On parametric Gevrey asymptotics for singularly perturbed partial differential equations with delays

We study a family of singularly perturbed $q-$difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter $\epsilon$. Moreover, we achieve the existence of a common formal power series in $\epsilon$ which represents each actual solution, and establish $q-$Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya Theorem regarding $q-$Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

From now on, q stands for a fixed real number with 0 < q < 1. We construct actual holomorphic solutions X(ǫ, t, z) for the previous Cauchy problem in E × T × C, where E is a bounded open sector in the complex plane with vertex at the origin, and T is an unbounded well-chosen open set. The procedure is based on the use of the map t → t/ǫ which was firstly considered by M. Canalis-Durand, J. Mozo-Fernandez and R. Schäfke in [10] to transform a singularly perturbed equation into an auxiliary regularly perturbed equation, easier to handle. This celebrated technique has also been used in the study of singularly perturbed partial differential equations (see [29] and [33] for example), q−difference-differential equations (like in [32] or [28]), and more recently to the study of difference-differential equations (see [34]).
Indeed, the present work is motivated by a previous work [34], where the second author studies a singularly perturbed difference-differential equation with small delay. This work can be seen as a continuation of that one. The dynamics appearing in that previous work involve a small shift in variable t with respect to ǫ, meaning that they are of the form (ǫ, t, z) → (ǫ, t + κ 2 ǫ, z), whereas the actual work deals with a shrinking behaviour in both t and z variables.
In [34], a Gevrey 1+ phenomenon, with estimates associated to the sequence ( n log n ) n n≥0 , is observed for the series solution of the problem. This sequence naturally appears when working with difference equations (see [9], [8] for example). Now, a q−Gevrey like behaviour, related to the sequence of estimates (q −n 2 ) n≥0 , appears. This behaviour comes up in the context of q−difference equations (see [17], [36]). One can observe that 1+ sequence is asymptotically upper bounded by Gevrey sequence (n!) n≥0 , and this one is upper bounded by q−Gevrey sequence (q −n 2 ) n≥0 .
The main aim of this work is to construct actual holomorphic solutions X(ǫ, t, z) of (1)+(2) and obtain sufficient conditions for the existence and unicity of a formal power series in the parameter ǫ,X(ǫ, t, z) = β≥0X β (t, z) ǫ β β! , owing its coefficients in an adequate functional space, and such that X is represented byX in a sense to precise (see Theorem 5). This representation is measured in terms of q−Gevrey bounds due to the appearance of q−difference operators on the right-hand side in (1).
The Cauchy problem (1)+(2) we consider in this paper comes also within the framework of the asymptotic analysis of linear differential and partial differential equations with multiplicative delays.
In the context of differential equations most of the statements in the literature are dedicated to linear problems of the form (3) x ′ (t) = F (t, x(λ 1 t), . . . , x(λ n t), x ′ (λ 1 t), . . . , x ′ (λ n t)) where F are vector valued polynomial functions in t and linear in its other arguments, where 0 < λ j < 1, for 1 ≤ j ≤ n are real numbers, and concern the study of asymptotic behaviour of some of their solutions x(t) as t tends to infinity for given initial data x(0). When F is real or matrix valued and with constant coefficients, we quote [11], [21], [24], [25]. For polynomial F in t, we notice [14], [20]. For studies in a complex variable t, we refer to [19], [39]. For more general delay functional equations, we indicate [13].
In the framework of linear partial differential equations, we mention a series of papers devoted to general results on the existence and unicity of holomorphic solutions to generalized Cauchy-Kowalevski type problems with shrinkings of the form ∂ m t u(t, x) = f (t, x, u(t, x), (∂ l x u(t, x), ∂ p x u(α(t)t, x), ∂ q x u(t, β(t, x)x)) (l,p,q)∈I ) for some integer m ≥ 1, a finite set I, and where f is analytic or of Gevrey type function and such that the functions α(t) and β(t, x) satisfy the shrinking constraints |α(t)| < 1 and |β(t, x)| < 1 for given initial data (∂ j t u)(0, x), 0 ≤ j ≤ m − 1 that belong to some functional space. We refer to [3], [26], [27]. For partial differential problems with contractions dealing with less regular solution spaces like Sobolev spaces, we quote [37], for instance. a n e −λnz have been throughly studied in the case when (λ n ) n≥0 is an increasing sequence of real numbers to ∞ (see [22], [38], [2]) or a sequence of complex numbers with |λ n | → ∞ (see [31]). This theory has also been developed when working with almost periodic functions, introduced by H. Bohr (see [6], [5], [15]), which are the uniform limits in R of exponential polynomials n k=1 a k e is k x , where the values s k belong to the so-called spectrum Λ ⊆ R. However, we are more interested in the behaviour of the sum when x tends to ∞ in he positive imaginary axis. Our technique rests on Euler-Mac-Laurin formula, Watson's Lemma and the equivalence between null q−Gevrey asymptotics and the fact of being q−exponentially small.
In [29], we solve the problem by means of a Dirichlet series with a spectrum being of the form ( 1 (k+1) α ) k≥0 . Now, the spectrum which helps us to achieve our purpose is of geometric nature (see Lemma 10).
The growth properties of W β for β ≥ 0 allow us to apply a Laplace like transform on each of them with respect to the variable τ in order to provide a holomorphic solution X i (ǫ, τ, z) of the main problem, defined in E i × T × C, for some appropriate unbounded open set T . In addition to this, one has null q−Gevrey asymptotic bounds for the difference of X i and X i+1 when the domain of the variable z is restricted to a bounded set, meaning that for every ρ > 0, there exist L 1 , L 2 > 0 such that Finally, a novel version regarding q−Gevrey asymptotics of Malgrange-Sibuya Theorem (Theorem 4) leads us to the main result in the present work (Theorem 5), where we guarantee the existence of a formal power series in ǫ, with coefficients in the Banach space of bounded holomorphic functions defined in T × D(0, ρ), which is common for every 0 ≤ i ≤ ν −1, and such that X i admitsX as its q−Gevrey asymptotic expansion of some positive type in he variable ǫ (see (60)).
It is worth pointing out that a q−Gevrey version of Malgrange-Sibuya Theorem was already obtained in [28], when dealing with q ∈ C, |q| > 1. There, the type in the asymptotic expansion involved suffers some increasement. This is so due to the need of extension results in ultradifferentiable classes of functions (see [7], [12]) to be applied along the proof. Here, the geometry of the problem changes so that we are able to maintain the type q−Gevrey. The proof rests on the classical Malgrange-Sibuya Theorem (see [23]).
The paper is organized as follows.
In Section 2 and Section 3, we introduce Banach spaces of formal power series in order to solve auxiliary Cauchy problems with the help of fixed point results involving complete metric spaces. In Section 2, this result is achieved when dealing with formal power series with holomorphic coefficients in a product of a finite sector with vertex at the origin times an infinite sector, while in Section 3 the result is obtained when dealing with a product of two punctured discs at 0.
In Section 4, we first recall the definition and main properties of a Laplace like transform, and q−Gevrey asymptotic expansions (Subsection 4.1). Next, we construct analytic solutions for the main problem and determine flat q−Gevrey bounds for the difference of two solutions when the intersection of the domains in the perturbation parameter is not empty (Subsection 4.2). In the proof, a Dirichlet type series is studied. The section is concluded proving the existence of a formal power series in the perturbation parameter which represents every solution in some sense which is specified (Subsection 4.3).

A Cauchy problem in weighted Banach spaces of Taylor power series
M, A 1 , C, δ 1 > 0 are fixed positive real numbers throughout the present work. Let q ∈ R with 0 < q < 1 and (R β ) β≥0 be a sequence of positive real numbers.
We consider an open and bounded sector E with vertex at the origin and we fix an open and unbounded sector S with vertex at the origin having positive distance to a fixed complex number a ∈ C ⋆ , it is to say, there exists M 1 > 0 such that |τ − a| > M 1 for every τ ∈ S. We write S β for the subset of S defined by The incoming definition of Banach spaces of functions and formal power series turns out to be an adaptation of the corresponding one in [28]. Here, the symmetry of these norms at 0 and the point of infinity in the τ variable has to be removed, so that a Laplace like transform of the elements in these Banach spaces makes sense.
For our purposes, the elements in the sequence (R β ) β≥0 are chosen to be related to the ones in a q−Gevrey sequence. This choice would provide that S β tends to S when β → ∞.
Let (E β ) β≥0 be a family of complex functional Banach spaces.
Lemma 1 Let s, ℓ 0 , ℓ 1 , m 1 , m 2 ∈ N, δ > 0 and ǫ ∈ E. We assume that In addition to this, we consider the elements in (R β ) β≥0 are such that for every β ≥ ℓ 1 + s. Moreover, we assume there exist constants d 1 , d 2 > 0 such that for every β ≥ 0. In addition to this, we assume Under the previous assumptions, there exists a positive constant C 11 , which does not depend on ǫ nor δ, such that for every v ∈ H(ǫ, δ, S).
Proof Direct calculations on the definition of the norms in the space H(ǫ, δ, S) allow us to conclude when taking C 12 := max{|F (ǫ, τ )| : ǫ ∈ E, τ ∈ S}. ✷ Let S ≥ 1, and N be a finite subset of N 2 . We also fix a ∈ C \ R + , where R + stands for the set {z ∈ C : Re(z) ≥ 0, Im(z) = 0}.
Theorem 1 Let Assumption (A) and Assumption (B) be fulfilled. We assume that the initial conditions in (13) verify there exist ∆ > 0 and 0 <M < M such that for every 0 ≤ j ≤ S − 1 . Then, there exist positive constants C 13 , and C 14 (only depending on q, d 1 , d 2 , C, S, δ 1 , A 1 ), and δ > 0 such that For an appropriate choice of δ, ∆ > 0, the map A ǫ turns out to be a Lipschitz shrinking map.
Let us fix κ = (κ 0 , κ 1 ) ∈ N and s ∈ I κ . Taking into account the definition of H(ǫ, δ, S)), we derive for some C 14 > 0 which only depends on the parameters defining equation (12). The terms of the form |ǫ| Cj in the previous expression can be upper bounded by an adequate constant. Taking into account (14), usual estimates in (18) derive for some C 15 depending on the parameters defining the equation, and such that tends to 0 whenever both ∆ and δ tend to 0. An appropriate choice for these constants allow us to conclude the first part of the proof. The second part of the lemma follows similar arguments as before.
The result is achieved with an adequate choice of δ > 0. ✷ Let R, ∆ and δ be as in the previous lemma. Bearing in mind Lemma 3 one can apply the shrinking map theorem on complete metric spaces to guarantee the existence of a fixed point for A ǫ in B(0, R) ⊆ H(ǫ, δ, S), sayW ǫ , which verifies W ǫ (τ, z) . Then, W (ǫ, τ, z) can be written as a formal power series in z, is a formal solution of (12)+ (13). Moreover, from the domain of holomorphy of the initial conditions in (13) and the recursion formula satisfied by the coefficients in W (ǫ, τ, z): Finally, the estimates in (15) are obtained for every β ≥ 0 from the fact thatW ǫ ∈ B(0, R) ⊆ H(ǫ, δ, S). The definition of the elements in H(ǫ, δ, S) lead us to for every β ≥ S. In addition to this, Assumption (B) and usual estimates allow us to refine the previous estimates leading to for some constants C 13 > 0 and C 14 > 0 which only depend on q, d 1 , d 2 , C, S, δ 1 and A 1 . This is valid for every ǫ ∈ E and τ ∈ S β . The hypothesis (14) in the enunciate allows us to affirm that (15) is also valid for 0 ≤ β ≤ S − 1. ✷ Remark: One derives holomorphy of W β in the variable τ in the whole sector S, and not only in S β for every β ≥ S whilst the estimates are only given for τ ∈ S β . It is also worth saying that R > 0 can be arbitrarily chosen whenever s > 0 for every s ∈ I κ , κ ∈ N .

Second Cauchy problem in a weighted Banach space of Taylor series
We provide the solution of a Cauchy problem with analogous equation as the one studied in the previous section, written as a formal power series in z with coefficients in an appropriate Banach space of functions in the variable τ and the perturbation parameter ǫ. In Section 2, the domain of holomorphy of the coefficients remains invariant from the domain of holomorphy of the initial conditions. This happens so because the dilation operator τ → q −1 τ sends points in any infinite sector in the complex plane with vertex at the origin into itself. Now, the domain of holomorphy of the coefficients for the formal solution of the Cauchy problem under study depends on the index considered. More precisely, if the initial conditions present a singularity at some point a ∈ C in the variable τ , the coefficients of the formal solution of the Cauchy problem have singularities in τ that tend to 0, providing a small divisor phenomenon. For every ρ > 0,Ḋ ρ stands for the set D(0, ρ) \ {0}. We preserve the value of the positive constants M, A 1 , C and δ 1 from the previous section. Let r 0 > 0 with E ⊆ D(0, r 0 ) and (R β ) β≥0 be a sequence of positive real numbers.
From (22), one derives that for every τ ∈ḊR The result follows provided that one is able to estimate the expression From the first of the hypotheses made in (21), |ǫ| C(ℓ 1 +s)−ℓ 0 is upper bounded by a constant. Also, taking into account (22), there existsR > 0 such that |τ | ≤R for every τ ∈ ∪ β≥0ḊR β , so that for some positive constant D 21 . The result immediately follows from (21) that guarantees that β!/(β − s)!q p 1 (β) is bounded from above. ✷ LetR > 0 be as in the proof of the previous lemma, i.e.R ≥R β for every β ≥ 0.
As it has been pointed out before, the Assumption (B') is substituted in the present work by Assumption (B") with the cost of losing some generality, but giving concrete values forR β , for every β ≥ 0. The incoming theorem is valid when considering any other choice of the elements in (R β ) β≥0 satisfying Assumption (B').
Proof The proof follows analogous steps as the one of Theorem 1, so we do not enter into details not to repeat arguments. Let ǫ ∈ D(0, r 0 ) \ {0} and 0 < δ < 1. The set E is taken to be {O(ḊR β ) : β ≥ 0}. We consider the map A ǫ from E[[z]] into itself defined in the same way as in (16).

Laplace transform and q−Gevrey asymptotic expansion
In this subsection, we recall some identities for the Laplace transform, and state some definitions and first results on q−Gevrey asymptotic expansions. The next lemma can be found in [34].

Lemma 9
Let m ∈ N, and w 1 (τ ) be a holomorphic function in an unbounded sector U such that there exist C, K > 0 with for every τ ∈ U . Let D be an unbounded sector with vertex at 0 which veryfies that d + arg(t) ∈ (− π 2 , π 2 ), cos(d + arg(t)) ≥ δ 2 , for some d ∈ R and δ 2 > 0. Then, is a holomorphic and bounded function defined for t ∈ D ∩ {|t| > K/δ 2 }. Moreover, the following identities hold: In the sequel, we work with functions which satisfy more restrictive bounds that the ones in (30). Indeed, we deal with bounds of the form C exp(K log 2 |τ |), for some C, K > 0. This alters the asymptotic behaviour of the Laplace transform and cause the appearance of q−Gevrey asymptotic expansions, associated to estimates related to the sequence (q −n 2 ) n≥0 .
H stands for a complex Banach space. We preserve the Definition of q−Gevrey asymptotic expansion established in [28], in order to be coherent with the definitions in that work.
Definition 3 Let S be a sector in C ⋆ with vertex at the origin, and A > 0. We say a holomorphic function f : S → H admits the formal power seriesf = n≥0 f n ǫ n ∈ H[[ǫ]] as its q−Gevrey asymptotic expansion of type A in S if for everyS ≺ S there exist C 1 , H > 0 such that for every ǫ ∈S.
The next proposition, detailed in [28] in the more general geometry of q−spirals, characterises null q−Gevrey asymptotic expansion.

Analytic solutions in a parameter of singularly perturbed Cauchy problem
We recall the definition of a good covering.

2.
T is an unbounded subset of an open sector with vertex at the origin. We assume |t| ≥ r T for every t ∈ T .

For every
Under the previous settings, we say the family {{S i } 0≤i≤ν−1 , T } is associated to the good covering Let us consider a good covering in C ⋆ , {E i } 0≤i≤ν−1 . Let S ≥ 1 and a ∈ C \ R + . We consider a finite subset of N 2 , N . For every κ = (κ 0 , κ 1 ) ∈ N , let m κ,1 , m κ,2 ∈ N, and b κ (ǫ, z) a holomorphic and bounded function on D(0, r 0 ) × C, for some r 0 > 0. For each 0 ≤ i ≤ ν − 1, we consider the main Cauchy problem in the present work: with initial conditions where the functions φ i,j (ǫ, t) are constructed as follows. Let {{S i } 0≤i≤ν−1 , T } be a family of open sets associated to the good covering {E i } 0≤i≤ν−1 .
We put We fist check that X i is, at least formally, a solution of (33)+ (34). From (31), one can check by inserting the formal power series X i in (33), that it turns out to be a formal solution in the variable z of (33)+ (34) if and only if W (ǫ, τ, z) is a formal solution of (12)+(13) and (26)+ (27).
Bearing in mind that W β verifies (38) and (39), one derives X i,β is well defined in E i × T , for every β ≥ 0. We now state a proof for the fact that (ǫ, τ, z) → X i (ǫ, τ, z) is indeed a holomorphic solution of (33)+(34) in E i × T × C. Let ǫ ∈ E i , t ∈ T , and β ∈ N. One has (40) We only give details on the first and second integrals appearing on the right-hand side of the previous inequality. The first integral on the right-hand side of (40) can be upper bounded by means of (39), and the choice of direction γ i .
for some constants C 31 , C 32 > 0 only depending on δ, r 0 , R 0 , q, S, A 1 , C, δ 1 , M , r T , δ 2 . We now consider the second integral appearing on the right-hand side of (40). From (37) and similar estimates as before we get (43) The function x → g 1 (x) = e M log 2 (x+δ 1 ) x Cβ , x ≥ 0 is such that g 1 (x) ≤ g 2 (x) for all x ≥ 0, where g 2 (x) =C 13 e M log 2 (x) x Cβ , for some positive constantC 13 , not depending on β. g 2 attains its maximum value at From (43) we derive (44) for someC 13 > 0. From (42) and (44), we lead to the existence of positive constants C 41 , C 42 , not depending on β, such that for every z ∈ C. This allows us to conclude the first part of the proof. Let 0 ≤ i ≤ ν − 1 and ρ > 0. For every (ǫ, t, z) We can write where L γ i+1 ,4 − L γ i ,4 stands for the path consisting of two parts: the first one going from R β e √ −1γ i+1 to 0 along the segment [0, R β e √ −1γ i+1 ] and the path going from 0 to R β e √ −1γ i following direction γ i . This integral has already been estimated in (44), for the first part of the proof, so we omit the details. We also omit the details on the integral concerning the path L γ i+1 ,2 which is analogous.
In order to estimate the integral along the path L γ i+1 ,4 − L γ i ,4 , one can observe that the function involved in the integrand does not depend on the index i considered, for this function is well defined for (ǫ, τ ) ∈ (D(0, r 0 ) \ {0}) ×ḊR β . One can apply Cauchy Theorem to derive for someĈ 23 ,C 23 > 0. It only rests to take into account that the function x ∈ (0, r 0 ) → can be included in the constantsC 23 and C 24 .
From (44) and (45) one gets the existence of positive constants C 6 , C 7 such that for every (ǫ, t) ∈ (E i ∩ E i+1 ) × T . Taking this last estimate into the expression of X i+1 − X i one can conclude that for every (ǫ, t, z) ∈ (E i ∩ E i+1 ) × T × D(0, ρ). The proof of the second statement in the theorem leans on the incoming lemma whose proof is left until the end of the current section. It provides information on the estimates for a Dirichlet type series. A similar argument concerning a Dirichlet series of different nature can be found in [29], Lemma 9, when dealing with Gevrey asymptotic expansions.

✷
The proof of Lemma 10 heavily rests on the q−Gevrey version of some preliminary results which are classical in Gevrey case (see [29] and the references therein). Their proofs do not differ from the classical ones, so we omit them.
Lemma 11 Let b > 0 and f : [0, b] → C a continuous function having the formal expansion n≥0 a n t n ∈ C[[t]] as its q-asymptotic expansion of type A 1 > 0 at 0, meaning there exist C, H > 0 such that for every N ≥ 1 and t ∈ [0, δ], for some 0 < δ < b. Then, the function admits the formal power series n≥0 a n n!ǫ n+1 ∈ C[[ǫ]] as its q−Gevrey asymptotic expansion of type A 1 at 0. It is to say, there existC,H > 0 such that One can adapt the proof of Proposition 4 in [28] in our framework.
where B 1 (s) = s − 1 2 is the Bernoulli polynomial and ⌊·⌋ stands for the floor function, to f (s) = D s 1 q A 1 s 2 e −D 2 q d 2 s ǫ . One leads to Taking the limit when n tends to infinity in the previous expression we arrive at an equality for a convergent series: From the fact that B 1 (t − ⌊t⌋) ≤ 1/2 for every t ≥ 0 and the change of variable D 2 q d 2 t = u, one gets Bearing in mind that f 3 (u) < 0 for u ∈ (0, D 2 ], and from usual estimates we derive The proof is complete if one can estimate e −D 2 /ǫ , I 1 and 1/ǫI 1 appropriately. The first expression is clearly upper bounded according to (46).

Existence of formal series solutions in the complex parameter
In this last subsection we obtain a q−Gevrey version of a Malgrange-Sibuya type Theorem. A result in this direction has already been obtained by the authors in [28] when dealing with q ∈ C, |q| > 1. In that work, the geometry of the problem differs from the one in the present work. Indeed, the result is settled in terms of discrete q−spirals tending to the origin, and with q ∈ C.
Given q ∈ C with 0 < |q| < 1 and a nonempty open subset U ⊂ C ⋆ , the discrete q−spiral associated to U and q consists of the products of an element in U and q m , for some m ∈ N. For our purpose, q is a real number and U is chosen in such a way that the discrete q−spiral turns out to be a sector with vertex at the origin.
The proof of the q−Gevrey version of Malgrange-Sibuya Theorem in [28] is based on the use of extension results on ultradifferential spaces of weighted functions which preserve the information of q−Gevrey bounds but causes that the q−Gevrey type involved in the q−Gevrey asymptotic suffers an increasement. Here, one can follow similar steps as for the classical proof Malgrange-Sibuya theorem based on Cauchy-Heine transform, so that the q−Gevrey type is preserved. In [34], an analogous demonstration for the Gevrey version of the result can be found. We have decided to include the whole proof of the result in order to facilitate comprehension and clarity of the result.

Theorem 4 (q-MS)
Let (E, ||.|| E ) be a Banach space over C and {E i } 0≤i≤ν−1 be a good covering in C * . For all 0 ≤ i ≤ ν − 1, let G i be a holomorphic function from E i into the Banach space (E, ||.|| E ) and let the cocycle ∆ i (ǫ) = G i+1 (ǫ) − G i (ǫ) be a holomorphic function from the sector Z i = E i+1 ∩ E i into E (with the convention that E ν = E 0 and G ν = G 0 ). We make the following assumptions.
1) The functions G i (ǫ) are bounded as ǫ ∈ E i tends to the origin in C, for all 0 ≤ i ≤ ν − 1.

Proof
We first state an auxiliary result.

Proof
We follow analogous arguments as in Lemma XI-2-6 from [23] with appropriate modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform.
For all 0 ≤ l ≤ ν − 1, we choose a segment These ν segments divide the open punctured disc D(0, r) \ {0} into ν open sectorsẼ 0 , . . . ,Ẽ ν−1 whereẼ for all ǫ ∈Ẽ l , for 0 ≤ l ≤ ν − 1, be defined as a sum of Cauchy-Heine transforms of the functions ∆ h (ǫ). By deformation of the paths C l−1 and C l without moving their endpoints and letting the other paths C h , h = l − 1, l untouched (with the convention that C −1 = C ν−1 ), one can continue analytically the function Ψ l onto E l . Therefore, Ψ l defines a holomorphic function on E l , for all 0 ≤ l ≤ ν − 1. Now, take ǫ ∈ E l ∩ E l+1 . In order to compute Ψ l+1 (ǫ) − Ψ l (ǫ), we write where the pathsĈ l andČ l are obtained by deforming the same path C l without moving its endpoints in such a way that: (a)Ĉ l ⊂ E l ∩ E l+1 andČ l ⊂ E l ∩ E l+1 , (b) Γ l,l+1 := −Č l +Ĉ l is a simple closed curve with positive orientation whose interior contains ǫ. Therefore, due to the residue formula, we can write for all ǫ ∈ E l ∩ E l+1 , for all 0 ≤ l ≤ ν − 1 (with the convention that Ψ ν = Ψ 0 ). In a second step, we derive asymptotic properties of Ψ l . We fix an 0 ≤ l ≤ ν − 1 and a proper closed sector W contained in E l . LetC l (resp.C l−1 ) be a path obtained by deforming C l (resp. C l−1 ) without moving the endpoints in order that W is contained in the interior of the simple closed curveC l−1 + γ l −C l (which is itself contained in E l ), where γ l is a circular arc joining the two points re √ −1θ l−1 and re √ −1θ l . We get the representation for all ǫ ∈ W. One assumes that the pathC l is given as the union of a segment L l = {te √ −1w l : t ∈ [0, r 1 ]} where r 1 < r and w l > θ l and a curve Γ l = {µ l (τ ) : τ ∈ [0, 1]} such that µ l (0) = r 1 e √ −1w l , µ l (1) = re √ −1θ l and r 1 ≤ |µ l (τ )| < r for all τ ∈ [0, 1). We also assume that there exists a positive number σ < 1 with |ǫ| ≤ σr 1 for all ǫ ∈ W. By construction of the path Γ l , we get that the function ǫ → 1 2π √ −1 Γ l ∆ l (ξ) ξ−ǫ dξ defines an analytic function on the open disc D(0, r 1 ). It remains to give estimates for the integral ξ−ǫ dξ. Let M ≥ 0 be an integer. From the usual geometric series expansion, one can write for all ǫ ∈ W.
Moreover, as above, one can choose a positive number η > 0 (depending on W) such that |ξ − ǫ| ≥ |ξ| sin(η) for all ξ ∈ L l and all ǫ ∈ W. Again by (48) and (55), and following analogous calculations as before we obtain (58) ||E l,M +1 (ǫ)|| E ≤ K l 2π sin(η) for all ǫ ∈ W. Using comparable arguments, one can give analogous estimates when estimating the other integrals for all h = l, l − 1. As a consequence, for any 0 ≤ l ≤ ν − 1, there exist ϕ l,m ∈ E, for all m ≥ 0 and a constant K l > 0 such that for all M ≥ 2, all ǫ ∈ W.
Taking into account Proposition 1, we deduce that for everyÊ i,i+1 ≺ E l ∩ E l+1 and for everŷ L > L, the function Ψ l+1 (ǫ) − Ψ l (ǫ) has the formal series0 as q−Gevrey asymptotic expansion of typeL inÊ i,i+1 . From the unicity of the asymptotic expansions on sectors, we deduce that all the formal series m≥0 ϕ l,m ǫ m , 0 ≤ l ≤ ν − 1, are equal to some formal series denoted We consider now the bounded holomorphic functions for all 0 ≤ i ≤ ν − 1, all ǫ ∈ E i . By definition, for any i ∈ {0, ..., ν − 1}, we have that for all ǫ ∈ Z i . Therefore, each a i (ǫ) is the restriction on E i of a holomorphic function a(ǫ) on D(0, r) \ {0}. Since a(ǫ) is moreover bounded on D(0, r) \ {0}, the origin turns out to be a removable singularity for a(ǫ) which, as a consequence, defines a convergent power series on D(0, r). Finally, one can write G i (ǫ) = a(ǫ) + Ψ i (ǫ) for all ǫ ∈ E i , all 0 ≤ i ≤ ν − 1. Moreover, a(ǫ) is a convergent power series, and for everyL > L, Ψ i (ǫ) has the seriesĜ(ǫ) = m≥0 ϕ m ǫ m as q−Gevrey asymptotic expansion of typeL on E i , for all 0 ≤ i ≤ ν − 1. ✷ We are under conditions to enunciate the main result in the present work.
Moreover, for every 0 ≤ i ≤ ν − 1 and every L 2 > d 2 2 A 1 , the function X i (ǫ, t, z) constructed in Theorem 3 admitsX(ǫ, t, z) as its q−Gevrey asymptotic expansion of type L 2 in E i , meaning