Régis De La Bretèche
email: regis.de-la-breteche@imj-prg.fr

Gérald Tenenbaum
email: gerald.tenenbaum@univ-lorraine.fr

Remarks on the Selberg-Delange method

Keywords: Averages of multiplicative functions, Selberg-Delange method, Dirichlet series, powers of the Riemann zeta function. 2020 Mathematics Subject Classification: 11N37

Let be a complex number and let f be a multiplicative arithmetic function whose Dirichlet series takes the form ζ(s) G(s), where G is associated to a multiplicative function g. The classical Selberg-Delange method furnishes asymptotic estimates for the averages of f under assumptions of either analytic continuation for G, or absolute convergence of a finite number of derivatives of G(s) at s = 1. We consider different set of hypotheses, not directly comparable to the previous ones, and investigate how they can yield sharp asymptotic estimates for the averages of f .

Introduction and statement of results

In a series of papers published in 1953 and 1954, [START_REF] Sathe | On a problem of Hardy on the distribution of integers having a given number of prime factors, I, II[END_REF], [START_REF] Sathe | On a problem of Hardy on the distribution of integers having a given number of prime factors, III, IV[END_REF], L. G. Sathe studied the local laws of the arithmetic functions counting the number of prime factors, with or without multiplicity, a problem previously considered by Hardy and Ramanujan. Sathe's results provided asymptotic formulae while only upper and lower bounds were previously known. However Sathe's method, based on induction formulae, involved very long and technical estimates. In the same year 1954, Selberg devised a fruitful method based on the idea that the Dirichlet series of the search for probabilities may be expressed through Taylor coefficients of powers of the Riemann zeta function. This idea was then systematically developed by Delange [START_REF] Delange | Sur des formules dues à Atle Selberg[END_REF], [START_REF] Delange | Sur des formules de Atle Selberg[END_REF]. In the second author's book [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF] (latest edition, first in 1990), the results were generalized and made effective regarding various parameters-the method being there named after Selberg and Delange.

This theory provides estimates for counting functions associated to Dirichlet series of the form

(1•1) F (s) = ζ(s) G(s),
where is a complex number, ζ(s) is Riemann's zeta function, and G(s) satisfies suitable regularity conditions. Usually, G is associated with a multiplicative arithmetic function g, but this need not be so -see, e.g., [START_REF] Hanrot | Moyennes de certaines fonctions multiplicatives sur les entiers friables[END_REF]. In the sequel, we shall however concentrate on the case when g is indeed multiplicative. Then F (s) is the Dirichlet series of a multiplicative function f . In [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF], two types of assumptions on G(s) are considered : (a) analytic continuation at the left of the vertical line σ = e s = 1; (b) absolute convergence at s = 1 for a finite number of derivatives.

In a recent work [START_REF] Granville | Beyond the LSD method for the partial sums of multiplicative functions[END_REF], Granville and Koukoulopoulos, propose a third type of condition, implying mere convergence, instead of absolute convergence, for a finite number of right-derivatives of G(s) at s = 1 . However, their analysis actually rests upon a much stronger assumption, viz., for some constant A > 0, not necessarily an integer, (1•2)

p x g(p) log p = n x {f (p) -} log p x/(log x) A (x 2).
Some extra, secondary hypotheses are also needed for the values f (p ν ) at prime powers. Here and in the sequel, the letter p denotes a prime number. For the sake of comparison, it is worthwhile to note that hypothesis (b) above essentially amounts to p |g(p)|(log p) j /p < ∞ for a finite number of exponents j.

Similar, but weaker, conditions have been considered by Wirsing [START_REF] Wirsing | Das asymptotische Verhalten von Summen über multiplikative Funktionen[END_REF], in the frame of comparison theorems, evaluating the ratio of averages of f and of a non-negative majorant. Via a further weakening of the hypotheses, such results were improved in [START_REF] Wirsing | Das asymptotische Verhalten von Summen über multiplikative Funktionen, II[END_REF]. In [START_REF] Tenenbaum | Moyennes effectives de fonctions multiplicatives complexes[END_REF], the second author considered generalizations, and obtained effective forms of the results. In the present work, the viewpoint is quite different: taking advantage of the strength of assumptions like (1•2), one aims at directly deriving an asymptotic estimate for the averages of f . As in the classical Selberg-Delange approach, this also enters in the frame of comparison theorems, but the average of f (n) is now compared with that of τ (n)-the nth coefficient in the Dirichlet series expansion of ζ(s) -instead of being compared with that of a majorant.

Assuming (1•1), (1•2), and |f | τ r for some parameter r > 0, the main result in [START_REF] Granville | Beyond the LSD method for the partial sums of multiplicative functions[END_REF] states that, with J := A -1 and suitable coefficients {λ j (f )} J j=0 , we have, for x 3,

(1•3) M (x; f ) := n x f (n) = x(log x) -1 0 j J λ j (f ) (log x) j + O x(log 2 x) δ A,J+1
(log x) A+1-r , with Kronecker's δ-notation. Here and henceforth, log k denotes the k-fold iterated logarithm. Formula (1•3) is specially interesting when A is small, for less is then required on g. However, it furnishes no more than an upper bound when A r -e . Incidentally, under the weaker assumption that the series p g(p) p converges, and arguing as in [START_REF] Hall | Effective mean value estimates for complex multiplicative functions[END_REF] (see also [18; th. III.4.14]), theorem 1.1 of [START_REF] Tenenbaum | Moyennes effectives de fonctions multiplicatives complexes[END_REF] readily yields, for real f ,

(1•4) M (x; f ) x(log x) r-1-min{1,K(r-)} (x 2),
where K ≈ 0.32867 is optimal. Moreover, in the case = 0, the same technique furnishes

(1•5) M (x; f ) x(log x) r-1-min{1,(1-2/π)r} (x 2).
In particular, (1•4) supersedes (1•3) as soon as A < K(r -) 1. Analogous upper bounds are available for complex f , under suitable hypotheses upon f (p): see e.g. [START_REF] Hall | A sharp inequality of Halász type for the mean value of a multiplicative arithmetic function[END_REF].

The purpose of the present work is two-fold: (a) to investigate refinements of (1•2) enabling an improvement of the error term in (1•3) by replacing the exponent r of log x by e , as expected in view of standard estimates in the theory; (b) to propose a simpler and more natural (i.e. relying on a direct convolution argument) proof of (1•3), with possibly weaker hypotheses.

In the first direction, we meet the set goal when e 0, where is the exponent appearing in the generic assumption (1•1). This latter restriction is actually necessary for achieving target (a): we construct a family of counter-examples in Section 5.1.

In the following statement, hypothesis (1•2) is replaced by a short interval version, with the same value of A. The other assumptions concern |f |: we use those now classical introduced by Shiu [START_REF] Shiu | A Brun-Titchmarsh theorem for multiplicative functions[END_REF], although they could be somewhat weakened if needed. Accordingly, we define the class S(B) of those multiplicative functions f such that:

(i) |f (p ν )| B ν (ν 0), (ii) ∀ε > 0 ∃C = C ε : |f (n)| C ε n ε (n 1),
and, for r > 0, we consider the subclass S(B, r) of those f satisfying the extra assumption:

(iii) p x |f (p)| p r log 2 x + O(1) (x 3). Theorem 1.1. Let A > 0, B > 0, 0 < α < 1, r > 0, ∈ C, J := A -1 , and let f ∈ S(B, r) verify (1•6) x<p x+z f (p) log p = z + O z (log x) A (x 2, x 1-α z x).
Then, for suitable constants {λ j (f )} 0 j J and with ϑ := B + | | + 1, we have

(1•7) M (x; f ) = x(log x) -1 0 j J λ j (f ) (log x) j + O x(log 2 x) ϑ (log x) A+1-max(0, e ) (x 3).
The coefficients λ j (f ) may be described as follows. Representing f = τ * g consistently with (1•1), we prove the convergence of the series

(1•8) γ j (g) := n 1 g(n)(log n) j n (0 j J),
and derive

(1•9) λ j (f ) = 1 Γ( -j) +h=j α ( )γ h (g) !h! (0 j J),
where Γ is Euler's function and α ( )/ ! is the -th Taylor coefficient at the origin of {sζ(s + 1)} /(s + 1).

Alternatively, we also have

(1•10) λ j (f ) = d j {s F (s + 1)/(s + 1)} j!Γ( -j) ds j (0+) (0 j J).
In particular, λ j (f ) = 0 for all j if in an integer 0. As for our second aim, namely goal (b) described above, we apply friable convergence (see below) to show the following result, essentially equivalent, in the intersection of the respective validity domains, to [8; th. 1]. For r > 0, σ ∈]0, 1[, we define the class F(r, σ) comprising those complex multiplicative functions f such that

(1•11) v<p w |f (p)| p r log log w log v + O(1) (w v 2), p |f (p)| 2 p 2σ + ν 2 |f (p ν )| p νσ < ∞.
These conditions are weaker than those of [8; th. 1] and not directly comparable to those described in [8; § 7]. For instance, letting p k denote the kth prime number, conditions (1•11) allow f (p k 2 ) k/ log 2k, whereas the latter do not. Contrary to that of [START_REF] Granville | Beyond the LSD method for the partial sums of multiplicative functions[END_REF], our analysis does not ascribe a special role to the case when A is an integer. Theorem 1.2. Let A > 0, J := A -1 , r > 0, ∈ C, σ ∈]0, 1[, and let f ∈ F(r, σ) be a multiplicative function such that p → g(p) := f (p) -verifies (1•2). Then, for suitable constants β > 0 and {λ j (f )} 0 j J , we have

(1•12) M (x; f ) = x(log x) -1 0 j J λ j (f ) (log x) j + O x(log 3 x) β (log x) A+1-r .
We shall see that the λ j (f ) are still given by (1•10). In Section 5.2, we show that, even when e 0, the remainder term of (1•12) cannot be sharpened so as to meet that of (1•7) -although a possibility of some improvement remains open if f is real. To this end, we construct a family of counterexamples f ∈ F(σ, r) satisfying (1•2) with = 0, 0 < A < r (resp. 0 < A < 2r/π in the real case), but contravening the short interval condition (1•6), and for which the exponent r in (1•12) cannot be replaced by cr if c < 1 (resp. c < 2/π).

As a final remark, we note the following: as that of [START_REF] Granville | Beyond the LSD method for the partial sums of multiplicative functions[END_REF], our analysis heavily depends on the asymptotic expansion for

(1•13) T (x) := M (x; τ ) = n x τ (n)
provided by the Selberg-Delange method; since the arithmetical functions under consideration are of the form f = g * τ , refinements regarding hypotheses on g or on its generating series G(s) may hence be regarded as genuine parts of this theory.

On the case = r

The bound (1•4) shows that the error term of (1•3) is not optimal in the general case. Moreover, Theorem 1.1 provides fairly standard assumptions under which an essentially optimal remainder is achieved. However, inasmuch the power of the logarithm is concerned, (1•3) is expected to be sharp when = r. This latter case is discussed in [START_REF] Granville | Beyond the LSD method for the partial sums of multiplicative functions[END_REF], where it is confirmed that the error term may be replaced by a quantity x(log x) r-A-1 when = r 1, A / ∈ Z. Assuming = r essentially amounts to considering f 0. In this latter circumstance, an asymptotic formula with optimal remainder may be obtained in a very simple way under lighter hypotheses. We present the details below. Theorem 2.1. Let A > 0, σ ∈]0, 1[, r > 0, and assume f is a non-negative multiplicative function such that

(i) p x f (p) log p = rx + O x (log x) A (x 2), (ii) p f (p) 2 p 2σ + ν 2 f (p ν ) p νσ < ∞.
We then have

(2•1) M (x; f ) = λ 0 (f )x(log x) r-1 1 + O (log 2 x) δ 1,A (log x) min(1,A) (x 2), with λ 0 (f ) := 1 Γ(r) p 1 - 1 p r ν 0 f (p ν ) p ν • Proof. We have (2•2) M (x; f ) log x = n x f (n) log n + n x f (n) log(x/n) = m x f (m) p ν x/m p m f (p ν ) log p ν + x 1 M (t; f ) t dt = rx m x f (m) m + O R + S + x(log x) r-1 , with R := mp x p | m f (m)f (p) log p + p ν m x ν 2 f (m)f (p ν ) log p ν , S := m x xf (m) m(log 2x/m) A ,
and where the last integral has been estimated by an Halberstam-Richert type bound-see, e.g., [18; th. III.3.5]. Now, with α := (σ + 1)/2 < 1, we have

R m x f (m) p ν m x ν 2 {f (p)f (p ν-1 ) + f (p ν )} log p ν m x f (m)x α m α p ν 2 f (p)f (p ν-1 ) + f (p ν ) p νσ m x f (m)x α m α x(log x) r-1 ,
where we used assumption (ii). Moreover, partial summation furnishes

S x(log x) r-1-min(1,A) (log 2 x) δ 1,A .
It remains to note that, by [20; th. 3.3] (with κ = r and y = x), we have

m x f (m) m = 1 + O 1 log x λ 0 (f ) r (log x) r (x 2),
and carry back into (2•2).

3. Proof of Theorem 1.1

3•1. Preparation

Given ∈ C, let M( ) denote the class of those multiplicative functions f satisfying

(3•1) f (p ν ) = τ (p ν ) + τ (p ν-1 ){f (p) -} (p 2, ν 1).
When f ∈ S(B, r), we can write f = f * h where f ∈ M( ), h is multiplicative, supported on the set of squareful integers, and such that the series

n 1 h(n)(log n) j n ( j

3•2. Main estimates

The main part of the argument consists in providing effective estimates for averages of the function g defined above. Lemma 3.1. For a suitable constant c 0 and uniformly for x 3, 0 j J, k log 2 x, we have

(3•2) G j,k (x) := n x ω(n)=k g(n)(log n) j x(B log 3 x + c 0 ) k-1 (k -1)!(log x) A+1-j •
For further reference we note right away that a consequence of (3•2) is that

G j (x) := n x g(n)(log n) j j x(log 2 x) B (log x) A+1-j (j 0, x 3), (3•3) g j (x) := n x g(n)(log n) j n = γ j (g) + O (log 2 x) B (log x) A-j (0 j J, x 3). (3•4)
Indeed, it suffices to apply, e.g., lemma 1 from [START_REF] Tenenbaum | A rate estimate in Billingsley's theorem for the size distribution of large prime factors[END_REF] in order to note that, for large, constant D, the contribution of those n with ω(n) > D log 2 x is negligible and then appeal to (3•2) for k D log 2 x.

In the following proof and henceforth, we let P + (n)-resp. P -(n)-denote the largest-resp. the smallest-prime factor of an integer n > 1, and make the standard convention that P + (1) = 1,

P -(1) = ∞.
Proof of Lemma 3.1. By partial summation, it is enough to consider j = 0. We may also plainly restrict to bounding the subsum over n ∈] √ x, x]. Given a suitably large parameter K, set X := (log x) K and represent each n arising in (3•2) as n = md with P + (m) X, P -(d) > X. By, for instance, [16; lemma 2] (a Rankin-type bound), we see that the contribution to G 0,k (x) of d x 1/4 is, for suitable positive constants c j ,

d x 1/4 |g(d)| x 1/4 <m x/d P + (m) X |g(m)| d x 1/4 |g(d)| d x 1-c1/ log X x 1-c2/ log X . Whence (3•5) G 0,k (x) = s+t=k t 1 m x 3/4 ω(m)=s P + (m) X g(m)G t x m ; X + O x 1-c3/ log 2 x , with (3•6) G t (w; X) := x 1/4 <d w ω(d)=t P -(d)>X g(d) 1 t k, x 1/4 < w x .
Let H := αK, where α > 0 is the constant appearing in (1•6), and let δ := 1/(log x) H . Put I( ) :=]e δ , e δ( +1) ] ( > L := (K log 2 x)/δ). The contribution to (3•6) from those integers d having at least two prime factors in a same interval I( ) is

B t 1 j<t X<p1<•••<pt p1•••pt w pj+1<pj e δ w p 1 • • • p t δw(log w) B w (log x) A+2 ,
for a suitable choice of K and hence of H.

For the remaining integers, split the prime factors of the summation variable d in (3•6) among the various I( ) and consider the multiple sum over all hypercubes that are hit. If 1 < 2 < . . . < t is a sequence of admissible indexes, then δ( 1 + . . . + t ) log w since d w in (3•6). If some product v := t j=1 p j happens to be > w, we must have w < v we tδ . Therefore the total contribution of those admissible hypercubes containing at least one product > w is

w<v we tδ B ω(v) µ(v) 2 w(log w) B-1 δt w (log x) A+2 •
We have shown so far that, for relevant values of t and w, G t (w; X) = S t + O w (log x) A+2 , with, for some absolute constants C j (j = 0, 1),

S t := L< 1 <•••< t (log x)/4δ< 1+...+ t (log w)/δ 1 j t p∈I( j ) g(p) L< 1<•••< t (log x)/4δ< 1 +...+ t (log w)/δ 1 j t C 0 e j δ δ A A+1 j C t 1 t A+1 (t -1)!(Lδ) A(t-1) (log x)/4δ< (log w)/δ e δ δ A A+1 C t 1 t A+1 w (t -1)!(δL) A(t-1) (log x) A+1 w (t -1)!(κ log 2 x) t-1 (log x) A+1 ,
where κ = κ(K) may be chosen as large as we wish. Here, we made use of the short-interval assumption (1•6) at the very first step. In the third line, we set := 1 + . . . + t and bounded 1/ A+1 t by t A+1 / A+1 . Carrying back into (3•5), we obtain

G 0,k (x) s+t=k t 1 x(B log 3 x + c 3 ) s s!(t -1)!(κ log 2 x) t-1 (log x) A+1 x(B log 3 x + c 4 ) k-1 (k -1)!(log x) A+1 ,
provided K, and therefore κ, is suitably chosen. This is the required estimate.

3•3. Completion of the argument

It is now a simple matter to derive (1•7). Recalling definition (1•13), we have

M (x; f ) = n x g(n)T x n .
By (3•4) and partial summation, we have

(3•7) g j (y) = n y g(n)(log n) j n j(log y) j-A (log 2 y) B+δ A,J+1 (y 3, j > J).
Recall the definition of the coefficients α h ( ) appearing in (1•9) and put

ν h := α h ( )/h!Γ( -h) (h 0).
By [18; th. II.5.2], for a suitable constant b, we have ν h (bh + 1) h and

(3•8) T (x) = x(log x) -1 0 h H ν h (log x) h + O (bH + 1) H+1 (log x) H+1 (x 2),
uniformly for H (log x) 1/3 , say. Now Dirichlet's hyperbola formula provides

M (x; f ) = U + V + O x(log x) -1-A , with U := n √ x g(n)T x n , V := d √ x τ (d) √ x<n x/d g(n).
The sum V may be treated as an error term: for d √ x, we have, by (3•3),

√ x<n x/d g(n) x(log 2 x) B d(log x) A+1 •
By distributing the variable d into intervals ]e -δ(j+1) √

x, e -δj √ x] and approximating the corresponding n-range by [ √ x, √ xe δj ] with δ := 1/(log x) K for suitably large K, we obtain

V j 0 e jδ √ x G 0 e δj √ x -G 0 √ x e -δ(j+1) √ x<d e -δj √ x τ (d) + x (log x) A+1 xδ(log 2 x) B (log x) A+1 j 0 e jδ √ x {(log x) -2jδ + 1} e -1
x(log 2 x) B (log x) A+1 1 + (log 2 x) δ 0, e (log x) e , with Kronecker's notation. Finally, we apply (3•8) with H = A + 2r to get

U = x n √ x g(n) n 0 h H ν j log x n -h-1 + O (log x) e -H-2 .
From the expansion log x/n log x

-h-1 = 0 k K c kh log n log x k + O K | -1-j| log n log x K (1 n √ x)
with c kh k | -1-h| , we get, for each h ∈ [0, H],

(3•9) n √ x g(n) n log x/n log x -h-1 = 0 k K c kh g k √ x (log x) k + O K | -1-h| 2 K (log 2 x) B ,
where we used the trivial estimate |g| B ω to bound the error term. Select K := 2(A + B + 1) log 2 x . Applying (3•4) for j J, and (3•7) when J < j H, we get

U = x(log x) -1 0 j J λ j (f ) (log x) j + O (log 2 x) B+δ A,J+1
(log x) A , the exponent δ A,J+1 arising from that in (3•7). This completes the proof.

Proof of Theorem 1.2-friable summation

4•1. Setting

An integer n such that P + (n) y is said to be y-friable. Friable summability of series, was defined in [START_REF] Duffin | Representation of Fourier integrals as sums[END_REF], [START_REF] Fouvry | Entiers sans grand facteur premier en progressions arithmétiques[END_REF], and has been employed systematically in [START_REF] De La Bretèche | P -régularité de sommes d'exponentielles[END_REF], [START_REF] De La Bretèche | Séries trigonométriques à coefficients arithmétiques[END_REF], [START_REF] De La Bretèche | Théorème de Jordan friable[END_REF]. A series n 1 a n is said to be friably summable to a (or is said to have friable sum a) if the subseries

n 1 P + (n) y a n
converges for each y 2 and tends to a as y tends to infinity. We then write n 1 a n = a (P ). Letting ζ(s) denote Riemann's zeta function, it is well known that, for any given real number τ = 0, the series n 1 1/n 1+iτ has friable sum ζ(1 + iτ ) (by the convergence of the Eulerian product on the pointed line 1 + iτ , τ = 0), while being divergent in the ordinary meaning.

We shall show that, representing f as a Dirichlet convolution f = τ * g, then, for 0 j J, we have

(4•1) γ j (g) := n 1 g(n)(log n) j n (P ).
With this notation, the coefficients λ j (f ) appearing in (1•12) are given by

(4•2) λ j (f ) = 1 Γ( -j) +h=j α ( )γ h (g) !h! (0 j J),
while (1•10) remains valid.

4•2. Reduction

Let g be exponentially multiplicative, i.e. such that g(p ν ) = g(p) ν /ν! for all primes p and all integers ν 0, (1) and define g(p) := f (p) -. Put f := τ * g. Then f = f * h where h is multiplicative, supported on the set of squareful integers, and, for each p, the values h(p ν ) are given by the power series expansion

ν 0 h(p ν )ξ ν = (1 -ξ) e -ξg(p) ν 0 f (p ν )ξ ν (|ξ| < 1/p σ ).
Thus, n 1 |h(n)|/n τ < ∞ for all τ > σ and so, by a standard convolution argument left to the reader, we may restrict to proving (1•12) for f. We note right away that g ∈ F(2r, σ) and that (1•2) holds. Introducing the series

P + (n) y g(n) n s = exp p y g(p)
p s ( e s 1), we readily see by partial summation that the right-derivatives of any order j J at s = 1 of the left-hand side converge to a limit as y → ∞, in other words that (4•1) holds. Moreover partial summation yields that, for integer j and y 2,

(4•3) γ j (y; g) := P + (n) y g(n)(log n) j n = γ j (g) + O (log y) j-A (0 j J),
O j!(log y) j-A (log 2 y) δ j,A (j > J).

To prove this, observe that, for |w| 1/ log y, we have

p y g(p) p 1-w = k 0 p y g(p)w k (log p) k k!p = k J µ k w k k! + O 1 k!(log y) A + O (log 2 y) δ A,J+1 (log y) A = k J µ k w k k! + O (log 2 y) δ A,J+1 (log y) A ,
with µ k := p g(p)(log p) k /p (0 k J). Estimate (4•3) then follows from Cauchy's formula.

4•3. An auxiliary estimate

For f = τ * g and g as above, let us write f = τ * g y * h y , where g y and h y are the multiplicative functions defined by

g y (p ν ) := 1 {p y} g(p ν ), h y (p ν ) = 1 {p>y} g(p ν ) (p 2, ν 1).
Our first goal is to estimate

H y (x) := y<n x h y (n) (2 y x).
Lemma 4.1. Uniformly for x y 2, and with u := (log x)/ log y, we have

(4•4) H y (x) xu 2r (log x) A+1 • Proof. We have H y (x) = y<p x p ν x g(p ν ) + y<m x/y h y (m) p ν x/m p>P + (m) g(p ν ) x (log x) A+1 + S, with S := y<m x/y mP + (m) x |h y (m)|x m{log(x/m)} A+1 .
Writing m = nq ν with P + (n) < q, where, here and in the sequel of this proof, q denotes a prime number, we have

S = q>y ν 1 |g(q)| ν ν!q ν n x/q ν+1 P + (n)<q |h y (n)|x n{log(x/nq ν )} A+1 , .
Let S -denote the contribution to S of n = 1, and S + that of n > y. We plainly have

S - ν 1 y<q x 1/(ν+1) |g(q)| ν x ν!q ν (log x/q ν ) A+1 ν 1 y<q x 1/(ν+1) x|g(q)| ν (ν + 1) A+1 ν!q ν (log x) A+1 x log(2u) (log x) A+1 •
In order to majorize S + , we appeal to the Rankin-type bound

y<n t P + (n) q |h y (n)| t 1-c/ log q (log t) 2r-1 (log y) 2r (t, q > y),
where c > 0 is an abolute constant. It follows that

S + ν 1 y<q x 1/(ν+1) x|g(q)| ν ν!q ν x/q ν+1 y 1 t(log x/tq ν ) A+1 dO t 1-c/ log q (log t) 2r-1 (log y) 2r ν 1 y<q x 1/(ν+1)
x|g(q)| ν ν!q ν (log y) 2r

x/q ν+1 y (log t) 2r-1 dt t 1+c/ log q (log x/tq ν ) A+1 ν 1 y<q x 1/(ν+1) x|g(q)| ν (log q) 2r-A-1 ν!q ν (log y) 2r (log x)/(log q)-ν-1 0 e -cw w 2r-1 dw {(log x)/(log q) -ν -w} A+1

ν 1 y<q x 1/(ν+1) x|g(q)| ν (log q ν ) 2r ν!q ν (log y) 2r (log x) A+1 xu 2r (log x) A+1 •

4•4. Completion of the argument

Put f y := τ * g y and let y := x c/ log 2 x , where c is a sufficiently small constant. We have

M (x; f y ) = V y (x) + E y (x) with V y (x) := n √ x g y (n)T x n , E y (x) := d √ x τ (d) √ x<n x/d g y (n).
We immediately note that a Rankin-type estimate such as [16; lemma 2] provides

(4•5) E y (x) x 1-1/ log y d x τ r (d) d x(log x) -1-N ,
for any given integer N .

From (3•8), we deduce, for any fixed integer H, that H+1 .

V y (x) = x n √ x g y (n)(log x/n) -1 n 0 h H ν h (log x/n) h + O H 1 (log x)
Using the bound provided by (1•11) and selecting H sufficiently large in terms of r, we get that the overall contribution of the above remainder term is

x(log x) +2r-H-2 x(log x) -1-N ,
for any given integer N .

Appealing to (3•9) for g y , we may write

(4•6) n √ x g y (n) n log x/n log x -h-1 = 0 k K c kh n √ x g y (n) n log n log x k + O 1 (log x) N +r+1 ,
for sufficiently large K. Let v := 1/ log y. For all 0 k K, we have, by Rankin's device again, provided c is suitably chosen,

n> √ x |g y (n)|(log n) k n k! v k x v/4 exp p y |g y (p)| p 1-v/2 k! x v/4 (log y) k+2r 1 (log x) N +r+1 •
We may therefore extend, in (4•6), the inner n-sum to all y-friable integers without perturbing the error. Taking (4•5) into account, we thus obtain N , which, by rearranging the double sum, yields

M (x; f y ) = x(log x) -1 0 h H ν h (log x) h 0 k K c kh γ k (y; g) (log x) k + O 1 (log x)
(4•7) M (x; f y ) = x(log x) -1 0 j K+H λ j (y; f ) (log x) j + O 1 (log x) N
where, in view of (4•3), (4•8) λ j (y; f ) = λ j (f ) + O j j (log y) j-A (log 2 y) δ A,j (0 j K + H), with the convention that, say, λ j (f ) = 0 for j > J. By Dirichlet's hyperbola formula, we have

(4•9) M (x; f -f y ) = M (x; f y * h y -f y ) = S + U -W, with S := n √ x f y (n)H y x n , U := y<m √ x h y (m)M x m ; f y , W := M √ x; f y H y √ x . Now S n √ x |f y (n)|x(log 2 x) 2r n(log x) A+1
x(log 2 x) 2r (log x) A+1-r , and similarly W x(log x) -A-2 (log 2 x) 2r

x(log 2 x) 2r (log x) A+1-r .

To estimate the sum U , we insert (4•7) and evaluate the contribution of the main terms by partial summation from (4•4). Taking (4•8) into acount, we obtain

U xu 2r (log x) -A-1 .
Recalling (4•7), (4•8) and (4•9), we have thus almost reached (1•12), but with a remainder term

R(x; f ) x(log 2 x) 2r (log x) A+1-r •
We now show that an inductive argument enables us to replace (log 2 x) 2r by a power of log 3 x. Indeed, under assumption (1•11), consider the exponentially multiplicative function ϕ defined by

ϕ(p ν ) = f (p) ν /2 ν ν! (p 2, ν 1).
Then ϕ ∈ F(r/2, σ). Moreover, we may write f = ϕ * ϕ * ψ with

ψ(p ν ) = 0 j ν (-1) j f (p) j f (p ν-j ) j! (ν 1).
Thus ψ(p) = 0 and, writing

ε p := |f (p)| 2 /p 2σ + ν 2 |f (p ν )|/p νσ , we have ν 2 |ψ(p ν )| p σν j+k 2 |f (p)| j |f (p k )| j!p σ(k+j) ε p + |f (p)| p σ ε p + ε p e |f (p)|/p σ 1 + |f (p)| p σ + ε p ε p .
Applying the estimate already proved for ϕ and writing the hyperbola formula for f furnishes R(x; f ) x(log 2 x) r (log x) A+1-r • After k + 1 iterations, the exponent of log 2 x becomes r/2 k , while the implicit constant in the upper bound for R(x; f ) gets multiplied by C k for a suitable constant C. Selecting k log 4 x yields (1•12).

Limitations

5•1. Optimatility of Theorem 1.1

We show here that it is not possible, in general, to replace max(0, e ) by e in the error term of (1•7). To this end, we initially consider the completely multiplicative function g 0 defined by g 0 (p ν ) := 1/(log p) Aν (p 2, ν 1) with 0 < A < 1-note however that, at the cost of slightly more complicated computations, our approach would work for any positive A. As a preliminary step we show that, for a suitable constant C, we have

(5•1) M (x; g 0 ) = Cx (log x) A+1 1 + O 1 (log x) A (x 2).
Indeed, on the one hand, since g 0 (n)/n < ∞, we have

(5•2) M (x; g 0 ) x/ log x (x 2),
and, on the other hand,

(5•3) M (x; g 0 ) log x- x 1 M (t; g 0 ) t dt = m x g 0 (m) d x/m g 0 (d)Λ(d) = x m x g 0 (m) m(log 2x/m) A + O x m x g 0 (m) m(log 2x/m) A+1 .
From this and (5•2), we deduce that M (x; g 0 )

x/(log x) A+1 (x 2). Carrying this estimate back into (5•3) yields

M (x; g 0 ) = x log x m x g 0 (m) m(log 2x/m) A + O x (log x) A+2 . = x log x m √ x g 0 (m) m(log x/m) A + O x (log x) 2A+1 = x (log x) A+1 m √ x g 0 (m) m 1 + O log m log x + O x (log x) 2A+1 = x (log x) A+1 m √ x g 0 (m) m + O x (log x) 2A+1 , which implies (5•1) with C := m 1 g 0 (m)/m.
Next, define g(n) := g 0 (n)n i (n 1). By partial summation, we have

(5•4) M (x; g) = Cx 1+i (1 + i)(log x) A+1 1 + O 1 (log x) A (x 2).
Now, select = -r, with A < r < 1 and define f = g * τ . This function satisfies the hypotheses of Theorem 1.1. By the hyperbola formula, we have, for 1 y x,

(5•5) M (x; f ) = U + V -W with U := n x/y g(n)T x n , V := n y τ (n)M x n ; g , W := T (y)M x y ; g .
We choose y := e a(log 2 x) 2 , where a is a sufficiently large constant. We plainly have

W x (log y) r+1 (log x) A+1 .
The sum U may be evaluated by the Selberg-Delange method in the form given in [18; § II.5.4, Notes]. For suitable positive constants b, c, we have

T (x) = b 0 α(t)x 1-t t r dt + O xe -c √ log x (x 2),
where α is continuous on [0, b]. Since we may deduce from (5•4) that

Z(t) := n x/y g(n) n 1-t (x/y) t (log x) A+1 (0 t b),
we get, with an appropriate choice of a,

U = b 0 α(t)x 1-t Z(t)t r dt + O x (log x) A+2 x (log x) A+1 (log y) r+1 • Finally V = Cx 1+i (1 + i)(log x) A+1 n y τ (n) n 1+i 1 + O log n log x + O x (log x) 2A+1 n y τ r (n) n = {1 + o(1)}Cx 1+i (1 + i)ζ(1 + i) r (log x) A+1 • Carrying back into (5•5), we obtain M (x; f ) x (log x) A+1 ,
which implies the stated property.

5•2. Optimality of Theorem 1.2

We show here that the exponent r appearing in the remainder term of (1•12) cannot in general be replaced by e , even when e 0. To this end we exhibit a function f belonging to F(σ, r) for 1/2 < σ < 1 and for which: (i) condition (1•6) fails, (ii) = 0, 0 < A < r, and (iii) given any c < 1, one cannot replace r by cr in (1•12). If f happens to be real, the conditions become 0 < A < 2r/π and c < 2/π. Our construction relies on exploiting possible resonance between f (p) and p it for certain values of the real parameter t.

Let A > 0, r > 0, C > 0 (large), and let x k := exp exp C k , t k := (log x k+1 ) A (k 1). Put ϕ(v) := e iv and define f as the exponentially multiplicative function such that f (p) := rϕ(t k log p) whenever x k < p x k+1 and, say, f (p) = 0 when p x 1 . Observe that y 0 ϕ(tw) dw 1/t (t 1, y > 0). Due to the rapid increase of x k , partial summation hence yields |M (y; f )| y 2 dy (log x) r(1-1/C)-A+o (1) . and so, as x → ∞, provided A < r(1 -1/C),

(5•8) M (x; f ) = Ω x (log x) A+1-r(1-1/C)+o (1) .

Since C may be chosen arbitrarily large, this furnishes the required result.

If we require f to be real, we define ϕ(v) := sgn(cos v) (v ∈ R). Since ϕ(v) cos v has mean value 2/π, we obtain S k = (2r/π)C k+1 (1 -1/C) + O(1) instead of (5•7), appealling, e.g., to [START_REF] Tenenbaum | Introduction to analytic and probabilistic number theory[END_REF]Lemma III.4.13]. The remainder of the analysis is essentially identical and so we get (5•8) with 2r/π in place of r. It is noticeable that, provided A is suitably chosen, the implied lower bound and (1•5) may agree to an arbitrary small power of log x.

Remark. The above construction shows that assumptions f ∈ F(σ, r) (σ < 1, r > 1) and (1•2) with = 0, 0 < A < r -1, are insufficient to imply the convergence of the series n 1 f (n)/n.

  0) are absolutely convergent and uniformly bounded in terms of B for bounded j. This is enough to yield the required estimate, arguing as in [18; th. II.5.4]. Therefore, it will be sufficient to prove that (1•7) holds when f ∈ M( ), |f (p)| B, and f satisfies (1•6). As a consequence of this reduction, we can assume that f = τ * g, where g is supported on the set of squarefree integers. Moreover, g ∈ S(B, r + | |) with B := B + | |, and satisfies (1•2).

1 |Mrlog log x k+1 log x k + O( 1 )

 11 x := x k+1 , t := t k , σ := 1 + 1/ log x, (y; f )| y 2 dy with S h := r x h <p x h+1 e {f (p)/p it } p (1 h k). = rC k+1 (1 -1/C) + O(1).

S

  h rC k+1 (1 -1/C) + O k 2 log C = r(1 -1/C) log 2 x + O (log 3 x) 2 log C .Carrying back into (5•6) yields x 1

  Next, for 2 log k < h k -1, we have, by the prime number theorem in a strong form,

	S h = r	x h+1 x h	cos{(t k -t h ) log v)} v log v	dv + O(1)
	= r	log x h+1 log x h	cos{(t k -t h )w} w	dw + O(1)	1 t k log x h	+ 1	1.

Bounding S h trivially for h 2 log k, we finally get h k

This concept has been extensively used in the literature, in particular by Wirsing[START_REF] Wirsing | Das asymptotische Verhalten von Summen über multiplikative Funktionen, II[END_REF].


Chaouki Aouiti
email: chaouki.aouiti@fsb.rnu.tn

Farah Dridi
email: farah.dridi@fsb.rnu.tn

Qing Hui
email: qing.hui@unl.edu

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

pseudo almost automorphic solutions of neutral type Clifford-valued high-order Hopfield neural networks with D operator

Keywords: Clifford algebra, High-order Hopfield neural network, (µ, ν)-pseudo almost automorphic function, D operator

published or not. The documents may come

(µ, ν)-pseudo almost automorphic solutions of neutral type Clifford-valued high-order Hopfield neural networks with D operator

Introduction

Artificial Neural Networks (NNs) are a computational technique that belongs to the field of Machine Learning (ML). Their goal is to achieve a fairly simplified model of the brain. High-Order Hopfield Neural Network (HOHNN) is one of the most powerful and efficient types of NNs. The key factors that affect its success are its strong approximation ability, its fast convergence rate and its high fault tolerance capability. HOHNNs have widespread applications in various fields such as associative memory, pattern recognition, signal processing, robotics, medical image edge detection, medical event detection in electronic health records, diagnosis prediction in health care and many others. The study of high-order NNs has attracted considerable multidisciplinary research. For instance, the exponential convergence of high-order cellular NNs (CNNs) with time-varying leakage delays has been obtained in [START_REF] Xu | New Stability Criteria for High-Order Neural Networks with Proportional Delays[END_REF]; the authors of [START_REF] Brahmi | On the dynamics of the highorder type of neural networks with time varying coefficients and mixed delay[END_REF] discussed the existence and uniqueness of pseudo almost periodic solutions of high-order type of NNs; the authors of [START_REF] Xu | Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays[END_REF] studied the existence and exponential stability of the weighted pseudo almost periodic solutions of high-order cellular NNs with mixed delays; the dynamics of the pseudo almost automorphic solutions of HOHNNs with mixed delays has been investigated in [START_REF] Alimi | Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays[END_REF]; to name but a few.

One can easily see that all the NN models considered in the above-mentioned works are under time-delay effect. The investigation of delayed NNs has become an interesting world-wide focus and several types of delays such as discrete, distributed, proportional and leakage delay have been used, see for instance [START_REF] Aouiti | Piecewise asymptotically almost automorphic solutions for impulsive non-autonomous high-order Hopfield neural networks with mixed delays[END_REF][START_REF] Aouiti | Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays[END_REF][START_REF] Aouiti | Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms[END_REF][START_REF] Ren | Periodic oscillation of higher-order bidirectional associative memory neural networks with periodic coefficients and delays[END_REF]. To precisely describe the dynamics of complex neural reactions, systems must contain information about the derivative of the past states. Here, we are talking about another type called "neutral-type delay". It should be mentioned that neutral type NN models can be classified into two categories: Non-Operator-Based Neutral Functional Differential Equations (NOBNFDEs) and D-Operator-Based Neutral Functional Differential Equations (DOBNFDEs). It is important to notice that neutral type NNs with D operator have more realistic significance than non-operator-based ones in many practical applications of NN dynamics. Recently, new success stories of neutral type NNs with D operator have been provided. In [START_REF] Chen | Global exponential stability of anti-periodic solutions for neutral type CNNs with D operator[END_REF], the global exponential stability of the anti-periodic solutions for neutral type cellular NNs with D operator has been studied. In [START_REF] Xu | On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator[END_REF], the anti-periodic solutions for neutral shunting inhibitory cellular NNs with time-varying delays and D operator have been investigated. Reference [START_REF] Yao | Global convergence of CNNs with neutral type delays and D operator[END_REF] dealt with the global convergence of CNNs with neutral type delays and D operator. In [START_REF] Yao | Global exponential convergence of neutral type shunting inhibitory cellular neural networks with D operator[END_REF], the authors analyzed the global exponential convergence of neutral type shunting inhibitory cellular NNs with D operator. Zhang studied the oscillation dynamics of almost periodic solutions for shunting inhibitory CNNs with neutral type proportional delays and D operator in [START_REF] Zhang | Almost periodic solutions for SICNNs with neutral type proportional delays and D operators[END_REF] and extended the results to the pseudo almost periodic solutions of the same model in [START_REF] Zhang | Pseudo almost periodic solutions for neutral type SICNNs with D operator[END_REF].

On the one hand, to follow real phenomena in biological systems, researchers have proposed several classes of functions such as the class of Almost Automorphic (AA) functions in [START_REF] Veech | Almost automorphic functions on groups[END_REF], the class of Pseudo Almost Automorphic (PAA) functions in [START_REF] Diagana | Almost automorphic type and almost periodic type functions in abstract spaces[END_REF] and the class of Weighted Pseudo Almost Automorphic (WPAA) functions which have extended to the class of (µ, ν)-Pseudo Almost Automorphic ((µ, ν)-PAA) functions [START_REF] Ait Dads | µ, ν)-Pseudo almost automorphic solutions for some non-autonomous differential equations[END_REF]. (µ, ν)-PAA functions have been rarely used in NN theory where the main task consists of finding an an-swer to the following problem: "what will be the nature of output when all the parameters of the NN model are (µ, ν)-PAA functions?". In [START_REF] Aouiti | ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks[END_REF], we found an answer to this problem by studying the dynamic oscillations of (µ, ν)-PAA solutions of bidirectional associative memory NNs. On the other hand, Clifford introduced Clifford algebra in the 19th century. It is important associative algebras within the theories of quadratic forms, orthogonal groups and theoretical physics. As an extension of real value models, Clifford-value NNs have become active research domain due to their powerful applications in many fields such as neural computing, robotic vision, image processing, control problems and other areas. Success stories of Clifford-value NNs are reported in the following. The existence and global exponential stability of the equilibrium point of Clifford-valued recurrent NNs have been studied in [START_REF] Zhu | Global exponential stability of Clifford-valued recurrent neural networks[END_REF]; sufficient conditions ensuring the existence and global stability of Clifford-valued NNs with timevarying delays have been derived in [START_REF] Liu | Global stability of Clifford-valued recurrent neural networks with time delays[END_REF][START_REF] Li | Existence and global exponential stability of pseudo almost periodic solution for Clifford-valued neutral high-order Hopfield neural networks with leakage delays[END_REF]; the globally asymptotic almost automorphic synchronization of almost automorphic solutions of Clifford-valued recurrent NNs with mixed delays is developed in [START_REF] Li | Globally asymptotic almost automorphic synchronization of Clifford-valued RNNs With Delays[END_REF].

In this brief, we tried to manipulate neutral type HOHNNs in Clifford algebra with (µ, ν)-PAA parameters. To our best knowledge, there are no public results considering the dynamic behavior of (µ, ν)-PAA solutions of neutral type Clifford-valued NNs.

The main aim of this work is to obtain new sufficient conditions for ensuring the existence and global exponential stability of (µ, ν)-pseudo almost automorphic solutions of neutral-type Clifford-valued HOHNNs. Most of the published articles on neutral type NNs focused on the first-order systems and analyzed real-valued, complex-valued and quaternion-valued NNs. So, this article brings several advancements listed below:

1. The study of the existence and uniqueness of the (µ, ν)-pseudo almost automorphic solutions for neutral type HOHNN. 2. The analysis of the global exponential stability of the (µ, ν)-pseudo almost automorphic solutions for the considered model without using the Lyapunov functional method. 3. The class of (µ, ν)-PAA functions covers larger classes of functions that are very sophisticated and difficult to handle. We generalize many earlier publications [START_REF] Alimi | Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays[END_REF][START_REF] Aouiti | Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays[END_REF][START_REF] Brahmi | On the dynamics of the highorder type of neural networks with time varying coefficients and mixed delay[END_REF]. 4. The parameters are (µ, ν)-PAA functions which have been considered in Clifford algebra for the first time in such context. Some previous works in the literature are significantly extended and complemented, such as [START_REF] Huo | AntiPeriodic Solutions for CliffordValued HighOrder Hopfield Neural Networks with StateDependent and Leakage Delays[END_REF][START_REF] Li | Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays[END_REF][START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF]. 5. Via direct method, we study the (µ, ν)-PAA solutions for Clifford-valued HOHNNs without decomposing them into real-valued systems. Compared with real-valued, complex-valued and quaternion-valued HOHNNs [START_REF] Liu | Global µ-stability criteria for quaternionvalued neural networks with unbounded time-varying delays[END_REF][START_REF] Liu | Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays[END_REF], the dynamical behaviors of Clifford-valued HOHNNs are the most complicated.

The outline of this paper is arranged as follows. In Section 2, we establish useful definitions, assumptions and lemmas. Section 3 is devoted to establish new criteria for the existence, uniqueness and global exponential stability of (µ, ν)-PAA solutions of HOHNNs. In section 4, a numerical example is given to illustrate the feasibility of the obtained results. Conclusion and meaningful remarks are drawn in Section 5.

Preliminaries

Real Clifford algebra

In this subsection, we recall some results about real Clifford algebra. For more details, the reader may refer to [START_REF] Delanghe | Clifford algebra and spinor-valued functions: a function theory for the Dirac operator[END_REF] and the references therein. Let us denote R m the m-dimensional real vector space. The real Clifford algebra over R m is defined as

A = A⊆{1•••m} a A e A ; a A ∈ R , where e A = e h1 • • • e h ζ with A = h 1 • • • h ζ , 1 ≤ h 1 < h 2 < • • • < h ζ ≤ m and 1 ≤ ζ ≤ m.
A equipped with m generators is defined as the Clifford algebra over the real number R with m multiplicative generators e 1 , • • • , e m such that e i ∈ R m , e ∅ = e 0 = 1, e 2 0 = 1 and

   e 0 e i = e i e 0 = e i , i = 1, 2, • • • , m, e i e j + e j e i = 0, i = j, i, j ∈ {1, • • • , m}, e 2 i = -1, i = 1, 2, • • • , m.
When one element is the product of multiple Clifford generators, we will write its subscripts together such as e h1 e h2 = e h1h2 and e h1 e h2 e h5 = e h1h2h5 . It is

easy to see that dim R A = m k=0 m k = 2 m .
We also define the norm on A by

x A = max 1≤i≤n {|x A |}, for x = A x A e A ∈ A,
and the norm on A n by

x A n = max 1≤i≤n { x i A }, for x = (x 1 , x 2 , • • • , x n ) T ∈ A n .
In the following, A n denotes the n-dimensional real Clifford vector space.

Model description

In this article, we deal with the following neutral type Clifford-valued HOHNNs with mixed delays:

x i (t) -q i (t)x i (t -r i (t)) = -c i (t)x i (t) + n j=1 a ij (t)f j (x j (t -τ ij (t))) + n j=1 n l=1 α ijl (t)g j (x j (t -σ ijl (t)))g l (x l (t -ν ijl (t))) + n j=1 β ij (t) +∞ 0 G ij (s)h j (x j (t -s))ds + n j=1 n l=1 p ijl (t) +∞ 0 P ijl (t -s)k j (x j (s))ds × +∞ 0 Q ijl (t -s)k l (x l (s))ds + I i (t), (1)
in which n corresponds to the number of units in the NN, x i (•) ∈ A corresponds to the state vector of the i th unit, c i (•) represents the rate with which the i th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, q i (•) ∈ A is the connection weights, a ij (•), β ij (•) ∈ A are the synaptic connection weight of the j th unit on the i th unit, α ijl (•), p ijl (•) ∈ A represent the second-order synaptic weights of the NNs, f j (•), g j (.), h j (•), k j (•) ∈ A represent the activation functions of signal transmission, G ij (•), P ijl (•), Q ijl (•) ∈ A are the transmission delay kernels, r i (•), τ ij (•), σ ijl (•), ν ijl (•) ∈ R + are the transmission delays, I i (•) ∈ A denotes the external inputs. The initial conditions associated with (1) are of the form:

x i (s) = φ i (s), s ∈ (-∞, 0], i = 1, 2, . . . , n, (2)
where φ i ∈ C (-∞, 0], A which is the set of continuous functions from (-∞, 0] to A.

Remark 1 In (1), the functions r i (•), τ ij (•), σ ijl (•) and ν ijl (•) correspond to the transmission delays. In fact, time-delays exist in most NN systems because neurons cannot communicate or respond instantaneously. Sometimes they make the dynamic behaviors more complex and may destabilize the stable equilibria (see [START_REF] Achouri | Bogdanov-Takens Bifurcation in a Neutral Delayed Hopfield Neural Network with Bidirectional Connection[END_REF][START_REF] Aouiti | New results on impulsive Cohen-Grossberg neural networks[END_REF][START_REF] Aouiti | Global dissipativity of fuzzy cellular neural networks with inertial term and proportional delays[END_REF][START_REF] Aouiti | Global dissipativity of fuzzy bidirectional associative memory neural networks with proportional delays[END_REF]). Definition 3 For µ, ν ∈ M, the measures µ and ν are said to be equivalent if there exist constants a 0 , a 1 > 0 and a bounded interval Ω ⊂ R such that

Notations and definitions

Definition 2 ([20]) Let f = (f 1 , f 2 , • • • , f n ) T : R → A n where f i = A f A i e A . f A : R → R is called almost automorphic if for every i = 1, • • • , n we have f A i ∈ AA(R, R n ). Denote by AA(R, A n ) the
a 0 ν(A) ≤ µ(A) ≤ a 1 ν(A) for all A ∈ B satisfying A ∩ Ω = ∅.
Now, we introduce a new concept of ergodicity, which generalizes those previously given in the literature.

Definition 4 Let µ, ν ∈ M. A bounded continuous function f : R → A n is said to be (µ, ν) ergodic if lim r-→+∞ 1 ν([-r, r]) [-r,r] f (t) A dµ(t) = 0.
We denote the collection of all such functions by ξ(R, A n , µ, ν).

Let us denote BC(R, A n ) the set of bounded continued functions from R to A n , then (BC(R, A n ), • * ) is a Banach space where • * is the norm

f * := sup t∈R max 1≤i≤n { f i (t) A }. Definition 5 ([7]) Let µ, ν ∈ M, f ∈ BC(R, A n ) is (µ, ν)-pseudo almost automorphic if it can be expressed as f = f 1 + f 2 where f 1 ∈ AA(R, A n ) and f 2 ∈ ξ(R, A n , µ, ν).
The collection of such functions will be denoted by P AA(R, A n , µ, ν).

The following assumptions are fundamental in this function space: (A 1 ) For all µ, ν ∈ M, we have lim sup

n→∞ µ([-r,r]) ν([-r,r]) < ∞. (A 2 ) For all τ ∈ R, there exist β > 0 and a bounded interval I such that µ({a + τ : a ∈ A}) ≤ βµ(A) when A ∈ B satisfies A ∩ I = ∅.
Let us state two useful theorems proved in [START_REF] Aouiti | ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks[END_REF] as follows.

Theorem 1 ([7]) Let µ, ν ∈ M satisfy (A 2 ). Then the decomposition of a (µ, ν)-pseudo almost automorphic function of the form f = f 1 + f 2 where f 1 ∈ AA(R, A n ) and f 2 ∈ ξ(R, A n , µ, ν) is unique. Theorem 2 ([7]) Let µ, ν ∈ M satisfy (A 1
) and (A 2 ). Then P AA(R, A n , µ, ν) is a Banach space.

Technical lemmas

For 1 ≤ i, j, l ≤ n, we denote q * i = sup

t∈R q i (t) A , c * i = sup t∈R c i (t) A , c i * = inf t∈R c i (t) A , a * ij = sup t∈R a ij (t) A , α * ijl = sup t∈R α ijl (t) A , β * ij = sup t∈R β ij (t) A , p * ijl = sup t∈R p ijl (t) A , I * ij = sup t∈R I ij (t) A ,r * i = sup t∈R r i (t),τ * ij = sup t∈R τ ij (t), σ * ijl = sup t∈R σ ijl (t), ν * ijl = sup t∈R ν ijl (t).
Moreover, assume that for all 1 ≤ i, j, l ≤ n, we have

c i (•), r i (•), τ ij (•), σ ijl (•), ν ijl (•) ∈ AA(R, R n ) and q i (•), a ij (•), α ijl (•), β ij (•), p ijl (•), I i (•) ∈ P AA(R, A n , µ, ν).
Assumption 1 For all 1 ≤ j ≤ n, and all u, v ∈ R there exist nonnegative constants

L f j , L g j , L h j , L k j , d g j , d k j such that ||f j (u) -f j (v)|| A ≤ L f j ||u -v|| A , ||g j (u) -g j (v)|| A ≤ L g j ||u -v|| A , ||g j (u)|| A ≤ d g j , ||h j (u) -h j (u)|| A ≤ L h j ||u -v|| A , ||k j (u) -k j (u)|| A ≤ L k j ||u -v|| A , ||k j (u)|| A ≤ d k j .
For simplicity of calculation and without loss of generality, we assume that

f j (0) = g j (0) = h j (0) = 0. Assumption 2 G ij : [0, +∞) → R is continuous and |G ij (t)|e κ1t is integrable on [0, +∞) for a certain positive constant κ 1 . P ijl : [0, +∞) → R is continuous and |P ijl (t)|e κ2t is integrable on [0, +∞) for a certain positive constant κ 2 . Q ijl : [0, +∞) → R is continuous and |Q ijl (t)|e κ3t is integrable on [0, +∞) for a certain positive constant κ 3 .
The following Lemma 1 and Lemma 2 are proved in [START_REF] Aouiti | ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks[END_REF].

Lemma 1 ([7]) If φ(•) ∈ P AA(R, A, µ, ν) then we have φ(•-a) ∈ P AA(R, A, µ, ν). Lemma 2 ([7]) If ϕ, ψ ∈ P AA(R, A, µ, ν) then we have ϕ×ψ ∈ P AA(R, A, µ, ν).
By using Lemma 1 and Lemma 2, it is possible to prove the following lemmas.

Lemma 3 ([7]) If f (•) ∈ C(A, A) satisfies the l f -Lipschitz condition, φ(•) ∈ P AA(R, A, µ, ν) and τ (•) ∈ AA(R, A, µ, ν) then we have f (φ(• -τ (•))) ∈ P AA(R, A, µ, ν). Lemma 4 ([7]) Assume that Assumptions 1-2 hold. For all 1 ≤ i, j ≤ n, if φ j (•) ∈ P AA(A, A, µ, ν) then we have t → +∞ 0 G ij (s)h j (φ j (• -s))ds ∈ P AA(R, A, µ, ν).
Definition 6 Let x(t) = (x 1 (t), . . . , x n (t)) T be a (µ, ν)-pseudo almost automorphic solution of the system (1) with the initial value ϕ(t) = (ϕ 1 (t), . . . , ϕ n (t)) T ∈ C (-∞, 0], A and y(t) = (y 1 (t), . . . , y n (t)) T be an arbitrary solution of the system (1) with the initial value

ψ(t) = (ψ 1 (t), . . . , ψ n (t)) T ∈ C (-∞, 0], A
respectively. If there exist positive constants λ and N such that

x(t) -y(t) A n ≤ N φ e -λt , ∀t > 0,
where

φ ε = max 1≤i≤n sup t∈(-∞,0] [φ i (t) -d i (t)φ i (t -τ i (t))] -[x * i (t) -d i (t)x * i (t -τ i (t))]
A (µ, ν)-pseudo almost automorphic solution of the system ( 1) is said to be globally exponentially stable.

Assumption 3 For each 1 ≤ i ≤ n, t ∈ R, M [c i ] = lim T →+∞ 1 T t+T t c i (s)ds > 0
and there exist a bounded and continuous function ci : R → (0, +∞) and a positive constant K i such that e -t s ci(u)du ≤ K i e -t s ci(u)du , for all t, s ∈ R, t -s ≥ 0.

Lemma 5 For φ = (φ 1 , . . . , φ n ) T ∈ P AA(R, A n , µ, ν), we define the nonlinear operator Θ = Θ φ as follows

Θ φ (t) =          t -∞ e - t s c1(u)du F 1 (s)ds . . . t -∞ e - t s cn(u)du F n (s)ds         
where

F i (s) = c i (s)q i (s)φ i (s -r i (s)) + n j=1 a ij (s)f j (φ j (s -τ ij (s))) + n j=1 n l=1 α ijl (s)g j (φ j (s -σ ijl (s)))g l (φ l (s -ν ijl (s))) + n j=1 β ij (s) +∞ 0 G ij (m)h j (φ j (s -m))dm + n j=1 n l=1 p ijl (s) +∞ 0 P ijl (m)k j (φ j (s -m))dm +∞ 0 Q ijl (m)k l (φ l (s -m))dm + I i (s).
Then Θ φ maps P AA(R, A n , µ, ν) into itself.

Proof Let φ ∈ P AA(R, A, µ, ν). Using Lemma 3, we obtain f j (φ j (s -τ ij (s))), g j (φ j (s-σ ijl (s))) and g l (φ l (s-ν ijl (s))) in P AA(R, A, µ, ν). By using Lemma 4, we have

+∞ 0 G ij (m)h j (x j (s -m))dm, +∞ 0 P ijl (m)k j (x j (s -m))dm and +∞ 0 Q ijl (m)k l (x l (s -m))dm in P AA(R, A, µ, ν). Then, F i is a (µ, ν)-PAA function. Via Theorem 1, we have F i = F 1 i + F 2 i with F 1 i ∈ AA(R, A n ) and F 2 i ∈ ξ(R, A n , µ, ν). Noting that M [c i ]
> 0 from Assumption 3 and using the theory of exponential dichotomy, we obtain that

t -∞ e - t s ci(u)du F 1 i (s)ds ∈ AA(R, A n ) (3)
is a solution of the following almost automorphic differential equation

ẏ(t) = -c i (t)y(t) + F 1 i (t), 1 ≤ i, j ≤ n. Now, let us show that t -∞ e - t s ci(u)du F 2 i (s)ds ∈ ξ(R, A n , µ, ν).
From Assumption 3, one has lim

r-→∞ 1 ν([-r, r]) [-r,r] F 2 i (t) A dµ(t) = lim r-→∞ 1 ν([-r, r]) [-r,r] t -∞ e - t s ci(u)du F 2 i (s)ds A dµ(t) ≤ lim r-→∞ K i ν([-r, r]) [-r,r] t -∞ e -(t-s)ci * F 2 i (s) A dsdµ(t).
Let

E 1 = lim r-→∞ K i ν([-r, r]) [-r,r] t -r e -(t-s)ci * F 2 i (s) A ds dµ(t), E 2 = lim r-→∞ K i ν([-r, r]) [-r,r]   -r -∞ e -(t-s)ci * F 2 i (s) A ds   dµ(t)
and m = t -s, then by Fubini's theorem we obtain

E 1 = lim r-→∞ K i ν([-r, r]) [-r,r] t -r e -(t-s)ci * F 2 i (s) A ds dµ(t) ≤ lim r-→∞ K i ν([-r, r]) [-r,r]   +∞ 0 e -mci * F 2 i (t -m) A dm   dµ(t) ≤ +∞ 0 e -mci * lim r-→∞ K i ν([-r, r]) [-r,r] F 2 i (t -m) A dµ(t) dm = +∞ 0 e -mci *   lim r-→∞ K i ν([-r, r]) r-m -r-m F 2 i (t) A dµ m (t)   dm ≤ K i +∞ 0 e -mci * lim r-→∞ ν([-r -m, r + m]) ν([-r, r] β ν([-r -m, r + m]) × [-r-m,r+m] F 2 i (t) A dµ(t) dm.
On the one hand we have F 2 i ∈ ξ(R, R, µ, ν) then E 1 = 0. On the other hand, we have

E 2 = lim r-→∞ K i ν([-r, r]) [-r,r]   -r -∞ e -(t-s)ci * F 2 i (s) A ds   dµ(t) = lim r-→∞ K i ν([-r, r]) [-r,r]   -r -∞ e -(t-s)ci * F 2 i (s) A ds   dµ(t) ≤ lim r-→∞ K i ν([-r, r]) -r -∞ e sci * F 2 i (s) A ds r -r e -tci * dµ(t) = lim r-→∞ K i F 2 i A ci * e -2rci * = 0.
Combining with (3) it leads to Θ φ maps P AA(R, A n , µ, ν) into itself.

Main results

In this section, we establish new results for the existence, uniqueness and stability of (µ, ν)-PAA solution of the system (1). For (φ 1 , φ 2 , • • • , φ n ) T ∈ P AA(R, A n , µ, ν), we define the norm of φ as

φ * := max 1≤i≤n sup t∈R φ i (t) A .
And we consider the following assumption: Assumption 4 For 1 ≤ i, j, l ≤ n, there exist strictly positive constants M, Λ i and Ω i such that

M = max 1≤i≤n K i ci * I * i Λ = max 1≤i≤n q * i + K i ci * c * i q * i + n j=1 a * ij L f j + n j=1 n l=1 α * ijl L g j L g l + n j=1 β * ij L h j +∞ 0 |G ij (u)|du + n j=1 n l=1 p * ijl L k j L k l +∞ 0 |P ijl (u)|du +∞ 0 |Q ijl (u)|du < 1, Ω = max 1≤i≤n q * i + K i ci * c * i q * i + n j=1 a * ij L f j + n j=1 n l=1 α * ijl (L g j d g l + L g l d g j ) + n j=1 β * ij L h j +∞ 0 |G ij (u)|du + n j=1 n l=1 p * ijl +∞ 0 |P ijl (u)|du +∞ 0 |Q ijl (u)|du(L k j d k l + L k l d k j ) < 1
and a strictly negative constant Υ 0 such that

Υ 0 = sup t∈R -ci (t) + K i 1 1 -q * i c * i q * i + n j=1 a * ij L f j 1 1 -q * j + n j=1 n l=1 α * ijl L g j d g l 1 1 -q * j + L g l d g j 1 1 -q * j + n j=1 β * ij L h j +∞ 0 |G ij (u)|du 1 1 -q * j + n j=1 n l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|du + L k l d k j +∞ 0 |Q ijl (u)|du 1 1 -q * j .
Theorem 3 Suppose that Assumptions 1-4 hold. Then the system (1) has only one (µ, ν)-PAA solution in the region

S = φ ∈ P AA(R, A n , µ, ν) : φ -φ 0 * ≤ Λ 1 -Λ M ,
where

φ 0 (t) =     t -∞ e -t s c1(u)du I 1 (s)ds . . . t -∞ e -t s cn(u)du I n (s)ds     .
which is globally exponentially stable.

Proof First part: existence and uniqueness of (µ, ν)-PAA solutions Let

Y i (t) = x i (t) -q i (t)x i (t -r i (t)), for all 1 ≤ i ≤ n. (4)
Then, we have

Y i (t) = x i (t) -q i (t)x i (t -r i (t)) = -c i (t)Y i (t) -c i (t)q i (t)x i (t -r i (t)) + n j=1 a ij (t)f j (x j (t -τ ij (t))) + n j=1 n l=1 α ijl (t)g j (x j (t -σ ijl (t)))g l (x l (t -ν ijl (t))) + n j=1 β ij (t) +∞ 0 G ij (s)h j (x j (t -s))ds + n j=1 n l=1 p ijl (t) +∞ 0 P ijl (s)k j (x j (t -s))ds +∞ 0 Q ijl (s)k l (x l (t -s))ds +I i (t), i = 1, . . . , n. (5)
By using Lemma 5, we define an operator Θ as follows:

Θ : S → S (φ 1 • • • , φ n ) T → (Θ φ1 • • • , Θ φn ) T such that Θ φi = {q i (t)φ i (t -r i (t))} + Θ φ (t) for all φ ∈ S. One has φ 0 * = max 1≤i≤n sup t∈R t -∞ e -t s ci(u)du I i (s)ds A ≤ max 1≤i≤n K i sup t∈R t -∞ e -t s ci(u)du I i (s)ds A ≤ max 1≤i≤n K i ci * I * i = M
is the unique (µ, ν)-pseudo almost automorphic solution of the following differential equations:

Y i (t) = -c i (t)Y i (t) + I i (t), i = 1, . . . , n. The set S = φ ∈ P AA(R, A n , µ, ν) : φ -φ 0 * ≤ Λ 1-Λ M is a closed convex subset of P AA(R, A n , µ, ν). If φ ∈ S, then φ * ≤ φ -φ 0 * + φ 0 * ≤ φ -φ 0 * +M. ( 6
)
We claim that for any φ ∈ S the mapping Θ φ ∈ S. Note that

Θ ϕ -φ 0 * = max 1≤i≤n sup t∈R q i (t)φ i (t -r i (t)) + t -∞ e -t s ci(u)du -c i (s)q i (s)φ i (s -r i (s)) + n j=1 a ij (s)f j (φ j (s -τ ij (s))) + n j=1 n l=1 α ijl (s)g j (φ j (s -σ ijl (s)))g l (φ l (s -ν ijl (s))) + n j=1 β ij (s) +∞ 0 G ij (u)h j (φ j (s -u))du + n j=1 n l=1 p ijl (s) +∞ 0 P ijl (u)k j (φ j (s -u))du +∞ 0 Q ijl (u)k l (φ l (s -u))du ds
A and then we have

Θ ϕ -φ 0 * ≤ max 1≤i≤n sup t∈R q i (t)φ i (t -r i (t)) A + t -∞ e -t s ci(u)du K i sup t∈R -c i (s)q i (s) A sup t∈R φ i (s -r i (s)) A + n j=1 sup t∈R a ij (s) A sup t∈R f j (φ j (s -τ ij (s)) A + n j=1 n l=1 sup t∈R α ijl (s) A sup t∈R g j (φ j (s -σ ijl (s))) A sup t∈R g l (φ l (s -ν ijl (s)) A + n j=1 sup t∈R β ij (s) A +∞ 0 |G ij (u)| sup t∈R h j (φ j (s -u)) A du + n j=1 n l=1 sup t∈R p ijl (s) A +∞ 0 |P ijl (u)| sup t∈R k j (φ j (s -u)) A du × +∞ 0 |Q ijl (u)| sup t∈R k l (φ l (s -u)) A du ds .
It leads to

Θ ϕ -φ 0 * ≤ max 1≤i≤n q * i + t -∞ e -t s ci(u)du K i c * i q * i + n j=1 a * ij L f j + n j=1 n l=1 α * ijl L g j L g l + n j=1 β * ij L h j +∞ 0 |G ij (u)|du + n j=1 n l=1 p * ijl L k j L k l +∞ 0 |P ijl (u)|du +∞ 0 |Q ijl (u)|du ds φ * = Λ φ * .
As Λ < 1, it implies that Θ ϕ ∈ S. Next, we prove that the mapping Θ is a contraction mapping of S. For φ, ψ ∈ S, we have

Θ φ -Θ ψ * ≤ max 1≤i≤n sup t∈R q i (t) φ i (t -r i (t)) -ψ i (t -r i (t)) + t -∞ e -t s ci(u)du -c i (s)q i (s) φ i (s -r i (s)) + ψ i (s -r i (s)) + n j=1 a ij (s) f j (φ j (s -τ ij (s))) -f j (ψ j (s -τ ij (s))) + n j=1 n l=1 α ijl (s) g j (φ j (s -σ ijl (s)))g l (φ l (s -ν ijl (s))) -g j (ψ j (s -σ ijl (s)))g l (ψ l (s -ν ijl (s))) + n j=1 β ij (s) +∞ 0 G ij (u) h j (φ j (s -u)) -h j (ψ j (s -u) du + n j=1 n l=1 p ijl (s) +∞ 0 P ijl (u)h j (φ(s -u))du +∞ 0 Q ijl (u)k l (φ(s -u))du - +∞ 0 P ijl (u)h j (ψ(s -u))du +∞ 0 Q ijl (u)k l (φ(s -u))du + +∞ 0 P ijl (u)h j (ψ(s -u))du +∞ 0 Q ijl (u)k l (φ(s -u))du - +∞ 0 P ijl (u)h j (ψ(s -u))du +∞ 0 Q ijl (u)k l (ψ(s -u))du ds A .
It leads to

Θ φ -Θ ψ * ≤ max 1≤i≤n q * i + t -∞ e -t s ci(u)du K i c * i q * i + n j=1 a * ij L f j + n j=1 n l=1 α * ijl (L g j d g l + L g l d g j ) + n j=1 β * ij L h j +∞ 0 |G ij (u)|du + n j=1 n l=1 p * ijl +∞ 0 |P ijl (u)|du × +∞ 0 |Q ijl (u)|du(L k j d k l + L k l d k j ) ds φ -ψ *
and then

Θ φ -Θ ψ * ≤ max 1≤i≤n q * i + K i ci * c * i q * i + n j=1 a * ij L f j + n j=1 n l=1 α * ijl L g j d g l + L g l d g j + n j=1 β * ij L h j +∞ 0 |G ij (u)| du + n j=1 n l=1 p * ijl +∞ 0 |P ijl (u)| du × +∞ 0 |Q ijl (u)| du(L k j d k l + L k l d k j ) φ -ψ * = Ω φ -ψ * .
As Ω < 1, then we obtain that Θ is a contraction. Via Theorem 2, P AA(R, A n , µ, ν) is a Banach space. Then due to Banach's fixed point principle, Θ possesses one and only one fixed point x * = {x * i (t)} ∈ S, such that

x * i (t) = {q i (t)x * i (t -r i (t))} + Θ x * (t) = q i (t)x * i (t -r i (t)) + t -∞ e -t s ci(u)du -c i (s)q i (s)x * i (s -r i (s)) + n j=1 a ij (s)f j (x * j (s -τ ij (s))) + n j=1 n l=1 α ijl (s)g j (s -x * j (σ ijl (s)))g l (x * l (s -ν ijl (s))) + n j=1 β ij (s) +∞ 0 G ij (u)h j (x * j (s -u))du + n j=1 n l=1 p ijl (s) +∞ 0 P ijl (u)k j (x j (s -u))du +∞ 0 Q ijl (u)k l (x l (s -u))du + I i (s) ds.
Then it leads to

x * i (t) -q i (t)x * i (t -r i (t)) = -c i (t)x * i (t) + n j=1 a ij (s)f j (x * j (t -τ ij (t))) + n j=1 n l=1 α ijl (t)g j (x * j (t -σ ijl (t)))g l (x * l (t -ν ijl (t))) + n j=1 β ij (t) +∞ 0 G ij (u)h j (x * j (t -u))du + n j=1 n l=1 p ijl (t) +∞ 0 P ijl (s)k j (x * j (t -s))ds × +∞ 0 Q ijl (s)k l (x * l (t -s))ds + I i (t).
Thus the system (1) has an unique (µ, ν)-PAA solution x * (t).

Second part: global exponential stability of the (µ, ν)-PAA solution In view of the first part, the system (1) has only one (µ, ν)-pseudo almost automorphic solution denoted by x * (t) = (x * 1 (t), . . . , x * n (t)) T and satisfying (2). Let x(t) = (x 1 (t), • • • , x n (t)) T be an arbitrary solution of the system (1) with initial value φ(t) = (φ

1 (t), • • • , φ n (t)) T satisfying (2). Let z i (t) = x i (t) -x * i (t), Z i (t) = x i (t) -q i (t)x i (t -r i (t)) -x * i (t) -q i (t)x * i (t -r i (t)) .
We have

Z i (t) = -c i (t)Z i (t) -c i (t)q i (t)z i (t -r i (t)) + n j=1 a ij (t) f j (x j (t -τ ij (t)) -f j (x * j (t -τ ij (t)) + n j=1 n l=1 α ijl (t) g j (x j (t -σ ijl (t))g l (x l (t -ν ijl (t)))) -g j (x * j (t -σ ijl (t)))g l (x * l (t -ν ijl (t))) + n j=1 β ij (t) +∞ 0 G ij (u) h j (x j (t -u)) -h j (x * j (t -u)) du + n j=1 n l=1 p ijl (t) +∞ 0 P ijl (s)k j (x j (t -s))ds +∞ 0 Q ijl (s)k l (x l (t -s))ds - +∞ 0 P ijl (s)k j (x * j (t -s))ds +∞ 0 Q ijl (s)k l (x * l (t -s))ds . (7)
Set

Υ (γ) = sup t∈R γ -ci (t) + K i e γr * i 1 -q * i e γr * i c * i q * i + n j=1 a * ij L f j e γτ * ij 1 -q * j e γr * j + n j=1 n l=1 α * ijl L g j d g l e γσ * ijl 1 -q * j e γr * j + L g l d g j e γν * ijl 1 -q * j e γr * j + n j=1 β * ij L h j +∞ 0 |G ij (u)|e γu du 1 1 -q * j e γr * j + n j=1 n l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|e γu du +L k l d k j +∞ 0 |Q ijl (u)|e γu du 1 1 -q * j e γr * j , γ ∈ [0, min 1≤ι≤3 κ ι ]. ( 8
)
From Assumption 4 and by continuity of Υ (γ) we have

Υ (0) = sup t∈R -ci (t) + K i 1 1 -q * i c * i q * i + n j=1 a * ij L f j 1 1 -q * j + n j=1 n l=1 α * ijl L g j d g l 1 1 -q * j + L g l d g j 1 1 -q * j + n j=1 β * ij L h j +∞ 0 |G ij (u)|du 1 1 -q * j + n j=1 n l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|du +L k l d k j +∞ 0 |Q ijl (u)|du 1 1 -q * j < 0. (9)
We can choose a positive constant λ such that 0 < λ < min{κ 1 , κ 2 , κ 3 , , c1 * , . . . , cn * } satisfying 1 -q * i e λr * j > 0, and

Υ (λ) = sup t∈R λ -ci (t) + K i e λr * i 1 -q * i e λr * i c * i q * i + n j=1 a * ij L f j e λτ * ij 1 -q * j e λr * j + n j=1 n l=1 α * ijl L g j d g l e λσ * ijl 1 -q * j e λr * j + L g l d g j e λν * ijl 1 -q * j e λr * j + n j=1 β * ij L h j +∞ 0 |G ij (u)|e λu du 1 1 -q * j e λr * j + n j=1 n l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|e λu du +L k l d k j +∞ 0 |Q ijl (u)|e λu du 1 1 -q * j e λr * j < 0. ( 10
)
Let

φ ε = max 1≤i≤n sup t∈(-∞,0] [φ i (t) -d i (t)φ i (t -τ i (t))] -[x * i (t) -d i (t)x * i (t -τ i (t))] A . ( 11
)
For any > 0, we obtain

Z(0) A < ( φ ε + ). ( 12
)
For all t ∈ (-∞, 0], we have

Z(t) A ≤ ( φ ε + )e -λt < N ( φ ε + )e -λt ( 13
)
where N is a constant satisfying N > max

1≤i≤n K i + 1. Let us now prove that Z(t) A < N ( φ ε + )e -λt , ∀t > 0. ( 14
)
If it is not the case, there exist i ∈ {1, . . . , n} and t 1 > 0 such that

Z(t 1 ) A = N ( φ ε + )e -λt1 (15)
and

Z(t) A < N ( φ ε + )e -λt ∀t ∈ (-∞, t 1 ). ( 16
)
Moreover, we have

e λ z j (η) A ≤ e λη z j (η) -q j (η)z j (η -r j (η)) A +e λη q j (η)z j (η -r j (η)) A ≤ e λη Z j (η) A +e λr * i q * j e λ(η-rj (η)) z j (η -r j (η)) A ≤ ( φ ε + )N + e λr * j q * j sup s∈(-∞,t]
e λs z j (s) A .

(

) 17
For all η ∈ (-∞, t], t ∈ (-∞, t 1 ), i ∈ {1, • • • , n}, it leads to e λt z j (t) A ≤ sup s∈(-∞,t]
e λs z j (s)

A ≤ N ( φ ε + ) 1 -e λq * j r * j . ( 18
)
Multiplying Equation ( 7) by e -s 0 ci(u)du and integrating on [0, t], we obtain

Z i (t) = Z i (0)e -t 0 ci(u)du + t 0 e -t s ci(u)du -c i (s)q i (s)z i (s -r i (s)) + n j=1 a ij (s) f j (x j (s -τ ij (s)) -f j (x * j (s -τ ij (s)) + n j=1 n l=1 α ijl (s)[g j (x j (s -σ ijl (s))g l (x l (s -ν ijl (s)))) -g j (x * j (s -σ ijl (s)))g l (x * l (s -ν ijl (s)))] + n j=1 β ij (s) +∞ 0 G ij (u) h j (x j (s -u)) -h j (x * j (s -u)) du + n j=1 n l=1 p ijl (s) +∞ 0 P ijl (u)k j (x j (s -u))du × +∞ 0 Q ijl (u)k l (x l (s -u))du - +∞ 0 P ijl (u)k j (x * j (s -u))du × +∞ 0 Q ijl (u)k l (x * l (s -u))du ds, t ∈ [0, t 1 ]. (19)
Using equations ( 8), ( 10)- [START_REF] Li | Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen-Grossberg neural networks with delays[END_REF], we obtain

Z i (t 1 ) A = Z i (0)e -t 1 0 ci(u)du + t1 0 e -t 1 s ci(u)du -c i (s)q i (s)z i (s -r i (s)) + n j=1 a ij (s)(f j (x j (s -τ ij (s))) -f j (x * j (s -τ ij (s)))) + n j=1 n l=1 α ijl (s)(g j (x j (s -σ ijl (s)))g l (x l (s -σ ijl (s))) -g j (x * j (s -σ ijl (s)))g l (x * l (s -σ ijl (s)))) + n j=1 β ij (s) +∞ 0 G ij (u)[h j (x j (s -u)) -h j (x * j (s -u))]du + n j=1 n l=1 p ijl (s)[ +∞ 0 P ijl (u)k j (x j (s -u))du × +∞ 0 Q ijl (u)k l (x l (s -u))du - +∞ 0 P ijl (u)k j (x * j (s -u))du × +∞ 0 Q ijl (u)k l (x * l (s -u))du ds A .
Then it leads to

Z i (t 1 ) A ≤ ( φ ε + )K i e - t 1 0 ci(u)du + t1 0 e - t 1 s ci(u)du K i c * i q * i x i (s -r i (s)) -x * i (s -r i (s)) A + n j=1 a * ij f j (x j (s -τ ij (s)) -f j (x * j (s -τ ij (s)) A + n j=1 n l=1 α * ijl g j (x j (s -σ ijl (s))g l (x l (s -ν ijl (s)))) -g j (x * j (s -σ ijl (s)))g l (x * l (s -ν ijl (s))) A + n j=1 β * ij +∞ 0 |G ij (u)| h j (x j (s -u)) -h j (x * j (s -u)) A du + n j=1 n l=1 p * ijl sup t∈R +∞ 0 P ijl (u)k j (x j (s -u))du +∞ 0 Q ijl (u)k l (x l (s -u))du - +∞ 0 P ijl (u)k j (x * j (s -u))du +∞ 0 Q ijl (u)k l (x * l (s -u))du
A ds and finally it yields

Z i (t 1 ) A ≤ ( φ ε + )e -λt1 K i e -t 1 0 [ci(u)-λ]du + t1 0 e -t 1 s [ci(s)-λ]du K i e λr * i 1 -q * i e λr * i c * i q * i + n j=1 a * ij L f j e λτ * ij 1 -q * j e λr * j + n j=1 n l=1 α * ijl L g j d g l e λσ * ijl 1 -q * j e λr * j + L g l d g j e λν * ijl 1 -q * j e λr * j + n j=1 β * ij L h j +∞ 0 |G ij (u)|e λu du 1 1 -q * j e λr * j + n j=1 n l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|e λu du +L k l d k j +∞ 0 |Q ijl (u)|e λu du 1 1 -q * j e λr * j dsN ( φ ε + )e -λt1
and then

Z i (t 1 ) A ≤ ( φ ε + )e -λt1 K i e -t 1 0 [ci(u)-λ]du + t1 0 e -t 1 s [ci(s)-λ]du ci (u) -λ dsN ( φ ε + )e -λt1 ≤ N ( φ ε + )e -λt1 ( K i N -1)e -t 1 0 (ci(u)-λ)du + 1 < N ( φ ε + )e -λt1
which contradicts equality [START_REF] Brahmi | Pseudo almost periodic solutions of impulsive recurrent neural networks with mixed delays[END_REF], so ( 14) holds. Letting → 0 + , then for all t > 0, we have Z(t) A ≤ N φ ε e -λt . Similarly, we have

e λt z j (t) A ≤ sup s∈(-∞,t] e λs z j (s) A ≤ N φ ε 1 -q * j e λr * j and z j (t) A ≤ N φ ε 1 -q * j e λr * j e -λt ∀t > 0, j ∈ {1, • • • , n}.
The (µ, ν)-PAA solution of the system (1) is global exponentially stable.

Remark 2 In this article, we not only consider the effects of the first-order terms a ij (•), β ij (•) on NNs but also the influences of the second-order terms α ijl (•) and p ijl (•). If α ijl (•) = p ijl (•) = 0 then the following classical (first order) neutral type Hopfiled NNs

x i (t) -q i (t)x i (t -r i (t)) = -c i (t)x i (t) + n j=1 a ij (t)f j (x j (t -τ ij (t))) + n j=1 β ij (t) +∞ 0 G ij (s)h j (x j (t -s))ds +I i (t), (20)
has a unique (µ, ν)-pseudo almost automorphic solution which is global exponential stable. Hence, our results generalize the results in [START_REF] Aouiti | On the differential equations of recurrent neural networks[END_REF].

Remark 3 If q i (t) = 0 then the following Hopfiled NN

ẋi (t) = -c i (t)x i (t) + n j=1 a ij (t)f j (x j (t -τ ij (t))) + n j=1 β ij (t) +∞ 0 G ij (s)h j (x j (t -s))ds + I i (t), (21)
has an unique (µ, ν)-pseudo almost automorphic solution which is global exponential stable. It is worth pointing out that our main results are also valid for the case of the non-neutral systems. Our results not only improve but also complement earlier results in [START_REF] Alimi | Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays[END_REF][START_REF] Aouiti | Pseudo Almost Automorphic Solutions of Recurrent Neural Networks with Time-Varying Coefficients and Mixed Delays[END_REF][START_REF] Aouiti | ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks[END_REF][START_REF] Aouiti | Existence and global exponential stability of pseudo almost periodic solution for neutral delay BAM neural networks with time-varying delay in leakage terms[END_REF][START_REF] Brahmi | On the dynamics of the highorder type of neural networks with time varying coefficients and mixed delay[END_REF][START_REF] Xu | Exponential stability of weighted pseudo almost periodic solutions for HCNNs with mixed delays[END_REF].

Remark 4

In [START_REF] Li | Existence and global exponential stability of pseudo almost periodic solution for Clifford-valued neutral high-order Hopfield neural networks with leakage delays[END_REF], the authors considered a class of Clifford-valued neutral HOHNN with leakage delays. They studied the existence and global exponential stability of pseudo-almost periodic solutions for this class of NNs. In [START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF], the authors dealt with a class of inertial quaternion-valued HOHNNs with state-dependent delays. They analyzed the existence of anti-periodic solutions of the NNs. In our article, we investigate the (µ, ν)-pseudo almost automorphic solution for neutral-type HOHNNs. This class of functions covers larger classes of functions such as almost periodic, pseudo almost periodic, almost automorphic and pseudo almost automorphic functions. Hence, our results are not only new but also most general.

Remark 5 Although the multiplication of Clifford numbers does not satisfy the commutativity, which brings great difficulties to the study of Clifford-valued systems, we have found a method that does not decompose Clifford-valued systems into real-valued systems.

Remark 6 Pseudo almost automorphic functions play an important role in describing the dynamics of differential equations. In [START_REF] Alimi | Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays[END_REF], the authors investigated the dynamics behavior of the pseudo almost automorphic solutions of a class of HOHNNs with mixed delays. Those results cannot be applicable for the systems studied in this article. Consequently, our analysis of dynamics behavior of neutral type Clifford-valued HOHNNs model with (µ, ν)-PAA functions as coefficients and mixed delays improve the previous study in [START_REF] Alimi | Dynamics and oscillations of generalized high-order Hopfield neural networks with mixed delays[END_REF]. If µ = ν = 1, then the system (1) has an unique PAA solution in Clifford algebra.

Remark 7 In [START_REF] Aouiti | ν)-Pseudo-almost automorphic solutions for high-order Hopfield bidirectional associative memory neural networks[END_REF], the authors studied a class of delayed high-order Hopfield bidirectional associative memory NNs. They used fixed delays because timevarying delays are difficult to handle when dealing with (µ, ν)-pseudo almost automorphic parameters. Hence, it is not obvious to prove the composition theorem of (µ, ν)-pseudo almost automorphic functions i.e. f (φ(• -τ (•))) ∈ P AA(R, R, µ, ν). In 2020, this problem has been solved in article [START_REF] Aouiti | On the differential equations of recurrent neural networks[END_REF].

Remark 8 In [START_REF] Aouiti | Dynamics behavior for second-order neutral Clifford differential equations: inertial neural networks with mixed delays[END_REF], the pseudo almost periodic solutions of Clifford-valued inertial neutral NNs with time-varying delays and infinite distributed delay are investigated. Then, by using the same approach we can extend the results to the space of pseudo almost automorphic functions. Our results can complement the results in [START_REF] Aouiti | Dynamics behavior for second-order neutral Clifford differential equations: inertial neural networks with mixed delays[END_REF].

Simulation results

In this section, we apply our main results to a specific system and demonstrate the efficiency of our new stability criteria. Consider the following neutral type Clifford-valued HOHNNs model

x i (t) -q i (t)x i (t -r i (t)) = -c i (t)x i (t) + 2 j=1 a ij (t)f j (x j (t -τ ij (t))) + 2 j=1 2 l=1 α ijl (t)g j (x j (t -σ ijl (t)))g l (x l (t -ν ijl (t))) + 2 j=1 β ij (t) +∞ 0 G ij (s)h j (x j (t -s))ds + 2 j=1 2 l=1 p ijl (t) +∞ 0 P ijl (t -s)k j (x j (s))ds × +∞ 0 Q ijl (t -s)k l (x l (s))ds + I i (t), (22)
where x i (t) = x 0 i (t)e 0 + x 1 i (t)e 1 + x 2 i (t)e 2 + x 12 i (t)e 12 and i = 1, 2. For 1 ≤ i ≤ 2, let e 0 = 1 0 0 1 , e 1 = 1 0 0 -1 , e 2 = 0 1 1 0 and e 12 = 0 1 -1 0 .

For µ, ν ∈ M, satisfying (A 1 ) and (A 2 ), we consider the measure µ(t) where its Radon-Nikodym derivative is ρ 1 (t) = e sin t , and the measure ν(t) where its Radon-Nikodym derivative is

ρ 2 (t) = e t if t ≤ 0, 1 if t > 0.
We take the following parameters:

f (x i ) = g(x i ) = h(x i ) = k(x i ), i = 1, 2, f (x i ) = |x 0 i + 1| -|x 0 i + 1| 22 e 0 + 1 20 cos x 1 i e 1 + 1 27 cos x 2 i e 2 + 1 21 cos x 12 i e 12 ⇒ L f j = L g j = L h j = L k j = 1, d g j = d k j = 0.1, r i (t) = 0.5, τ ij (t) = σ ijl (t) = ν ijl (t) = 0.5| cos t|. Let G ij (t) = P ijl (t) = Q ijl (t) = e -t , K i = e 0.
1 and κ ι = 0.5 for ι = 1, 2, 3. c 1 (t) = (4 + cos 2 t)e 0 , c 2 (t) = (4 + sin 2 t)e 0 , and q 1 (t) = 0.12 sin(t)e 0 , q 2 (t) = 0.12 cos(t)e 0 , and a 11 (t) = 0. We have

Λ = max 1≤i≤2 q * i + K i ci * c * i q * i + 2 j=1 a * ij L f j + 2 j=1 2 l=1 α * ijl L g j L g l + 2 j=1 β * ij L h j +∞ 0 |G ij (u)|du + 2 j=1 2 l=1 p * ijl L k j L k l +∞ 0 |P ijl (u)|du +∞ 0 |Q ijl (u)|du = 0.9509 < 1, Ω = max 1≤i≤2 q * i + K i ci * c * i q * i + 2 j=1 a * ij L f j + 2 j=1 2 l=1 α * ijl (L g j d g l + L g l d g j ) + 2 j=1 β * ij L h j +∞ 0 |G ij (u)|du + 2 j=1 2 l=1 p * ijl +∞ 0 |P ijl (u)|du +∞ 0 |Q ijl (u)|du(L k j d k l + L k l d k j ) = 0.8825 < 1 and Υ 0 = sup t∈R -ci (t) + K i 1 1 -q * i c * i q * i + 2 j=1 a * ij L f j 1 1 -q * j + 2 j=1 2 l=1 α * ijl L g j d g l 1 1 -q * j + L g l d g j 1 1 -q * j + 2 j=1 β * ij L h j +∞ 0 |G ij (u)|du 1 1 -q * j + 2 j=1 2 l=1 p * ijl L k j d k l +∞ 0 |P ijl (u)|du +L k l d k j +∞ 0 |Q ijl (u)|du 1 1 -q * j = -1.1779 < 0.
By a direct computation, we can check that all the conditions of Theorem 3 are satisfied. Therefore, the system ( 22) has an unique (µ, ν)-PAA solution which is represented in Figures 1, 2, 3 and4. Besides, the unique (µ, ν)-PAA solution of the system ( 22) is global exponential stable. Almost automorphy is not always as easy to identify visually. In the above example x 0 i , x 1 i , x 2 i and x 12 i with i = 1, 2 never exactly repeat themselves. They are not periodic. Figures 5,6 solution for the system [START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF]. Figures 1 to 8 confirm that the proposed conditions in our theoretical results are effective for the above example.

Remark 9 In the above example, r i (•), τ ij (•), σ ijl (•), ν ijl (•) represent the time-delay functions. The time-delay as an inherent feature of signal transmission between different neurons is one of the main sources for causing dynamic properties of the system [START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF]. It should be mentioned that the presence of time-delay is a particularly harmful source of potential instability. In this example, the established criteria are straightforward to test and independent of delays, that is, the stability of the considered NN models is insensitive to the presence of the delays.

Conclusion

In this manuscript, neutral type Clifford-valued HOHNNs with mixed delays and D operator have been studied. By employing the fixed point theorem and differential inequalities, new sufficient conditions for the existence, uniqueness and global exponential stability of the (µ, ν)-pseudo almost automorphic solutions have been established. To our best knowledge, this is the first paper studying the (µ, ν)-pseudo almost automorphic solutions in Clifford algebra for such kind of NNs. As future research, there are some paths in this article that can be explored further. For instance: , i=1,2. 1. In the model (1), the activation functions of signal transmission f j (•), g j (.), h j (•) and k j (•) are continuous functions. They can be considered as discontinuous functions due to the impulse behavior of firing neurons. 2. The concept of Stepanov like pseudo weighted almost automorphy (W P AAS p ) is quite sophisticated. However, the dynamic oscillations of delayed systems with W P AAS p parameters are still relatively new. Soon, we will try to investigate dynamic oscillations of HOHNNs with W P AAS p parameters in Clifford algebra. 3. The study of second-order such systems [START_REF] Zhang | Double-integrator dynamics for multiagent systems with antagonistic reciprocity[END_REF][START_REF] Zhang | Velocity Constraint on Double-Integrator Dynamics Subject to Antagonistic Information[END_REF]. , i=1,2. Fig. 3 Curves of x 2 i for i = 1, 2, of the system [START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

  set of all such functions. Let B the Lebesgue σ-field of R, M denotes the set of all positive measures µ on B satisfying µ(R) = +∞ and µ([a, b]) < +∞ for all a, b ∈ R (a ≤ b).

Fig. 1

 1 Fig.1Curves of x 0 i , i = 1, 2, of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

Fig. 2

 2 Fig.2Curves of x 1 1 and x 1 2 of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

Fig. 4

 4 Fig. 4 Curves of x 12i for i = 1, 2, of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

2 Fig. 5

 25 Fig. 5 Stability of x 0 i , i = 1, 2, of the system (22).

2 Fig. 6

 26 Fig. 6 Stability of x 1i for i = 1, 2, of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

2 Fig. 7

 27 Fig. 7 Stability of x 2i for i = 1, 2, of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

2 Fig. 8

 28 Fig. 8 Stability of x 12i for i = 1, 2, of the system[START_REF] Li | Existence and global exponential stability of anti-periodic solutions for quaternion-valued cellular neural networks with time-varying delays[END_REF].

  -s n ) = f (t) for each t ∈ R. Denote by AA(R, R n ) the set of all such functions.

	(s n ) n∈N such that g(t) = lim n→∞	f (t + s n ) is well defined for each t ∈ R and
	lim n→∞	g(t
	Definition 1 ([17]) A continuous function f : R → R n is called almost
	automorphic if for every real sequence (S n ) n∈N , there exists a subsequence