (smfc)k such that

S := lim TSm T exists in D&,

fc^+TO k

lim r-smfc S = T in D&.

fc^+TO

(4) There exists a sequence (fm)meN C Baa converging to T in B&. We denote by the space of almost automorphic distributions defined on r.

Proposition 11. The following is true: (1) If T €B^, then T(i) €B^a, Vi € z+.

(2) twB&aa C B&aa, Vw € r.

(3) B&aa X Baa C B&aa•

(4) Baa cB&&.

(5) B&a nB+o = {0} .

We now give the definition of asymptotically almost automorphic distributions.

Definition 7. A distribution T € B& is said asymptotically almost automorphic if there exist P € B&aa and Q € B+ 0 such that T = P + Q on j. We denote by B&aaa the space of asymptotically almost automorphic distributions.

Remark 6. The equality T = P + Q on j means that Vp € D+, (T, p) = (P, p) + (Q, p), where D+ := {p € D : supp p C j} .

Proposition 12. The decomposition of an asymptotically almost automorphic distribution is unique on j.

Proof. Let Pi, P2 € B&aa and Qi, Q2 € B+ 0 such that T = Pi + Qi = P2 + Q2 on j, then we obtain that P1 — P2 € B+ 0, by Proposition 11-(5), P1 — P2 = 0. Hence Q1 = Q2 on j. □

Notation 1. If T € B&aaa and T = P+Q on j, we call P the principal term and R the corrective term of T and we denote them respectively by Taa and Tcor. This is summarized by the notation

T = (Taa + Tcor) € Baaa.

Example 3.

Theorem 2. Let T € B&, the following assertions are equivalent:

(1) T € B&aa.

m)mgN C Baaa such that Hni — T in B .

(3) T * p € Caaa, Vp € D.

(4) 3k € z+ and / € Caaa, 0 < j < k, such that T = £ fj

Proof. (1) ^ (2) Let T € B&aaa, by definition T = Taa + Tcor on j. By the characterization of B& there exists (^>m)meN C Baa such that lim ^m = Taa in B&. It is easy to prove that

& m^+x

T — Taa € B+0, so by Proposition 10-(5) there exists (0m)meN C B+o such that lim 0m =

+& & m^+x

T — Taa in B&. Set 0m := <£m + 0m, m € n, then (0m)meN C Baaa and we have T — dm = (T — Taa) — 0m + (Taa — <^m). Hence we obtain lim 0m = T in B&.

n^+x

(2) ^ (3) As in the proof of Proposition 10-(5), if (<m)meN C Baaa is such that lim <m — T

meN m^+x

in B&, then for V^> € D we have

sup |(<m * <^)(x) — (T * <^)(x)| = sup |«m — T,T-x<£)| -► 0.

xeR xeR m^+x

That is (<m * ^)m€N C Caaa converges uniformly on r to (T * , it follows that T * ^ € Caaa, V^ € D.

(3) ^ (4) For n € z+, consider the function

x > 0.

Then En € Cn-2, suppEn C j and E^ = 5. Take a function 7 € D such that 7 = 1 in the neighborhood of 0, a direct calculus gives (7En)(n) = 5 + Zn, where

Zn = £(n)Y(n-fc) € D.

As T € B , we have

T = (7En * T)(n) — T * Zn,

where T * Zn € Caaa. It remains to show that 7En * T € Caaa for a suitable n. There exist m € Z+ and C > 0 such that

|(T,0)|< C |0|m,i , V0 €DL1. Take n = m + 2, then 7Em+2 € D^l, where

Dmx := {<£> € Cm : Vj < m, € L1} endowed with the norm | ■ |m>1.

We have D ^ Dli ^ D^l and there exists a sequence (0k)keN C D such that (0k)k converges to 7Em+2 with respect to the norm | ■ |m>1, so

| (T * 0k) (x) — (T * 7Em+2) (x) | = | (T, T-x9k — T-( (7^m+2) > | ,

< C 1 T-xk — T-x (7Em+2) |m,1 ,

< C |0k — YEm+2|m,1 ,

consequently,

sup |(T * 0k) (x) — (T * 7Em+2)(x)|< C |0k — 7Em+2|m1 "+ 0.

xeR & k^+x

i.e. the sequence of functions (T * 0k)keN C Caaa converges uniformly on r to T * 7Em+2, hence T * 7Em+2 € Caaa.

(4) ^ (1) Let T = ^ /j\\ where / € Caaa, j = 0,k, so j=o j

t = £ fijsl + e ft on j. j=0 j=0

Then by Theorem 1 and Definition 6, P = £ /j € B^ and Q = £ /j(jc)or € B+0. Therefore

j=o & j=o

T = P + Q on j, i.e. T € Baaa. □

Remark 7. Connected with this theorem, let us quote the preprint [10]. The authors thank the referee for pointing out the recent work [11].

We have the following properties of B^aa. Proposition 13. The following is true :

(1) If T € Baaa, then Vi € z+, T(i) = (Taa) + Ti^r) € Baaa.

(2) TwBaaa C Baaa, Vw € r+.

(3) Baaa X Baa C Baaa.

(4) Baaa * DLi C Baaa.

Proof. The proof of the assertions (1)-(3) follows from the definition, the uniqueness of the decomposition and the same properties satisfied by the space Baa.

(4) Let T € Baaa, by the previous Theorem, there exist (/i)i<k C Caaa such that T = E /i(i).

Let S € DL1, by [13, Theorem XXV, Section 8, Chapter VI], there exists (Vj)jXm C L1 such that

m , .. _

S = E Vj). Thus j=0

(T* S) = ££(/& * V)(i+j).

By Proposition 7-(4), /& * Vj € Caaa. By [13, Theorem XXVI, Section 8, Chapter VI] we have B& * DLi C B&, hence S * T € Baaa. □

A linear neutral difference differential equation is an equation

p q d Luu := J] J]+ * M =

where (aij)i<p, j<q C ^ K € L1 and W = (Wjj<q C r+.

By the properties of the space Baaa it is clear that Baaa C Baaa. To prove the main result of this section we need the following result.

Lemma 1. Let T € B&,g,g € B and (sm)meN a sequence of real numbers such that

T:= lim Tsm T in D&, (5.1)

g(j) (x) = lim tsmg(j) (x), Vx € r, Vj € z+, (5.2)

lim Tsm (gT) = gT in D&.

Proof. Let (sm)meN, T € B& and g, g € B such that (5.1) and (5.2) hold. As T € B&, 3C > 0, 31 € z+, such that

|<T,^}|< C , V^ €DL1.

So Vf € D,

|<Tsm(gT) -gT,f)| = |(T,gT-smf} — <TZs,gf>|, = |<TSm T,fTsm g} — <T\\gf)|,

< | <TsmT — T, f f ) | + | <TsmT, (Tsmg —- g)f ) |,

< |<Tsm T — T,gf)| + C |(TSm g — g)f|u,

< KTsmT — Tf,gf | + C £ II ((TSmg — g)f)(i) ||Ll.

|| \\\\&smy — y

The lemma is proved due to (5.1) and the following estimate

\\(0 II ^ ^

||((Tsmg — g)f)(i) ||L1 < ¿j / |g(j)(x + Sm) — g(j) (x)||f(i-j)(x)|dx —— 0,

j=o Vj/ R

which is due to the dominated convergence theorem. □

The main result of this section is the following.

The°rem 3.Let s€ Baaa,the equation t — s has a soiution t € Ba«a on j if and only if

V = S„„ on r (5.3)

there exist V € and W € B+0, such that

W = Scor on j. (5.4)

Proof. Suppose that equations (5.3) and (5.4) are satisfied, then

(V + W) = V + W = S„„ + Scor = S on j.

So T = V + W € BU, is a solution on j of T = S.

Conversely, let T € B^ be a solution on j of the equation T = S and let (sm)meN be a sequence of real numbers which converges to As Saa, Taa € and Scor, Tcor € B+0, and by Proposition 9, there is a subsequence (smk)k of (sm) converging to and functions a^- € B such that Vx € r, Vi < p, Vj < q, we have

lim TSmfcaij (x) = aij (x) exists and lim T_Smfcaj (x) = aj (x), (5.5)

and the following limits exist in D ,

lim Tsm Taa = V and lim r_Sm, V = Taa, (5.6)

k—+x k k—+x k

, lim Ts-k Saa = P and lim T-smfc P =Saai k—+x k k—+x k

lim Tsm TCOr = 0 and lim t« Scor = 0. (5.7)

k—+x k k—+x k

Let gD, we have

<Tsmfc (LWT),^ = £]T(-1)^t,t-wj (aijT-smfc^)(i)} + <K * TSmfcT,^,

i=0 j=0 p q

TsmfcT,T-wj.(^smfcaij)(i)) + <K * TsmfcT,p),

i=0 j=0 p q

= E E (Tsmfc aijTwj. (Tsmfc T)(i), ^ + <K * Tsmfc T, p), i=0 j=0

EE+*• i=0 j=0

On [—Smk, we have TSmfc S = TSmfc T, i.e.

TSmk Saa + TSmk Scor = TSmk (LwTaa) + TSmk (LWTcor) = (Lw,kTsmk Taa) + (Lw,kTSmk Tcor) ,

By (5.5), (5.6), (5.7) and Lemma 1, the limits

Jj+X (Ts»k Saa + TSmk Scor) = Jjmx (Lw,kTsmk Taa) + Jim^ (LW,kTsmk Tcor) ,

P = V on r,

Consequently by (5.6) we obtain

i=0 j=0 dx

which gives

Saa = Taa on r.

Finally, the equation Saa + Scor = Taa + Tcor on j implies

Scor - Tcor on j,

hence the conclusion is true. □

Remark 8. The proof of the theorem appeals to Lemma 1 and particulary to Proposition 9 characterisating the introduced space of smooth asymptotically almost automorphic functions.

Remark 9. The result of the theorem remains valid if we consider systems. Other problems can be tackled within the space of asymptotically almost automorphic distributions.

The following result concerns primitives.

Corollary 3. Let S € B&aaa, the following propositions are equivalent :

(1) T € B&aaa is a primitive of S on j .

(2) There exist V € B&aa a primitive on r of Saa and W € B+0 a primitive of Scor on j such that

T = V + W on j. REFERENCES