# On Parametric Gevrey Asymptotics for Singularly Perturbed Partial Differential Equations with Delays

Authors: Alberto Lastra; Stéphane Malek
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study a family of singularly perturbedq-difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter ϵ. Moreover, we achieve the existence of a common formal power series in ϵ 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.


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## Body

## 1. Introduction

We study a family ofq-difference-differential equations of the form
(1)ϵ∂t∂zSX(ϵ,t,z)+a∂zSX(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(∂tκ0∂zκ1X)(ϵ,qmκ_,1t,qmκ_,2z)
under appropriate initial conditions
(2)(∂zjX)(ϵ,t,0)=ϕj(ϵ,t),0≤j≤S-1.Here,S is an integer with S≥1 and a∈ℂ⋆:=ℂ∖{0}. 𝒩 stands for a finite subset of ℕ2, where ℕ:={0,1,2,…} is the set of nonnegative integers. For every (κ0,κ1)∈𝒩, bκ_(ϵ,z) turns out to be a polynomial in the variable z with holomorphic and bounded coefficients in a neighborhood of the origin in the parameter and mκ_,1,mκ_,2∈ℕ.From now on,q stands for a fixed real number with 0<q<1.We construct actual holomorphic solutionsX(ϵ,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/ϵ which was firstly considered by Canalis-Durand et al. in [1] 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 [2, 3], e.g.), q-difference-differential equations (like in [4] or [5]), and more recently to the study of difference-differential equations (see [6]).Indeed, the present work is motivated by a previous work [6], 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 ϵ, meaning that they are of the form (ϵ,t,z)↦(ϵ,t+κ2ϵ,z), whereas the actual work deals with a shrinking behaviour in both t and z variables.In [6], a Gevrey 1+ phenomenon, with estimates associated to the sequence ((n/logn)n)n≥0, is observed for the series solution of the problem. This sequence naturally appears when working with difference equations (see [7, 8], e.g.). Now, a q-Gevrey-like behaviour, related to the sequence of estimates (q-n2)n≥0, appears. This behaviour comes up in the context of q-difference equations (see [9, 10]). 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-n2)n≥0.The main aim of this work is to construct actual holomorphic solutionsX(ϵ,t,z) of (1)+(2) and obtain sufficient conditions for the existence and unicity of a formal power series in the parameter ϵ, X^(ϵ,t,z)=∑β≥0‍X^β(t,z)(ϵβ/β!), owing its coefficients in an adequate functional space and such that X is represented by X^ in a sense to precise (see Theorem 27). 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(λ1t),…,x(λnt),x′(λ1t),…,x′(λnt)),
where F are vector valued polynomial functions in t and linear in its other arguments, where 0<λj<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 [11–14]. For polynomial F in t, we notice [15, 16]. For studies in a complex variable t, we refer to [17, 18]. For more general delay functional equations, we indicate [19].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(4)∂tmu(t,x)=f(t,x,u(t,x),(∂xlu(t,x),∂xpu(α(t)t,x),∂xqu(t,β(t,x)x)∂xl)(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 (∂tju)(0,x), 0≤j≤m-1 that belong to some functional space. We refer to [20–22]. For partial differential problems with contractions dealing with less regular solution spaces like Sobolev spaces, we quote [23], for instance.Let us briefly reproduce the strategy followed. We consider a finite family of sectors with vertex at the origin(ℰi)0≤i≤ν which provides a good covering at 0 in the variable ϵ (see Definition 19). Let i∈{0,1,…,ν-1}. One can consider an auxiliary Cauchy problem as follows:
(5)∂zSW(ϵ,τ,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(a-τ)qmκ_,1(κ0+1)(-τϵ)κ0×(∂zκ1W)(ϵ,q-mκ_,1τ,qmκ_,2z)
with initial conditions (∂zjW)(ϵ,τ,0)=Wj(ϵ,τ), 0≤j≤S-1. We assume Wj is a holomorphic function in (D(0,r0)∖{0})×D(0,R^0) for some r0,R^0>0, for every 0≤j≤S-1, which is upper bounded in terms of q-Gevrey bounds (see (61)). Moreover, we assume each Wj can be extended to ℰi×𝒮, where 𝒮 is a sector with vertex at the origin and verifying q-Gevrey bounds in ℰi×S0 with S0:={z∈𝒮:|z|≥R0} (see (31)). Under these hypotheses, one can construct a formal solution to the auxiliary Cauchy problem, W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), where Wβ(ϵ,τ) turns out to be a holomorphic function in (D(0,r0)∖{0})×(Dβ∖{0}). Here, Dβ is a disc centered at the origin with radius decreasing to 0 whenever β tends to infinity and reproducing q-Gevrey bounds given by the initial conditions (see (62)). Moreover, each Wβ(ϵ,τ) can be extended to ℰi×Sβ under q-Gevrey bounds (see (32)), where Sβ:={z∈𝒮:|z|>Rβ}, with (Rβ)β≥0 being a sequence of positive numbers that decrease to 0. We assume Sβ∩Dβ≠∅ for every β≥0. The decrease rate of both Rβ and the radius of Dβ has to be chosen adequately, in accordance to the elements of a q-Gevrey sequence such as (qαβ2)β≥0 for some α>0.The main difficulty in this work is the occurrence of propagation of singularities in the coefficients of the auxiliary problem which leads to a small divisor phenomenon. The singular points form a sequence of complex numbers tending to 0. As a result, one can only obtain a formal solution for the auxiliary problem. In [24], a small divisor phenomenon comes from the Fuchsian operator studied in the main Cauchy problem. There, q∈ℂ is chosen to have |q|>1, whilst in the present work q∈ℝ with 0<q<1. A suchlike phenomenon also appears in [2], where the asymptotics in the parameter suffers the effect of a small divisor, and it is solved studying a Dirichlet-like series.General Dirichlet series of the form(6)∑n≥0‍ane-λnz
have been throughly studied in the case when (λn)n≥0 is an increasing sequence of real numbers to ∞ (see [25–27]) or a sequence of complex numbers with |λn|→∞ (see [28]). This theory has also been developed when working with almost periodic functions, introduced by Bohr (see [29–31]), which are the uniform limits in ℝ of exponential polynomials ∑k=1n‍akeiskx, where the values sk belong to the so-called spectrum Λ⊆ℝ. However, we are more interested in the behaviour of the sum when x tends to ∞ in the positive imaginary axis. Our technique rests on the Euler-Mac-Laurin formula, Watson's lemma, and the equivalence between null q-Gevrey asymptotics. The characterization of q—exponentially at functions is also considered on the way.In [2], 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 22).The growth properties ofWβ 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 Xi(ϵ,τ,z) of the main problem, defined in ℰi×𝒯×ℂ, for some appropriate unbounded open set 𝒯. In addition to this, one has null q-Gevrey asymptotic bounds for the difference of Xi and Xi+1 when the domain of the variable z is restricted to a bounded set, meaning that for every ρ>0, there exist L1,L2>0 such that
(7)supz∈D(0,ρ)t∈τ|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤L1e-L2(1/(-log(q))2)log2|ϵ|,
for every ϵ∈ℰi∩ℰi+1.Finally, a novel version regardingq-Gevrey asymptotics of Malgrange-Sibuya theorem (Theorem 25) leads us to the main result in the present work (Theorem 27), where we guarantee the existence of a formal power series in ϵ as follows:
(8)X^(ϵ,t,z)=∑β≥0‍Xβ(t,z)β!ϵβ∈ℍ𝒯,ρ[[ϵ]],
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 Xi admits X^ as its q-Gevrey asymptotic expansion of some positive type in the variable ϵ (see (135)).It is worth pointing out that aq-Gevrey version of Malgrange-Sibuya theorem was already obtained in [5], when dealing with q∈ℂ, |q|>1. The type in the asymptotic expansion suffers some increasement in that previous work. This is so due to the need of extension results in ultradifferentiable classes of functions (see [32, 33]) 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 [34]).The paper is organized as follows.In Sections2 and 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 Section4, we first recall the definition and main properties of a Laplace-like transform and q-Gevrey asymptotic expansions (Section 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 (Section 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 (Section 4.3).


## 2. A Cauchy Problem in Weighted Banach Spaces of Taylor Power Series

M, A1, C, and δ1>0 are fixed positive real numbers throughout the present work. Let q∈ℝ with 0<q<1 and let (Rβ)β≥0 be a sequence of positive real numbers.We consider an open and bounded sectorℰ 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 M1>0 such that |τ-a|>M1 for every τ∈𝒮. We write Sβ for the subset of 𝒮 defined by
(9)Sβ:={z∈𝒮:|z|>Rβ}.The incoming definition of Banach spaces of functions and formal power series turns out to be an adaptation of the corresponding one in [5]. 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.Definition 1.
Letϵ∈ℰ and β∈ℕ. Eβ,ϵ,Sβ denotes the vector space of functions v∈𝒪(Sβ) such that
(10)∥v(τ)∥β,ϵ,Sβ:=supτ∈Sβ{|v(τ)|eMlog2(|τ|/|ϵ|+δ1)|τϵ|-Cβ}q-A1β2
is finite.
Letδ>0. H(ϵ,δ,𝒮) denotes the complex vector space of all formal power series v(τ,z)=∑β≥0‍vβ(τ)(zβ/β!) with vβ∈𝒪(Sβ) for every β≥0 and such that
(11)∥v(τ,z)∥(ϵ,δ,𝒮):=∑β≥0‍∥vβ(τ)∥β,ϵ,Sβδββ!<∞.
It is straightforward to check that the pair(H(ϵ,δ,𝒮),∥·∥(ϵ,δ,𝒮)) is a Banach space.For our purposes, the elements in the sequence(Rβ)β≥0 are chosen to be related to the ones in a q-Gevrey sequence. This choice would provide that Sβ tends to 𝒮 when β→∞.Let(𝔼β)β≥0 be a family of complex functional Banach spaces. For every v(τ,z)=∑β≥0‍vβ(τ)(τβ/β!)∈(∪β≥0𝔼β)[[z]], we consider the formal integration operator ∂z-1 defined on (∪β≥0𝔼β)[[z]] by
(12)∂z-1(v(τ,z)):=∑β≥1‍vβ-1(τ)zββ!.Lemma 2.
Lets, ℓ0, ℓ1, m1, and m2∈ℕ, δ>0, and ϵ∈ℰ. We assume that
(13)C(ℓ1+s)-ℓ0-2m1M(-log(q))≥0.
In addition to this, we consider the elements in(Rβ)β≥0 are such that
(14)Rβ≥qm1Rβ-ℓ1-s,
for every β≥ℓ1+s. Moreover, we assume there exist constants d1,d2>0 such that
(15)Rβ≥d1qd2β,
for every β≥0. In addition to this, we assume
(16)m2-2A1(ℓ1+s)-m1C+d2×[ℓ0-2m1Mlog(q)-C(ℓ1+s)]>0.
Under the previous assumptions, there exists a positive constantC11, which does not depend on ϵ nor δ such that
(17)∥zs(-τϵ)ℓ01qm1(ℓ0+1)(∂z-ℓ1v)(τq-m1,zqm2)∥(ϵ,δ,𝒮)≤C11δℓ1+s∥v(τ,z)∥(ϵ,δ,𝒮),
for every v∈H(ϵ,δ,𝒮).Proof.
Letv(τ,z)=∑β≥0‍vβ(τ)(zβ/β!)∈H(ϵ,δ,𝒮). We have
(18)∥zs(-τϵ)ℓ01qm1(ℓ0+1)(∂z-ℓ1v)(τq-m1,zqm2)∥(ϵ,δ,𝒮)=∥∑β≥ℓ1+s‍vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)×β!(β-s)!(-τϵ)ℓ0zββ!∑β≥ℓ1+s‍∥(ϵ,δ,𝒮).
From (14), one derives that for every τ∈Sβ, vβ-ℓ1-s(τq-m1) is well defined and the function τ↦vβ-ℓ1-s(q-m1τ) is holomorphic in Sβ for every β≥ℓ1+s. The expression in (18) equals
(19)∑β≥ℓ1+s‍∥(-τϵ)ℓ0vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)×β!(β-s)!(-τϵ)ℓ0∥β,ϵ,Sβδββ!.
Letβ≥ℓ1+s. From the definition of the norm ∥·∥β,ϵ,Sβ, we get
(20)∥vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)β!(β-s)!(-τϵ)ℓ0∥β,ϵ,Sβ=supτ∈Sβ{|vβ-ℓ1-s(τq-m1)|(|τ|q-m1|ϵ|)-C(β-ℓ1-s)×e-Mlog2((|τ|/|ϵ|)q-m1+δ1)×|τϵ|ℓ0(|τ|q-m1|ϵ|)C(β-ℓ1-s)×eMlog2(|τ|q-m1/|ϵ|+δ1)×|τϵ|-Cβe-Mlog2(|τ|/|ϵ|+δ1)(|τ|q-m1|ϵ|)-C(β-ℓ1-s)}×β!(β-s)!qm2(β-s)-m1(ℓ0+1)q-A1β2×qA1(β-ℓ1-s)2q-A1(β-ℓ1-s)2.
Direct calculations allow us to obtain the following estimates(21)e-Mlog2((|τ|/|ϵ|)+δ1)+Mlog2((|τ|q-m1/|ϵ|)+δ1)≤C01e-Mlog2(|τ|/|ϵ|)+Mlog2(|τ|q-m1/|ϵ|)≤C02(|τ||ϵ|)-2m1Mlog(q),
for some positive constants C01 and C02 only depending on q, m1, and M. Moreover,
(22)(|τ|q-m1)C(β-ℓ1-s)=C03q-m1Cβ|τ|C(β-ℓ1-s),
for some constant C03>0 depending on q, m1, ℓ1, and  s. This last equality and (13) yield
(23)|τϵ|ℓ0(|τ|q-m1|ϵ|)C(β-ℓ1-s)|τϵ|-Cβ(|τ||ϵ|)-2m1Mlog(q)=C04(|ϵ||τ|-ℓ0+C(ℓ1+s)+2m1Mlog(q))≤C05|τ|ℓ0-2m1Mlog(q)-C(ℓ1+s)q-m1Cβ,
for some positive constants C04 and C05 depending on q, m, ℓ0, ℓ1, s, C, and ℰ. From the hypothesis (15) on Rβ, the last expression is upper bounded by
(24)C05q(-m1C+d2(ℓ0-2m1Mlog(q)-C(ℓ1+s)))β,
for some positive constant C05 only depending on q, m1, ℓ0, ℓ1, s, ℰ, C, and d1. Now, from (16) one gets that β!/(β-s)!qp1(β) is upper bounded by a constant C06>0 which does not depend on β, where
(25)p1(β)=m2(β-s)-m1(ℓ0+1)-A1β2+A1(β-ℓ1-s)2-m1Cβ+d2[ℓ0-2m1Mlog(q)-C(ℓ1+s)].
Taking into account all these computations, one achieves that (20) can be upper bounded by
(26)C05supτ∈Sβ{|vβ-ℓ1-s(τq-m1)|(|τ|q-m1|ϵ|)-C(β-ℓ1-s)×e-Mlog2(|τ|q-m1/|ϵ|+δ1)∑555}q-A1(β-ℓ1-s).
The lemma follows bearing in mind (14) and the definition of the norms in Eβ-(ℓ1+s),ϵ,Sβ-(ℓ1+s) and of H(ϵ,δ,𝒮).Remark 3.
The hypotheses made in (14), (15), and (16) are verified if one departs from Rβ=d1qd2β for small enough positive d2 and any d1>0 provided (13) is satisfied and m2-2A1(ℓ1+s)-m1C>0.Lemma 4.
LetF(ϵ,τ) be a holomorphic and bounded function defined on ℰ×𝒮. Then, there exists a constant C12=C12(F,ℰ,𝒮)>0 such that
(27)∥F(ϵ,τ)vϵ(τ,z)∥(ϵ,δ,𝒮)≤C12∥vϵ(τ,z)∥(ϵ,δ,𝒮)
for every ϵ∈ℰ, every δ>0 and all vϵ∈H(ϵ,δ,𝒮).Proof.
Direct calculations on the definition of the norms in the spaceH(ϵ,δ,𝒮) allow us to conclude when taking C12:=max{|F(ϵ,τ)|:ϵ∈ℰ,τ∈𝒮}.LetS≥1 and let 𝒩 be a finite subset of ℕ2. We also fix a∈ℂ∖ℝ+, where ℝ+ stands for the set {z∈ℂ:Re(z)≥0,Im(z)=0}.For everyκ_=(κ0,κ1)∈𝒩, let mκ_,1, mκ_,2 be nonnegative integers and bκ_(ϵ,z)∈𝒪(D(0,r0))[z], where r0>0 is such that ℰ¯⊆D(0,r0). We write bκ_(ϵ,z)=∑s∈Iκ_‍bκ_,s(ϵ)zs, where Iκ_ is a finite subset of ℕ for every κ_∈𝒩. We assume that 1≤κ1<S for every κ_=(κ0,κ1)∈𝒩.We consider the functional equation(28)∂zSW(ϵ,τ,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(a-τ)qmκ_,1(κ0+1)(-τϵ)κ0×(∂zκ1W)(ϵ,q-mκ_,1τ,qmκ_,2z)
with initial conditions
(29)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where the function (ϵ,τ)↦Wj(ϵ,τ) is an element in 𝒪(ℰ×𝒮) for every 0≤j≤S-1.We make the following assumptions.Assumption A.
For everyκ_=(κ0,κ1)∈𝒩 and every s∈Iκ_, we assume
(30)C(S-κ1+s)-κ0-2mκ_,1M(-log(q))≥0,[C(S-κ1+s)-κ0-2mκ_,1M(-log(q))]d2<mκ_,2-2A1(S-κ1+s)-mκ_,1C.Assumption B.
R
β
≥
q
m
κ
_
,
1
R
β
-
κ
1
-
s and there exist d1, d2>0 with Rβ≥d1qd2β, for every κ_=(κ0,κ1)∈𝒩 and every s∈Iκ_.Theorem 5.
Let Assumption A and Assumption B be fulfilled. We assume that the initial conditions in (29) verify there exist Δ>0 and 0<M~<M such that for every 0≤j≤S-1(31)|Wj(ϵ,τ)|≤ΔeM~log2(|τ|/|ϵ|+δ1)|ϵ|K0,
for every τ∈S0, ϵ∈ℰ, where K0=max{κ0:(κ0,κ1)∈𝒩}. Then, there exists W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), formal solution of (28)+(29), where Wβ∈𝒪(ℰ×Sβ).
Then, there exist positive constantsC13 and C14 (only depending on q, d1, d2, C, S, δ1, and  A1) and δ>0 such that
(32)|Wβ(ϵ,τ)|≤C13β!(C14δ)βeMlog2(|τ|/|ϵ|+δ1)×|τϵ|CβqA1β2,
for every β≥0, all ϵ∈ℰ and every τ∈Sβ.Proof.
Letϵ∈ℰ. We put 𝔼:={𝒪(Sβ):β≥0} and define the map 𝒜ϵ from 𝔼[[z]] into itself by
(33)𝒜ϵ(W~(τ,z)):=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(a-τ)qmκ_,1(κ0+1)(-τϵ)κ0×[(∂zκ1-SW~)(q-mκ_,1τ,qmκ_,2z)+∂zκ1wϵ(q-mκ_,1τ,qmκ_,2z)(∂zκ1-SW~)],
where wϵ(τ,z)=∑j=0S-1‍Wj(ϵ,τ)(zj/j!). For an appropriate choice of δ,Δ>0, the map 𝒜ϵ turns out to be a Lipschitz shrinking map.Lemma 6.
There existR,δ,Δ>0 (not depending on ϵ) such that (1)
∥𝒜ϵ(W~(τ,z))∥(ϵ,δ,𝒮)≤R for every W~(τ,z)∈B(0,R). B(0,R) denotes the closed ball centered at 0 with radius R in H(ϵ,δ,𝒮).(2)
(34)∥𝒜ϵ(W~1(τ,z))-𝒜ϵ(W~2(τ,z))∥(ϵ,δ,𝒮)≤12∥W~1(τ,z)-W~1(τ,z)∥(ϵ,δ,𝒮),for every W~1, W~2∈B(0,R).Proof.
LetR>0 and δ>0. In order to prove the first enunciate, we take W~(τ,z)∈B(0,R)⊆H(ϵ,δ,𝒮). From Lemmas 2 and 4 we deduce that
(35)∥𝒜ϵ(W~(τ,z)∥(ϵ,δ,𝒮)≤∑κ_=(κ0,κ1)∈𝒩‍∑s∈Iκ_‍Mκ_sM1[C01δS-κ1+s∥W~(τ,z)∥(ϵ,δ,𝒮)+∥zsqmκ_,1(κ0+1)(-τϵ)κ0∂κ1wϵ×(q-mκ_,1τ,qmκ_,2z)zsqmκ_,1(κ0+1)∥(ϵ,δ,𝒮)],
with Mκ_s=supϵ∈ℰ|bκ_s(ϵ)|<∞ for every κ_∈𝒩 and s∈Iκ_.
Let us fixκ_=(κ0,κ1)∈𝒩 and s∈Iκ_. Taking into account the definition of H(ϵ,δ,𝒮), we derive
(36)∥zsqmκ_,1(κ0+1)(-τϵ)κ0∂κ1wϵ(q-mκ_,1τ,qmκ_,2z)∥(ϵ,δ,𝒮)=∑j=sS-1-κ1-s‍∥Wj+κ1-s(ϵ,q-mκ_,1τ)(-τϵ)κ0∥j,ϵ,Sj×qmκ_,2(j-s)-mκ_,1(κ0+1)j!(j-s)!δjj!≤C14∑j=sS-1-κ1-s‍supτ∈Sj|Wj+κ1-s(ϵ,q-mκ_,1τ)|×|τϵ|κ0|τϵ|-Cje-Mlog2(|τ|/|ϵ|+δ1)δj,
for some C14>0 which only depends on the parameters defining (28). The terms of the form |ϵ|Cj in the previous expression can be upper bounded by an adequate constant. Taking into account (31), usual estimates in (36) derive
(37)∥𝒜ϵ(W~(τ,z))∥(ϵ,δ,𝒮)≤∑κ_=(κ0,κ1)∈𝒩‍∑s∈Iκ_‍Mκ_sM1[C01δS-κ1+s∥W~(τ,z)∥(ϵ,δ,𝒮)+C15],
for some C15 depending on the parameters defining the equation and such that it tends to 0 whenever both Δ and δ tend to 0. An appropriate choice for these constants allows us to conclude the first part of the proof.
The second part of the lemma follows similar arguments as before. LetW~1, W~2∈B(0,R)⊆H(ϵ,δ,𝒮). One has
(38)∥𝒜ϵ(W~1)-𝒜ϵ(W~2)∥(ϵ,δ,𝒮)≤∑κ_=(κ0,κ1)∈𝒩‍∑s∈Iκ_‍Mκ_sM1C01δS-κ1+s∥W~1-W~2∥(ϵ,δ,𝒮).
The result is achieved with an adequate choice of δ>0.LetR, Δ, and let δ be as in the previous lemma. Bearing in mind Lemma 6 one can apply the shrinking map theorem on complete metric spaces to guarantee the existence of a fixed point for 𝒜ϵ in B(0,R)⊆H(ϵ,δ,𝒮), say W~ϵ, which verifies ∥W~ϵ(τ,z)∥(ϵ,δ,𝒮)≤R, and 𝒜ϵ(W~ϵ(τ,z))=W~ϵ(τ,z). Let us define
(39)Wϵ(τ,z)=∂z-SW~ϵ(τ,z)+wϵ(τ,z).We putW~(ϵ,τ,z):=W~ϵ(τ,z), and W(ϵ,τ,z):=∂z-SW~(ϵ,τ,z)+wϵ(τ,z). Then, W(ϵ,τ,z) can be written as a formal power series in z as
(40)W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)zββ!,
where Wβ+S(ϵ,τ)=W~β,ϵ(τ) for every β≥0.From the construction ofW(ϵ,τ,z), we have W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!) is a formal solution of (28)+(29). Moreover, from the domain of holomorphy of the initial conditions in (29) and the recursion formula satisfied by the coefficients in W(ϵ,τ,z), we get
(41)Wh+S(ϵ,τ)h!=∑κ_=(κ0,κ1)∈𝒩‍∑h1+h2=h,h1∈Iκ_‍bκ_,h1(ϵ)(-τϵ)κ0×qmκ_,2h2(a-τ)h2!qmκ_,1(κ0+1)×Wh2+κ1(ϵ,q-mκ_,1τ).
We can conclude the function (ϵ,τ)↦Wβ∈𝒪(ℰ×𝒮) for every β≥0.Finally, the estimates in (32) are obtained for every β≥0 from the fact that W~ϵ∈B(0,R)⊆H(ϵ,δ,𝒮). The definition of the elements in H(ϵ,δ,𝒮) lead us to
(42)∥W~β,ϵ∥β,ϵ,Sβ≤Rβ!(1δ)β,
so that
(43)|Wβ(ϵ,τ)|=|W~β-S,ϵ(τ)|≤R(β-S)!(1δ)β-S×eMlog2(|τ|/|ϵ|+δ1)|τϵ|C(β-S)qA1(β-S)2,
for every β≥S. In addition to this, Assumption B and usual estimates allow us to refine the previous estimates leading to
(44)|Wβ(ϵ,τ)|≤C13β!(C14δ)βeMlog2((|τ|/|ϵ|)+δ1)|τϵ|CβqA1β2,
for some constants C13>0 and C14>0 which only depend on q, d1, d2, C, S, δ1, and A1. This is valid for every ϵ∈ℰ and τ∈Sβ. The hypothesis (31) in the enunciate allows us to affirm that (32) is also valid for 0≤β≤S-1.Remark 7.
One derives holomorphy ofWβ in the variable τ in the whole sector 𝒮 and not only in Sβ for every β≥S whilst the estimates are only given for τ∈Sβ. It is also worth saying that R>0 can be arbitrarily chosen whenever s>0 for every s∈Iκ_, κ_∈𝒩.


## 3. 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 inz with coefficients in an appropriate Banach space of functions in the variable τ and the perturbation parameter ϵ. In Section 2, the domain of holomorphy of the coefficients remains invariant from the domain of holomorphy of the initial conditions. This happens because the dilation operator τ↦q-1τ 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 everyρ>0, D˙ρ stands for the set D(0,ρ)∖{0}. We preserve the value of the positive constants M,A1,C, and δ1 from the previous section. Let r0>0 with ℰ¯⊆D(0,r0) and let (R^β)β≥0 be a sequence of positive real numbers.Definition 8.
Letβ∈ℕ. For r0>0 and ϵ∈D(0,r0)∖{0}; Eβ,ϵ,D˙R^β2 stands for the vector space of functions v∈𝒪(D˙R^β) such that
(45)|v(τ)|β,ϵ,D˙R^β:=supτ∈D˙R^β{|v(τ)||ϵ|CβeMlog2(|τ|+δ1)}q-A1β2
is finite. Let δ>0. We write H2(ϵ,δ) for the vector space of all formal power v(τ,z)=∑β≥0‍vβ(τ)zβ/β! such that vβ∈Eβ,ϵ,D˙R^β2 with
(46)|v(τ,z)|(ϵ,δ):=∑β≥0‍|vβ(τ)|β,ϵ,D˙R^βδββ!<∞.
The pair (H2(ϵ,δ),|·|(ϵ,δ)) is a Banach space.Lemma 9.
Lets, ℓ0, ℓ1, m1, andm2∈ℕ, δ>0, and ϵ∈D(0,r0)∖{0}. We assume that
(47)C(ℓ1+s)-ℓ0≥0,m2-2A1(ℓ1+s)>0.
Moreover, we assume that the elements of the sequence(R^β)β≥0 are such that
(48)R^β≤qm1R^β-ℓ1-s,
for every β≥ℓ1+s.
Under the previous assumptions, there exists a positive constantC21 which depends on C, q, m1, m2, s, ℓ0, ℓ1, M, A1, δ1, andr0 (not depending on ϵ nor δ) such that
(49)|zs(-τϵ)ℓ01qm1(ℓ0+1)(∂z-ℓ1v)(τq-m1,zqm2)|(ϵ,δ)≤C21δℓ1+s|v(τ,z)|(ϵ,δ),
for every v∈H2(ϵ,δ).Proof.
Letv(τ,z)=∑β≥0‍vβ(τ)(zβ/β!) be an element of H2(ϵ,δ). We have
(50)|zs(-τϵ)ℓ01qm1(ℓ0+1)(∂z-ℓ1v)(τq-m1,zqm2)|(ϵ,δ)=|∑β≥ℓ1+s‍vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)×β!(β-s)!(-τϵ)ℓ0zββ!∑β≥ℓ1+s‍|(ϵ,δ).
From (48), one derives that for every τ∈D˙R^β-ℓ1-s, vβ-ℓ1-s(τq-m1) is well defined. In addition to this, the function τ↦vβ-ℓ1-s(τq-m1) is holomorphic in D˙R^β for every β≥ℓ1+s. The expression in (50) equals
(51)∑β≥ℓ1+s‍|(-τϵ)ℓ0vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)×β!(β-s)!(-τϵ)ℓ0|β,ϵ,D˙R^βδββ!.
Letβ≥ℓ1+s. From the definition of the norm |·|β,ϵ,D˙R^β, we get
(52)|(-τϵ)ℓ0vβ-ℓ1-s(τq-m1)qm2(β-s)-m1(ℓ0+1)×β!(β-s)!(-τϵ)ℓ0|β,ϵ,D˙R^β=supτ∈D˙R^β{em(log)|vβ-ℓ1-s(τq-m1)|×|ϵ|C(β-ℓ1-s)eMlog2(|τ|/qm1+δ1)×|τϵ|ℓ0|ϵ|C(ℓ1+s)×eM(log2(|τ|/qm1+δ1)-log2(|τ|+δ1))}×qp1(β)β!(β-s)!q-A1(β-ℓ1-s)2,
with p1(β)=m2(β-s)-m1(ℓ0+1)-A1β2+A1(β-ℓ1-s)2.
The result follows provided that one is able to estimate the expression(53)qp1(β)β!(β-s)!|τ|ℓ0|ϵ|C(ℓ1+s)-ℓ0×eM(log2((|τ|/qm1)+δ1)-log2(|τ|+δ1)).
From the first of the hypotheses made in (47), |ϵ|C(ℓ1+s)-ℓ0 is upper bounded by a constant. Also, taking into account (48), there exists R^>0 such that |τ|≤R^ for every τ∈∪β≥0D˙R^β, so that
(54)|τ|ℓ0exp(M(log2(|τ|q-m1+δ1)-log2(|τ|+δ1)))≤D21(q,m1,δ1,M,ℓ0),
for some positive constant D21. The result immediately follows from (47) that guarantees that β!/(β-s)!qp1(β) is bounded from above.LetR^>0 be as in the proof of the previous lemma, that is, R^≥R^β for every β≥0.Lemma 10.
LetF(ϵ,τ) be a holomorphic and bounded function defined on D(0,r0)×D(0,R^).
Then, there exists a constantC22=C22(F)>0 such that
(55)|F(ϵ,τ)vϵ(τ,z)|(ϵ,δ)≤C22|vϵ(τ,z)|(ϵ,δ),
for every ϵ∈D(0,r0)∖{0}, every δ>0 and all vϵ∈H2(ϵ,δ).Proof.
Direct calculations on the definition of the norms in the spaceH2(ϵ,δ) allow us to conclude when taking C22:=max{|F(ϵ,τ)|:ϵ∈D(0,r0),τ∈D(0,R^)}.LetS≥1 and let 𝒩 be a finite subset of ℕ2. We also fix a∈ℂ∖ℝ+ such that |a|≥R^, with R^ as before.Letmκ_,1, mκ_,2, and let bκ_ be as in Section 2, for every κ_=(κ0,κ1).We consider the functional equation(56)∂zSW(ϵ,τ,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(a-τ)qmκ_,1(κ0+1)(-τϵ)κ0×(∂zκ1W)(ϵ,q-mκ_,1τ,qmκ_,2z)
with initial conditions
(57)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where the function (ϵ,τ)↦Wj(ϵ,τ) is an element in 𝒪((D(0,r0)∖{0})×D˙R^0) for every 0≤j≤S-1.We make the following assumptions.Assumption A
′. For every κ_=(κ0,κ1)∈𝒩 and every s∈Iκ_, we assume
(58)C(S-κ1+s)-κ0≥0,mκ_,2-2A1(S-κ1+s)>0.Remark 11.
Observe that Assumption A implies Assumption A′.Assumption B
′. We assume R^β≤qmκ_,1R^β-κ1-s for every κ_=(κ0,κ1)∈𝒩, every s∈Iκ_, and every β≥κ1+s.We first state a result which provides a concrete value for the elements in(R^β)β≥0 under Assumption B′. The choice is made in two respects: first, to clarify how the singularities suffer propagation in the formal solution of (56)+(57), with respect to the variable τ, and second, to provide acceptable domains of holomorphy for such coefficients when regarding this phenomenon of propagation of singularities. Any other appropriate choice for the elements in (R^β)β≥0 regarding these issues would also be fairish for our purpose.Lemma 12.
Letd^1,d^2>0, and ϵ∈D(0,r0)∖{0}.
We putR^β:=R^0 for β=0,1,…,S-1, and R^β=d^1qd^2β for every β≥S. Let us assume that (56)+(57) has a formal solution in z, W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!). Then, there exists d^20 such that for every d^2≥d^20, the function τ↦Wβ(ϵ,τ) belongs to 𝒪(D˙R^β) for every β≥S and all ϵ∈D(0,r0)∖{0}.Proof.
LetW(ϵ,τ,z) be a formal power series in z of the form ∑β≥0‍Wβ(ϵ,τ)(zβ/β!). One can plug the formal power series into (56) to obtain the recursion formula in (41) for the coefficients (Wβ)β≥S. From this recurrence, one derives that the domain of holomorphy for Wh+S in the variable τ depends on the domain of holomorphy on τ of Wh2+κ1 and also on q-mκ_,1 for every κ_=(κ0,κ1)∈𝒩 and every 0≤h2≤h such that h-h2∈Iκ_.
The initial conditionsW0,…,WS-1 are holomorphic functions in D˙R^0.Lemma 13.
For everyN≥1 the coefficients WNS-(N-1)κ10,…,W(N+1)S-Nκ10 turn out to be holomorphic functions in D˙qNm¯κ_,1R^0, for κ10:=max{κ1:(κ0,κ1)∈𝒩} and m¯κ_,1:=max{mκ_,1:κ_∈𝒩}.Proof.
We prove it by recurrence onN and regarding the recursion formula (41).
LetN=1. One has h2+κ1≤S-1 for any h2,κ1 as in (41) if and only if h2≤S-1-κ1 for every (κ0,κ1)∈𝒩, it is to say, if and only if h2≤S-1-κ10. In this case, Wh+S only depends on the initial conditions (Wj)0≤j≤S-1. Moreover,
(59)h+S∈{S,S+1,…,2S-κ10-1},
and the dilation on the variable τ allows us to obtain that WS,…,W2S-κ10-1 are holomorphic functions in D˙qm¯κ_,1R^0.
The proof can be followed recursively for everyN≥2 by considering analogous blocks of indices as before.Regarding Lemma13, the proof of Lemma 12 is concluded if one can check that for every N≥1, R^β≤R^0qNm¯κ_,1 whenever
(60)β∈{NS-(N-1)κ10,…,(N+1)S-Nκ10-1}={N(S-κ10)+κ10,…,N(S-κ10)+S-1}.LetN≥1 and β=N(S-κ10)+L, with κ10≤L≤S-1. Let d^1≤R^0. We have R^β=d~1qd~2[N(S-κ10)+L]≤R^0qNm¯κ_,1 if and only if Nm¯κ_,1≤d~2[N(S-κ10)+L]. The result follows for any d~2≥m¯κ_,1/(S-κ10).Lemma 14.
LetR^β be defined as in Lemma 12. Then, (R^β)β≥0 satisfies Assumption B′.Proof.
From the definition ofR^β, the lemma follows when taking d~2≥mκ_,1/(κ1+s) for every κ_=(κ0,κ1)∈𝒩 and every s∈Iκ_.Assumption
B
′
′. We assume R^β=R^0 for 0≤β≤S-1 and R^β:=d^1qd^2β for any d^2≥d^20, with d^20>0 as in Lemma 12.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′.Theorem 15.
Let Assumption A′ and Assumption B′′ be fulfilled. We also make the next assumption on the initial conditions (57); there exist Δ>0 and 0<M~<M such that
(61)|Wj(ϵ,τ)|≤ΔeM~log2(|τ|+δ1)|ϵ|K0,
for every τ∈D(0,R^0), ϵ∈D(0,r0)∖{0} and 0≤j≤S-1, where K0=max{κ0:(κ0,κ1)∈𝒩}. Then, there exists a formal power series W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), with Wβ(ϵ,τ)∈𝒪((D(0,r0)∖{0})×D˙R^β), which provides a formal solution of (56)+(57). Moreover, there exist positive constants C23 and C24 (only depending on r0, R^0, q, C, S, A1, δ1, and M) and 0<δ<1 such that
(62)|Wβ(ϵ,τ)|≤C23β!(C24δ)β|ϵ|-Cβ×eMlog2(|τ|+δ1)qA1β2,
for every ϵ∈D(0,r0)∖{0}, τ∈D˙R^β and for every β∈ℕ.Proof.
The proof follows analogous steps as the one of Theorem5, so we do not enter into details as to not to repeat arguments.
Letϵ∈D(0,r0)∖{0} and 0<δ<1. The set 𝔼 is taken to be {𝒪(D˙R^β):β≥0}. We consider the map 𝒜ϵ from 𝔼[[z]] into itself defined in the same way as in (33).
From Lemma12 and Assumption B′′, the unique formal solution of (56)+(57) determined by the recursion formula (41), W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), is such that Wβ(ϵ,τ)∈𝒪(D˙(0,R^β)) for every β≥0.
Regarding the initial conditions of the Cauchy problem, one can reduced^1, if necessary, so that R^j+sq-mκ_,1≤R^0 and so the map τ↦Wj+κ1(ϵ,q-mκ_,1τ) is well defined, for every κ_=(κ0,κ1)∈𝒩, every s∈Iκ_, and j=0,1,…,S-1-κ1. Moreover, from (61), the expression
(63)|zsqmκ_,1(κ0+1)(-τϵ)κ0∂zκ1wϵ(q-mκ_,1τ,qmκ_,2z)|(ϵ,δ)
can be estimated in an analogous manner as in the corresponding step of the proof of Theorem 5, for every κ_=(κ0,κ1)∈𝒩 and all s∈Iκ_.


## 4. Analytic Solutions in a Parameter of Singularly Perturbed Cauchy Problem

### 4.1. Laplace Transform andq-Gevrey Asymptotic Expansion

In this subsection, we recall some identities for the Laplace transform and state some definitions and first results onq-Gevrey asymptotic expansions. The next lemma can be found in [6].Lemma 16.
Letm∈ℕ and let w1(τ) be a holomorphic function in an unbounded sector U such that there exist C,K>0 with
(64)|w(τ)|≤Cexp(K|τ|),
for every τ∈U. Let 𝒟 be an unbounded sector with vertex at 0 which verifies that
(65)d+arg(t)∈(-π2,π2),cos(d+arg(t))≥δ2,
for some d∈ℝ and δ2>0. Then,
(66)t⟼∫Ld‍w(τ)e-tτdτ
is a holomorphic and bounded function defined for t∈𝒟∩{|t|>K/δ2}. Moreover, the following identities hold:
(67)∫Ld‍τme-tτdτ=m!tm+1,∂t(∫Ld‍w(τ)e-tτdτ)=∫Ld‍(-τ)w(τ)e-tτdτ,
where Ld=ℝ+eid⊆U∪{0}, for all t∈𝒟∩{|t|>K/δ2}.In the sequel, we work with functions which satisfy more restrictive bounds that the ones in (64). Indeed, we deal with bounds of the form Cexp(Klog2|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 with estimates related to the sequence (q-n2)n≥0.For any open sectorS={z∈ℂ:a<arg(z)<b,|z|<ρ} in the complex plane with vertex at 0 with ρ finite or infinite and 0≤a<b≤2π, we say the finite sector S~ with vertex at the origin is a proper subsector of S, and we denote it as S~≺S, if S~={τ∈ℂ⋆:a1<arg(τ)<b1,|z|<ρ~} for some 0≤a<a1<b1<b≤2π and some ρ~∈ℝ, 0<ρ~<ρ.ℍ stands for a complex Banach space.We preserve the Definition ofq-Gevrey asymptotic expansion established in [5], in order to be coherent with the definitions in that work.Definition 17.
LetS 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 series f^=∑n≥0‍fnϵn∈ℍ[[ϵ]] as its q-Gevrey asymptotic expansion of type A in S if for every S~≺S there exist C1,H>0 such that
(68)∥f(ϵ)-∑n=0N‍fnϵn∥ℍ≤C1HNq-A(N2/2)|ϵ|N+1(N+1)!,N≥0,
for every ϵ∈S~.The next proposition, detailed in [5] in the more general geometry of q-spirals, characterises null q-Gevrey asymptotic expansion.Proposition 18.
LetA>0 and f:S→ℍ a holomorphic function in a sector S with vertex at the origin. The following holds.(i)
Iffadmits the power series with null coefficients, which is denoted by0^, as itsq-Gevrey asymptotic expansion of typeA, then for everyS~≺Sthere existsC1>0with(69)∥f(ϵ)∥ℍ≤C1e-(1/a~)(1/2(-log(q)))log2|ϵ|,for everyϵ∈S~and everya~>A.(ii)
If for everyS~≺Sthere existsC1>0with(70)∥f(ϵ)∥ℍ≤C1e-(1/A)(1/2(-log(q)))log2|ϵ|,for everyϵ∈S,~thenfadmits0^as itsq-Gevrey asymptotic expansion of typea~inS, for everya~>A.


### 4.2. Analytic Solutions in a Parameter of Singularly Perturbed Cauchy Problem

We recall the definition of a good covering.Definition 19.
Let{ℰi}0≤i≤ν-1 be a finite family of open sectors with vertex at the origin and finite radius ϵ0. We assume that ℰi∩ℰi+1≠∅ for 0≤i≤ν-1 (we put ℰν:=ℰ0) and also that D(0,ν0)∖{0}⊆∪i=0ν-1ℰi for some ν0>0. Then, the family {ℰi}1≤i≤ν-1 is known as a good covering in ℂ⋆.Definition 20.
Let{ℰi}0≤i≤ν-1 be a good covering in ℂ⋆. We consider a family {{S~i}0≤i≤ν-1,𝒯} such that the following holds. (1)
There existdi∈[0,2π), 0<θi<π/2 such that
(71)S~i=S~i(di,θi):={t∈ℂ⋆:|arg(t)-di|<θi2},
for every 0≤i≤ν-1.(2)
𝒯 is an unbounded subset of an open sector with vertex at the origin. We assume |t|≥r𝒯 for every t∈𝒯.(3)
For every0≤i≤ν-1 and τ∈S~i, there exists δ3>0 such that |a-τ|>δ3.(4)
For every0≤i≤ν-1, t∈𝒯, and ϵ∈ℰi, one has t/ϵ∈S~i. Under the previous settings, we say the family {{S~i}0≤i≤ν-1,𝒯} is associated to the good covering {ℰi}0≤i≤ν-1.Let us consider a good covering inℂ⋆, {ℰi}0≤i≤ν-1.LetS≥1 and a∈ℂ∖ℝ+. We consider a finite subset of ℕ2, 𝒩. For every κ_=(κ0,κ1)∈𝒩, let mκ_,1,mκ_,2∈ℕ and let bκ_(ϵ,z) be a holomorphic and bounded function on D(0,r0)×ℂ, for some r0>0. For each 0≤i≤ν-1, we consider the following main Cauchy problem in the present work:
(72)ϵ∂t∂zSXi(ϵ,t,z)+a∂zSXi(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(∂tκ0∂zκ1Xi)×(ϵ,qmκ_,1t,qmκ_,2z)
with initial conditions
(73)(∂zjXi)(ϵ,t,0)=ϕi,j(ϵ,t)0≤j≤S-1,
where the functions ϕi,j(ϵ,t) are constructed as follows. Let {{S~i}0≤i≤ν-1,𝒯} be a family of open sets associated with the good covering {ℰi}0≤i≤ν-1.From now on, we assume the values of(Rβ)β≥0 and (R^β)β≥0 are those in the preceeding sections. If necessary, one can adjust the values of d1, d2, d^1, and d^2 so that Rβ<R^β for every β≥0, so that D˙R^β∩S~i,β≠∅ for every β≥0 and every 0≤i≤ν-1. Here, we have put
(74)S~i,β≔{τ∈ℂ⋆:τ∈S~i,|τ|>Rβ},β≥0,0≤i≤ν-1.For every0≤j≤ν-1, we assume that (ϵ,τ)↦Wj(ϵ,τ) is a bounded and holomorphic function on (D(0,r0)∖{0})×D˙R^j verifying
(75)|Wj(ϵ,τ)|≤ΔeM~log2((|τ|/|ϵ|)+δ1)|ϵ|K0,
for every (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^j. Here M~, K0, Δ, and δ1 are the constants provided in Theorem 5. Assume that Wj(ϵ,τ) can be extended to an analytic function (ϵ,τ)↦Wℰi,Sj,j(ϵ,τ) defined on ℰi×S~i,j and
(76)|Wℰi,S~i,j,j(ϵ,τ)|≤ΔeM~log2(|τ|+δ1)|ϵ|K0,
for every (ϵ,τ)∈ℰi×S~i,j.Takeγi such that ℝ+eγi-1⊆DR^j∪S~i,j. We put
(77)ϕi,j(t,ϵ):=∫Lγi‍Wℰi,Sj,j(ϵ,τ)e-tτ/ϵdτ,
for every (ϵ,t)∈ℰi×𝒯. One can check that ϕi,j is well defined and holomorphic in 𝒯×ℰi. Indeed, there exists δ2>0 such that cos(γj-arg(t/ϵ))>δ2 for every (t,ϵ)∈ℰi×𝒯. Moreover, from the growth properties of Wℰi,S~i,j,j(ϵ,τ), one deduces
(78)|∫Lγi‍Wℰi,Sj,j(ϵ,τ)e-tτ/ϵdτ|≤∫0∞‍|Wℰi,Sj,j(ϵ,se-1γi)|e-|t|δ2s/|ϵ|ds≤Δ|ϵ|K0∫0∞‍eM~log2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|ds,
which is convergent for every (ϵ,t)∈ℰi×𝒯.Theorem 21.
Let Assumptions A, B, and B′′ be fulfilled. For every0≤i≤ν-1, we consider the problem (72)+(73) with initial conditions constructed as above. Then, the problem (72)+(73) admits a solution Xi(ϵ,t,z) which is holomorphic and bounded in ℰi×𝒯×ℂ.
Moreover, for every0≤i≤ν-1 and for every Δ>1 there exists E1>0 (not depending on ϵ) such that
(79)supz∈D(0,ρ)t∈𝒯|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤E1e-((A1/(2d22Δ))(1/(-log(q))))log2|ϵ|,
for every ϵ∈ℰi∩ℰi+1 (where, by convention, Xν:=X0).Proof.
Let0≤i≤ν-1 and ϵ∈ℰi. We consider the Cauchy problem (28) with initial conditions given by
(80)(∂zjW)(ϵ,τ,0)=Wℰi,S~i,j,j(ϵ,τ),0≤j≤S-1.
Theorem 5 shows that the problem (31)+(80) has a formal solution W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), with Wβ∈𝒪(ℰi×S~i,β) for every β≥0. Moreover, for every β≥0 one has
(81)|Wβ(ϵ,τ)|≤C13β!(C14δ)β×eMlog2(|τ|/|ϵ|+δ1)|τϵ|CβqA1β2,
for every (ϵ,τ)∈ℰi×S~i,β, where C13,C14, and δ are positive constants provided in the proof of Theorem 5. In a parallel direction, one can consider the same Cauchy problem with initial conditions given by
(82)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where Wj∈𝒪((D(0,r0)∖{0})×D˙R^j) are as previously shown.
From Theorem15, one concludes that the formal power series W(ϵ,τ,z) is such that Wβ can be extended to a holomorphic function defined in (D(0,r0)∖{0})×D˙R^β, for every β≥0. We preserve notation for these extensions. Moreover, for every β≥0 one has
(83)|Wβ(ϵ,τ)|≤C23β!(C24δ)β|ϵ|-CβeMlog2(|τ|+δ1)qA1β2,
for every (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^β, and some positive constants C23 and C24 determined in the proof of Theorem 15.
We putXi(ϵ,τ,z)=∑β≥0‍Xi,β(t,ϵ)(zβ/β!), where
(84)Xi,β(ϵ,t):=∫Lγi‍Wβ(τ,ϵ)e-τt/ϵdτ.
We fist check thatXi is, at least formally, a solution of (72)+(73). From (67), one can check by inserting the formal power series Xi in (72), that it turns out to be a formal solution in the variable z of (72)+(73) if and only if W(ϵ,τ,z) is a formal solution of (28)+(29) and (56)+(57).
Bearing in mind thatWβ verifies (82) and (83), one derives Xi,β is well defined in ℰi×𝒯, for every β≥0. We now state a proof for the fact that (ϵ,τ,z)↦Xi(ϵ,τ,z) is indeed a holomorphic solution of (72)+(73) in ℰi×𝒯×ℂ. Let ϵ∈ℰi, t∈𝒯, and β∈ℕ. One has
(85)|∫Lγi‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤|∫Lγi,1‍Wβ(τ,ϵ)e-tτ/ϵdτ|+|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|+|∫Lγi,3‍Wβ(τ,ϵ)e-tτ/ϵdτ|,
where Lγi,1:=Lγi∩DR^β, Lγi,2:=Lγi∩S~i,β, and Lγi,3:=Lγi∩D˙R^β∩S~i,β. 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 (85) can be upper bounded by means of (83) and the choice of direction γi.
Consider(86)|∫Lγi,1‍Wβ(τ,ϵ)e-τ/ϵtdτ|≤∫0R^β‍|Wβ(se-1γi,ϵ)|e-(s|t|/|ϵ|)cos(γi-arg(t/ϵ))ds≤C23β!(C24δ)βqA1β2×∫0R^β‍|ϵ|-CβeMlog2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|ds.
One has(87)|ϵ|-CβeMlog2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|≤C~23|ϵ|-Cβe-sδ2r𝒯/2|ϵ|,
for some C~23>0. Now, the function x↦x-Cβe-sδ2r𝒯/|ϵ| attains its maximum at x=Cβ2/sδ2r𝒯. One can reduce r0, if necessary, to conclude that this function is increasing for x∈[0,ϵ0]. The expression in (86) is upper bounded by
(88)C23β!(C24δr0C)βqA1β2∫0∞‍e-sδ2r𝒯/2r0ds.
This yields(89)|∫Lγi,1‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C31β!C32βqA1β2,
for some constants C31,C32>0 only depending on δ, r0, R0, q, S, A1, C, δ1, M, r𝒯, and δ2. We now consider the second integral appearing on the right-hand side of (85). From (81) and similar estimates as before we get
(90)|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C13β!(C14δ)βqA1β2×∫Rβ∞‍eMlog2((s/|ϵ|)+δ1)(s|ϵ|)Cβe-(sδ2r𝒯)/|ϵ|ds.
The functionx↦g1(x)=eMlog2(x+δ1)xCβ, x≥0 is such that g1(x)≤g2(x) for all x≥0, where g2(x)=C~13eMlog2(x)xCβ, for some positive constant C~13, not depending on β. g2 attains its maximum value at x0=exp(-Cβ/2M) so that g(x)≤g(x0)=exp(-C2β2/4M), for every x>0. This implies
(91)eMlog2(s/|ϵ|+δ1)(s|ϵ|)Cβe-sδ2r𝒯/|ϵ|≤e-(C2/4M)β2e-sδ2r𝒯/|ϵ|.
From (90) we derive
(92)|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C13∫0∞‍e-sδ2r𝒯/2r0dsβ!qA1β2e-(C2/4M)β2e-Rβδ2r𝒯/2r0=C~14β!q(A1-C2/4Mlog(q))β2,
for some C~13>0.
From (89) and (92), we lead to the existence of positive constants C41,C42, not depending on β, such that
(93)|∑β≥0‍Xi,β(t,ϵ)zββ!|≤C41∑β≥0‍C42βqA1β2|z|β,
for every z∈ℂ. This allows us to conclude the first part of the proof.
Let0≤i≤ν-1 and ρ>0. For every (ϵ,t,z)∈(ℰi∩ℰi+1)×𝒯×D(0,ρ) we have
(94)|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤∑β≥0‍|Xi+1,β(ϵ,t)-Xi,β(ϵ,t)|ρββ!.
We can write(95)Xi+1,β(ϵ,t)-Xi,β(ϵ,t)=∫Lγi+1,2‍Wβ(τ,ϵ)e-tτ/ϵdτ-∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ+∫Lγi+1,4-Lγi,4‍Wβ(τ,ϵ)e-tτ/ϵdτ,
where Lγi+1,4-Lγi,4 stands for the path consisting of two parts: the first one going from Rβe-1γi+1 to 0 along the segment [0,Rβe-1γi+1] and the path going from 0 to Rβe-1γi following direction γi.
This integral has already been estimated in (92), for the first part of the proof, so we omit the details. We also omit the details on the integral concerning the path Lγi+1,2 which is analogous.
In order to estimate the integral along the pathLγi+1,4-Lγi,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 (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^β. One can apply Cauchy Theorem to derive
(96)∫L‍Wβ(τ,ϵ)e-tτ/ϵdτ=0,
where L=Lγi+1,4-Lγi,4-L1 is the closed path with s∈[γi,γi+1]→L1(s)=Rβe-1s. Moreover, |∫Lγi+1,4-Lγi,4‍Wβ(τ,ϵ)e-tτ/ϵdτ| equals
(97)|∫L1‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C23β!(C24δ)β|ϵ|-CβeMlog2(Rβ+δ1)qA1β2Rβ×∫γiγi+1‍e-Rβ|t|cos(θ-arg(t/ϵ))/|ϵ|dθ≤C^23β!(C24δ)β|ϵ|-CβqA1β2Rβ×∫γiγi+1‍e-Rβr𝒯δ2/2|ϵ|ds≤C˘23β!(C24δ)βqA1β2|ϵ|-Cβ×e-Rβr𝒯δ2/2|ϵ|e-Rβr𝒯δ2/2|ϵ|,
for some C^23,C˘23>0. It only rests to take into account that the function x∈(0,r0)↦x-Cβe-Rβr𝒯δ2/2x is monotonely increasing in (0,r0), so that |ϵ|-Cβe-Rβr𝒯δ2/2|ϵ| can be included in the constants C˘23 and C24.
From (92) and (97) one gets the existence of positive constants C6andC7 such that
(98)|Xi+1,β(ϵ,t)-Xi,β(ϵ,t)|≤C6β!C7βqA1β2e-(d1δ2r𝒯/2)(qd2β/|ϵ|),
for every (ϵ,t)∈(ℰi∩ℰi+1)×𝒯. Taking this last estimate into the expression of Xi+1-Xi one can conclude that
(99)|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤C6∑β≥0‍(C7ρ)βqA1β2e-(d1δ2r𝒯/2)(qd2β/|ϵ|),
for every (ϵ,t,z)∈(ℰi∩ℰi+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 [2], Lemma 16, when dealing with Gevrey asymptotic expansions.Lemma 22.
LetA1, D1, D2, and letd2 be positive constants, with D2>1. Then, for every Δ>1 there exist E1>0 and δ>0 such that
(100)∑β≥0‍D1βqA1β2e-D2(qd2β/ϵ)≤E1e-(A1/d22Δ)(1/(-log(q))2)log2ϵ,
for every ϵ∈(0,δ].The proof of Lemma22 heavily rests on the q-Gevrey version of some preliminary results which are classical in Gevrey case (see [2] and the references therein). Their proofs do not differ from the classical ones, so we omit them.Lemma 23.
Letb>0 and f:[0,b]→ℂ a continuous function having the formal expansion ∑n≥0‍antn∈ℂ[[t]] as its q-asymptotic expansion of type A1>0 at 0, meaning there exist C,H>0 such that
(101)|f(t)-∑n=0N-1‍antn|≤CHNq-A1N2/2|t|NN!,
for every N≥1 and t∈[0,δ], for some 0<δ<b.
Then, the function(102)I(x)=∫0b‍f(s)e-s/xds
admits the formal power series ∑n≥0‍ann!ϵn+1∈ℂ[[ϵ]] as its q-Gevrey asymptotic expansion of type A1 at 0. It is to say, there exist C~,H~>0 such that
(103)|I(x)-∑n=0N-1‍ann!xn+1|≤C~H~N+1q-A1(N+1)2/2|x|N+1(N+1)!,
for every N≥0 and x∈[0,δ′] for some 0<δ′<b.One can adapt the proof of Proposition  4 in [5] in our framework.Lemma 24.
LetA1,δ>0, and let ψ:[0,δ]→ℂ be a continuous function. The following holds. (1)
If there existC,H>0 such that |ψ(x)|≤CHnq-A1n2/2|x|n/n!, for every n∈ℕ, n≥0 and x∈[0,δ], then for every A~1>A1 there exists C~>0 such that
(104)|ψ(x)|≤C~e-((1/A~1)(1/(-log(q))2))log2|x|,
for every x∈(0,δ].(2)
If there existsC>0 such that |ψ(x)|≤Ce-((1/A1)(1/(-log(q))2))log2|x|, for every n∈ℕ, and x∈[0,δ], then for every A~1>A1 there exists C~,H~>0 such that
(105)|ψ(x)|≤C~H~nq-A~1n2/2|x|nn!,
for every n∈ℕ and for every x∈(0,δ].Proof of Lemma22.
Letf:[0,+∞)→ℝ be a 𝒞1 function. For every n∈ℕ, one can apply the Euler-Mac-Laurin formula
(106)∑κ=0n‍f(κ)=12(f(0)+f(n))+∫0n‍f(t)dt+∫0n‍B1(t-⌊t⌋)f′(t)dt,
where B1(s)=s-1/2 is the Bernoulli polynomial and ⌊·⌋ stands for the floor function to f(s)=D1sqA1s2e-D2(qd2s/ϵ). One has
(107)∑κ=0n‍D1κqA1κ2e-D2(qd2κ/ϵ)=12(e-D2/ϵ+D1nqA1n2e-D2(qd2n/ϵ))+∫0n‍D1tqA1t2e-D2(qd2t/ϵ)dt+∫0n‍B1(t-⌊t⌋)D1tqA1t2e-D2(qd2t/ϵ)×(log(D1)+log(q)A12t-D2log(q)d2ϵ)dt.
Taking the limit whenn tends to infinity in the previous expression we arrive at the following equality for a convergent series:
(108)∑κ=0∞‍D1κqA1κ2e-D2(qd2κ/ϵ)=12e-D2/ϵ+∫0∞‍D1tqA1t2e-D2(qd2t/ϵ)dt+∫0∞‍B1(t-⌊t⌋)D1tqA1t2e-D2(qd2t/ϵ)×(log(q)d2ϵlog(D1)+log(q)A12t-D2log(q)d2ϵ)dt.
LetI1:=∫0∞‍f(t)dt and I2:=∫0∞‍B1(t-⌊t⌋)(log(D1)+log(q)A12t-D2(log(q)d2/ϵ)qD2t/ϵ)f(t)dt.
From the fact thatB1(t-⌊t⌋)≤1/2 for every t≥0 and the change of variable D2qd2t=u, one gets
(109)I1=∫0D2‍f1(u)e-u/ϵdu,wheref1(u)≔D1log(u/D2)/log(q)d2qA1log2(u/D2)/log2(q)d22×1d2(-log(q))u,I2≤12∫0D2‍(f2(u)+f3(u)+f4(u))e-u/ϵdu,
with f2(u):=log(D1)f1(u), f3(u):=(2A1/d2)log(u/D2)f1(u), and f4:=((-log(q))d2/ϵ)eD2log(u/D2)/d2ϵf1(u), for every u∈(0,D2]. Bearing in mind that f3(u)<0 for u∈(0,D2] and from usual estimates we derive I2≤C1(1+1/ϵ)∫0D2‍f1(u)e-u/ϵdu for some C1>0. The proof is complete if one can estimate e-D2/ϵ, I1, and 1/ϵI1 appropriately. The first expression is clearly upper bounded according to (100).
From usual estimates we arrive at(110)I1≤C3∫0D2‍e-(A1/(-log(q))d22)log2(u/D2)e-u/ϵdu=C3∫0D3‍f~(u)e-u/ϵ,
for some C3>0. From Lemma 24, the function g~ defined by u∈[0,1]↦f~(D2u) (extended by continuity to u=0) is such that for every Δ~>1 there exist C~,H~>0 with
(111)|g~(u)|≤C~H~nq-((d22Δ~2)/A1)n2|u|nn!,
for every u∈[0,D2] and for every n≥0. From Lemma 23, the functions I1(ϵ) and (1/ϵ)I1(ϵ) admit the series with null coefficients as q-asymptotic expansion of type (d22Δ~)/A1. Again, from Lemma 24, one can conclude that for every Δ>Δ~ there exists C4>0 such that both I1 and (1/ϵ)I1 are upper bounded by C4e-((A1/d22Δ)(1/(-log(q))2))log2(ϵ), for every ϵ∈(0,ϵ1], for some ϵ1>0.


### 4.3. Existence of Formal Series Solutions in the Complex Parameter

In this last subsection we obtain aq-Gevrey version of a Malgrange-Sibuya type theorem. A result in this direction has already been obtained by the authors in [5] 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.Givenq∈ℂ with 0<|q|<1 and a nonempty open subset U⊂ℂ⋆, the discrete q-spiral associated with U and q consists of the products of an element in U and qm, 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 theq-Gevrey version of Malgrange-Sibuya theorem in [5] is based on the use of extension results on ultradifferential spaces of weighted functions which preserve the information of q-Gevrey bounds but causes the q-Gevrey type involved in the q-Gevrey asymptotic to suffer 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 [6], 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 25 (q-MS).
Let(𝔼,∥·∥𝔼) be a Banach space over ℂ and let {ℰi}0≤i≤ν-1 be a good covering in ℂ*. For all 0≤i≤ν-1, let Gi be a holomorphic function from ℰi into the Banach space (𝔼,∥·∥𝔼) and let the cocycle Δi(ϵ)=Gi+1(ϵ)-Gi(ϵ) be a holomorphic function from the sector Zi=ℰi+1∩ℰi into 𝔼 (with the convention that ℰν=ℰ0 and Gν=G0). We make the following assumptions.(1)
The functionsGi(ϵ) are bounded as ϵ∈ℰi tends to the origin in ℂ, for all 0≤i≤ν-1.(2)
Δi has a q-exponential decreasing of some type L>0, for every 0≤i≤ν-1, meaning there exists Ci>0 such that
(112)∥Δi(ϵ)∥𝔼≤Cie-((1/L)(1/(-log(q))2))log2|ϵ|,Then, there exists a formal power seriesG^(ϵ)∈𝔼[[ϵ]] such that Gi(ϵ) admits G^(ϵ) as its q-Gevrey asymptotic expansion of type L on ℰi, for every 0≤i≤ν-1.Proof.
We first state an auxiliary result.Lemma 26.
For all0≤i≤ν-1, there exist bounded holomorphic functions Ψi:ℰi→ℂ such that
(113)Δi(ϵ)=Ψi+1(ϵ)-Ψi(ϵ),
for all ϵ∈Zi, where by convention Ψν(ϵ)=Ψ0(ϵ). Moreover, there exist φm∈𝔼, m≥0, such that for each 0≤l≤ν-1 any L^>L and every 𝒲≺ℰl, there exist K^l,M^l>0 with
(114)∥Ψl(ϵ)-∑m=0M-1‍φmϵm∥𝔼≤K^l(M^l)Mq-L^((M-1)2/2)|ϵ|MM!
for all ϵ∈𝒲 and all M≥2.Proof.
We follow analogous arguments as in Lemma XI-2-6 from [34] with appropriate modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform.
For all0≤l≤ν-1, we choose a segment
(115)𝒞l={te-1θl,t∈[0,r]}⊂ℰl∩ℰl+1.
Theseν segments divide the open punctured disc D(0,r)∖{0} into ν open sectors ℰ~0,…,ℰ~ν-1, where
(116)ℰ~l={ϵ∈ℂ*θl-1<arg(ϵ)<θl,|ϵ|<r},0≤l≤ν-1,
where by convention θ-1=θν-1. Let
(117)Ψl(ϵ)=-12π-1∑h=0ν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ
for all ϵ∈ℰ~l, for 0≤l≤ν-1, be defined as a sum of Cauchy-Heine transforms of the functions Δh(ϵ). By deformation of the paths Cl-1 and Cl without moving their endpoints and letting the other paths Ch, h≠l-1,andl untouched (with the convention that C-1=Cν-1), one can continue analytically the function Ψl onto ℰl. Therefore, Ψl defines a holomorphic function on ℰl, for all 0≤l≤ν-1.
Now, takeϵ∈ℰl∩ℰl+1. In order to compute Ψl+1(ϵ)-Ψl(ϵ), we write
(118)Ψl(ϵ)=-12π-1∫C^l‍Δl(ξ)ξ-ϵdξ+-12π-1×∑h=0,h≠lν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,Ψl+1(ϵ)=-12π-1∫C˘l‍Δl(ξ)ξ-ϵdξ+-12π-1×∑h=0,h≠lν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,
where the paths C^l and C˘l are obtained by deforming the same path Cl without moving its endpoints in such a way that(a)
C^l⊂ℰl∩ℰl+1 and C˘l⊂ℰl∩ℰl+1,(b)
Γl,l+1:=-C˘l+C^l is a simple closed curve with positive orientation whose interior contains ϵ.
Therefore, due to the residue formula, we can write(119)Ψl+1(ϵ)-Ψl(ϵ)=12π-1∫Γl,l+1‍Δl(ξ)ξ-ϵdξ=Δl(ϵ),
for all ϵ∈ℰl∩ℰl+1 and for all 0≤l≤ν-1 (with the convention that Ψν=Ψ0).
In a second step, we derive asymptotic properties ofΨl. We fix an 0≤l≤ν-1 and a proper closed sector 𝒲 contained in ℰl. Let C~l (resp., C~l-1) be a path obtained by deforming Cl (resp., Cl-1) without moving the endpoints so that 𝒲 is contained in the interior of the simple closed curve C~l-1+γl-C~l (which is itself contained in ℰl), where γl is a circular arc joining the two points re-1θl-1 and re-1θl. We get the representation
(120)Ψl(ϵ)=-12π-1∫C~l‍Δl(ξ)ξ-ϵdξ+-12π-1×∫C~l-1‍Δl-1(ξ)ξ-ϵdξ+-12π-1∑h=0,h≠l,l-1ν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,
for all ϵ∈𝒲. One assumes that the path C~l is given as the union of a segment Ll={te-1wl:t∈[0,r1]}, where r1<r and wl>θl and a curve Γl={μl(τ):τ∈[0,1]} such that μl(0)=r1e-1wl, μl(1)=re-1θl, and r1≤|μl(τ)|<r for all τ∈[0,1). We also assume that there exists a positive number σ<1 with |ϵ|≤σr1 for all ϵ∈𝒲. By construction of the path Γl, we get that the function ϵ↦(1/2π)-1∫Γl‍(Δl(ξ)/(ξ-ϵ))dξ defines an analytic function on the open disc D(0,r1).
It remains to give estimates for the integral(1/2π-1)∫Ll‍(Δl(ξ)/(ξ-ϵ))dξ. Let M≥0 be an integer. From the usual geometric series expansion, one can write
(121)12π-1∫Ll‍Δl(ξ)ξ-ϵdξ=∑m=0M‍αl,mϵm+ϵM+1El,M+1(ϵ),
where
(122)αl,m=12π-1∫Ll‍Δl(ξ)ξm+1dξ,El,M+1(ϵ)=12π-1∫Ll‍Δl(ξ)ξM+1(ξ-ϵ)dξ,
for all ϵ∈𝒲.
Gathering (112) and (122), we get
(123)∥αl,m∥𝔼≤Kl2π∫0r1‍e-((1/L)(1/(-log(q))2))log2ττm+1dτ.
The changes of variable log(τ)=s first, and s=2L(-log(q))t afterwards, transform the right-hand side of (123) into
(124)Kl2π∫-∞log(r1)‍e-((1/L)(1/(-log(q))2))s2esmds=Kl2L(-log(q))2π×∫-∞log(r1)/2L(-log(q))‍exp(2L(-log(q))t-t2-m2L(-log(q))t)dt≤Kl2L(-log(q))2π×∫-∞∞‍exp(-t2-m2L(-log(q))t)dt.
The application of(125)ea2/4π=∫-∞∞‍e-x2-axdx,
for every a∈ℝ, which can be found in [35] (Chapter 10, page 498), leads us to
(126)∥αl,m∥𝔼≤2L(-log(q))Kl2πq-L(m2/2).
Moreover, as previously described, one can choose a positive numberη>0 (depending on 𝒲) such that |ξ-ϵ|≥|ξ|sin(η) for all ξ∈Ll and all ϵ∈𝒲. Again by (112) and (122) and following analogous calculations as before, we obtain
(127)∥El,M+1(ϵ)∥𝔼≤Kl2πsin(η)∫0r1‍e-((1/L)(1/2(-log(q))))log2ττM+2dτ≤2L(-log(q))Kl2sin(ν)πq-L((M+1)2/2),
for all ϵ∈𝒲. Using comparable arguments, one can give analogous estimates when estimating the other integrals
(128)-12π-1∫C~l-1‍Δl-1(ξ)ξ-ϵdξ,-12π-1∫Ch‍Δh(ξ)ξ-ϵdξ,
for all h≠l,l-1.
As a consequence, for any0≤l≤ν-1, there exist φl,m∈𝔼, for all m≥0 and a constant K^l>0 such that
(129)∥Ψl(ϵ)-∑m=0M-1‍φl,mϵm∥𝔼≤K^lq-L(M2/2)|ϵ|M,
for all M≥2 and all ϵ∈𝒲.
Taking into account Proposition18, we deduce that for every ℰ^i,i+1≺ℰl∩ℰl+1 and for every L^>L, the function Ψl+1(ϵ)-Ψl(ϵ) has the formal series 0^ as q-Gevrey asymptotic expansion of type L^ in ℰ^i,i+1. From the unicity of the asymptotic expansions on sectors, we deduce that all the formal series ∑m≥0‍φl,mϵm, 0≤l≤ν-1, are equal to some formal series denoted G^(ϵ)=∑m≥0‍φmϵm∈𝔼[[ϵ]].We consider now the bounded holomorphic functions(130)ai(ϵ)=Gi(ϵ)-Ψi(ϵ),
for all 0≤i≤ν-1 and all ϵ∈ℰi. By definition, for any i∈{0,…,ν-1}, we have that
(131)ai+1(ϵ)-ai(ϵ)=Gi+1(ϵ)-Gi(ϵ)-Δi(ϵ)=0,
for all ϵ∈Zi. Therefore, each ai(ϵ) is the restriction on ℰi of a holomorphic function a(ϵ) on D(0,r)∖{0}. Since a(ϵ) is bounded on D(0,r)∖{0}, the origin turns out to be a removable singularity for a(ϵ) which, as a consequence, defines a convergent power series on D(0,r).Finally, one can write(132)Gi(ϵ)=a(ϵ)+Ψi(ϵ),
for all ϵ∈ℰi, all 0≤i≤ν-1. Moreover, a(ϵ) is a convergent power series, and for every L^>L, Ψi(ϵ) has the series G^(ϵ)=∑m≥0‍φmϵm as q-Gevrey asymptotic expansion of type L^ on ℰi, for all 0≤i≤ν-1.We are under conditions to enunciate the main result in the present work.Theorem 27.
Letρ>0.Under the same hypotheses as in Theorem 21, we denote ℍ𝒯,ρ the Banach space of holomorphic and bounded functions in 𝒯×D(0,ρ) with the supremum norm. Then, there exists a formal power series
(133)X^(ϵ,t,z)=∑k≥0‍Xk(t,z)k!ϵk∈ℍ𝒯,ρ[[ϵ]],
which is formal solution of
(134)ϵ∂t∂zSX^(ϵ,t,z)+a∂zSX^(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(∂tκ0∂zκ1X^)×(ϵ,qmκ_,1t,qmκ_,2z).
Moreover, for every 0≤i≤ν-1 and every L2>d22/A1, the function Xi(ϵ,t,z) constructed in Theorem 21 admits X^(ϵ,t,z) as its q-Gevrey asymptotic expansion of type L2 in ℰi, meaning that for every 0≤i≤ν-1 and ℰ~i≺ℰi, there exist L0,L1>0 such that
(135)supz∈D(0,ρ)t∈𝒯|X(ϵ,t,z)-∑k=0N‍Xk(t,z)k!ϵk|≤L0L1Nq-L2(N2/2)|ϵ|N+1(N+1)!,
for every N≥0 and all ϵ∈ℰ~i.Proof.
Let us consider the family(Xi(ϵ,t,z))0≤i≤ν-1 constructed in Theorem 21. For every 0≤i≤ν-1, we define the function ϵ∈ℰi↦Gi(ϵ):=Xi(ϵ,t,z), which belongs to the space 𝔼𝒯,ρ. From (79), we derive the cocycle Δi:=Gi+1(ϵ)-Gi(ϵ) verifies (112) in Theorem 25, with 1/L:=A1/d22Δ for some fixed 0<Δ<1. Theorem 25 guarantees the existence of a formal power series X^(ϵ)∈ℍ𝒯,ρ[[ϵ]], such that for every L2>L, Gi(ϵ) admits G^(ϵ) as its q-Gevrey asymptotic expansion of type L2 on ℰi. This is valid for every 0≤i≤ν-1. This concludes the second part of the result.
It only rests to verify thatX^ is a formal solution of (72)+(73).
If we writeX^(ϵ,t,z):=∑k≥0‍(Xk(t,z)/k!)ϵk, we have
(136)limϵ∈ℰ~iϵ→0sup(t,z)∈𝒯×D(0,ρ)|∂ϵℓXi(t,z,ϵ)-Xℓ(t,z)|=0,
for every 0≤i≤ν-1 and all ℓ≥0.
Let0≤i≤ν-1. By construction, Xi satisfies (72)+(73). We differentiate in the equality (72) ℓ≥1 times with respect to ϵ. By Leibniz’s rule, we deduce that ∂ϵℓXi(ϵ,t,z) satisfies
(137)ϵ∂ϵℓ∂t∂zSXi(ϵ,t,z)+ℓ∂ϵℓ-1∂t∂zSXi(ϵ,t,z)+a∂zSXi(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍∑ℓ0+ℓ1=ℓ‍ℓ!ℓ0!ℓ1!∂ϵℓ1bκ_(ϵ,z)×(∂ϵℓ1∂tκ0∂zκ1Xi)(ϵ,qmκ_,1t,qmκ_,2z),
for every (ϵ,t,z)∈ℰi×𝒯×D(0,ρ). Let ϵ→0 in the previous expression. From (136) we obtain
(138)∂t∂zS(Xℓ-1(t,z)(ℓ-1)!)+a∂zS(Xℓ(t,z)ℓ!)=∑κ_=(κ0,κ1)∈𝒩‍∑ℓ0+ℓ1=ℓ‍(∂ϵℓ0bκ_(0,z)ℓ0!)×(∂tκ0∂zκ1Xℓ1(qmκ_,1t,qmκ_,2z)ℓ1!).bκ_(z,ϵ) is holomorphic wih respect to ϵ for every κ_∈𝒩. This entails bκ_(ϵ,z)=∑h≥0‍(∂ϵhbκ_(0,z)/h!)ϵh, for every (ϵ,z) in a neighborhood of the origin in ℂ2. From this and (138), we deduce X^(ϵ,t,z)=∑k≥0‍(Xk(t,z)/k!)ϵk∈ℍ𝒯,ρ[[ϵ]] is a formal solution of (72)+(73).


## 4.1. Laplace Transform andq-Gevrey Asymptotic Expansion

In this subsection, we recall some identities for the Laplace transform and state some definitions and first results onq-Gevrey asymptotic expansions. The next lemma can be found in [6].Lemma 16.
Letm∈ℕ and let w1(τ) be a holomorphic function in an unbounded sector U such that there exist C,K>0 with
(64)|w(τ)|≤Cexp(K|τ|),
for every τ∈U. Let 𝒟 be an unbounded sector with vertex at 0 which verifies that
(65)d+arg(t)∈(-π2,π2),cos(d+arg(t))≥δ2,
for some d∈ℝ and δ2>0. Then,
(66)t⟼∫Ld‍w(τ)e-tτdτ
is a holomorphic and bounded function defined for t∈𝒟∩{|t|>K/δ2}. Moreover, the following identities hold:
(67)∫Ld‍τme-tτdτ=m!tm+1,∂t(∫Ld‍w(τ)e-tτdτ)=∫Ld‍(-τ)w(τ)e-tτdτ,
where Ld=ℝ+eid⊆U∪{0}, for all t∈𝒟∩{|t|>K/δ2}.In the sequel, we work with functions which satisfy more restrictive bounds that the ones in (64). Indeed, we deal with bounds of the form Cexp(Klog2|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 with estimates related to the sequence (q-n2)n≥0.For any open sectorS={z∈ℂ:a<arg(z)<b,|z|<ρ} in the complex plane with vertex at 0 with ρ finite or infinite and 0≤a<b≤2π, we say the finite sector S~ with vertex at the origin is a proper subsector of S, and we denote it as S~≺S, if S~={τ∈ℂ⋆:a1<arg(τ)<b1,|z|<ρ~} for some 0≤a<a1<b1<b≤2π and some ρ~∈ℝ, 0<ρ~<ρ.ℍ stands for a complex Banach space.We preserve the Definition ofq-Gevrey asymptotic expansion established in [5], in order to be coherent with the definitions in that work.Definition 17.
LetS 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 series f^=∑n≥0‍fnϵn∈ℍ[[ϵ]] as its q-Gevrey asymptotic expansion of type A in S if for every S~≺S there exist C1,H>0 such that
(68)∥f(ϵ)-∑n=0N‍fnϵn∥ℍ≤C1HNq-A(N2/2)|ϵ|N+1(N+1)!,N≥0,
for every ϵ∈S~.The next proposition, detailed in [5] in the more general geometry of q-spirals, characterises null q-Gevrey asymptotic expansion.Proposition 18.
LetA>0 and f:S→ℍ a holomorphic function in a sector S with vertex at the origin. The following holds.(i)
Iffadmits the power series with null coefficients, which is denoted by0^, as itsq-Gevrey asymptotic expansion of typeA, then for everyS~≺Sthere existsC1>0with(69)∥f(ϵ)∥ℍ≤C1e-(1/a~)(1/2(-log(q)))log2|ϵ|,for everyϵ∈S~and everya~>A.(ii)
If for everyS~≺Sthere existsC1>0with(70)∥f(ϵ)∥ℍ≤C1e-(1/A)(1/2(-log(q)))log2|ϵ|,for everyϵ∈S,~thenfadmits0^as itsq-Gevrey asymptotic expansion of typea~inS, for everya~>A.


## 4.2. Analytic Solutions in a Parameter of Singularly Perturbed Cauchy Problem

We recall the definition of a good covering.Definition 19.
Let{ℰi}0≤i≤ν-1 be a finite family of open sectors with vertex at the origin and finite radius ϵ0. We assume that ℰi∩ℰi+1≠∅ for 0≤i≤ν-1 (we put ℰν:=ℰ0) and also that D(0,ν0)∖{0}⊆∪i=0ν-1ℰi for some ν0>0. Then, the family {ℰi}1≤i≤ν-1 is known as a good covering in ℂ⋆.Definition 20.
Let{ℰi}0≤i≤ν-1 be a good covering in ℂ⋆. We consider a family {{S~i}0≤i≤ν-1,𝒯} such that the following holds. (1)
There existdi∈[0,2π), 0<θi<π/2 such that
(71)S~i=S~i(di,θi):={t∈ℂ⋆:|arg(t)-di|<θi2},
for every 0≤i≤ν-1.(2)
𝒯 is an unbounded subset of an open sector with vertex at the origin. We assume |t|≥r𝒯 for every t∈𝒯.(3)
For every0≤i≤ν-1 and τ∈S~i, there exists δ3>0 such that |a-τ|>δ3.(4)
For every0≤i≤ν-1, t∈𝒯, and ϵ∈ℰi, one has t/ϵ∈S~i. Under the previous settings, we say the family {{S~i}0≤i≤ν-1,𝒯} is associated to the good covering {ℰi}0≤i≤ν-1.Let us consider a good covering inℂ⋆, {ℰi}0≤i≤ν-1.LetS≥1 and a∈ℂ∖ℝ+. We consider a finite subset of ℕ2, 𝒩. For every κ_=(κ0,κ1)∈𝒩, let mκ_,1,mκ_,2∈ℕ and let bκ_(ϵ,z) be a holomorphic and bounded function on D(0,r0)×ℂ, for some r0>0. For each 0≤i≤ν-1, we consider the following main Cauchy problem in the present work:
(72)ϵ∂t∂zSXi(ϵ,t,z)+a∂zSXi(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(∂tκ0∂zκ1Xi)×(ϵ,qmκ_,1t,qmκ_,2z)
with initial conditions
(73)(∂zjXi)(ϵ,t,0)=ϕi,j(ϵ,t)0≤j≤S-1,
where the functions ϕi,j(ϵ,t) are constructed as follows. Let {{S~i}0≤i≤ν-1,𝒯} be a family of open sets associated with the good covering {ℰi}0≤i≤ν-1.From now on, we assume the values of(Rβ)β≥0 and (R^β)β≥0 are those in the preceeding sections. If necessary, one can adjust the values of d1, d2, d^1, and d^2 so that Rβ<R^β for every β≥0, so that D˙R^β∩S~i,β≠∅ for every β≥0 and every 0≤i≤ν-1. Here, we have put
(74)S~i,β≔{τ∈ℂ⋆:τ∈S~i,|τ|>Rβ},β≥0,0≤i≤ν-1.For every0≤j≤ν-1, we assume that (ϵ,τ)↦Wj(ϵ,τ) is a bounded and holomorphic function on (D(0,r0)∖{0})×D˙R^j verifying
(75)|Wj(ϵ,τ)|≤ΔeM~log2((|τ|/|ϵ|)+δ1)|ϵ|K0,
for every (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^j. Here M~, K0, Δ, and δ1 are the constants provided in Theorem 5. Assume that Wj(ϵ,τ) can be extended to an analytic function (ϵ,τ)↦Wℰi,Sj,j(ϵ,τ) defined on ℰi×S~i,j and
(76)|Wℰi,S~i,j,j(ϵ,τ)|≤ΔeM~log2(|τ|+δ1)|ϵ|K0,
for every (ϵ,τ)∈ℰi×S~i,j.Takeγi such that ℝ+eγi-1⊆DR^j∪S~i,j. We put
(77)ϕi,j(t,ϵ):=∫Lγi‍Wℰi,Sj,j(ϵ,τ)e-tτ/ϵdτ,
for every (ϵ,t)∈ℰi×𝒯. One can check that ϕi,j is well defined and holomorphic in 𝒯×ℰi. Indeed, there exists δ2>0 such that cos(γj-arg(t/ϵ))>δ2 for every (t,ϵ)∈ℰi×𝒯. Moreover, from the growth properties of Wℰi,S~i,j,j(ϵ,τ), one deduces
(78)|∫Lγi‍Wℰi,Sj,j(ϵ,τ)e-tτ/ϵdτ|≤∫0∞‍|Wℰi,Sj,j(ϵ,se-1γi)|e-|t|δ2s/|ϵ|ds≤Δ|ϵ|K0∫0∞‍eM~log2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|ds,
which is convergent for every (ϵ,t)∈ℰi×𝒯.Theorem 21.
Let Assumptions A, B, and B′′ be fulfilled. For every0≤i≤ν-1, we consider the problem (72)+(73) with initial conditions constructed as above. Then, the problem (72)+(73) admits a solution Xi(ϵ,t,z) which is holomorphic and bounded in ℰi×𝒯×ℂ.
Moreover, for every0≤i≤ν-1 and for every Δ>1 there exists E1>0 (not depending on ϵ) such that
(79)supz∈D(0,ρ)t∈𝒯|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤E1e-((A1/(2d22Δ))(1/(-log(q))))log2|ϵ|,
for every ϵ∈ℰi∩ℰi+1 (where, by convention, Xν:=X0).Proof.
Let0≤i≤ν-1 and ϵ∈ℰi. We consider the Cauchy problem (28) with initial conditions given by
(80)(∂zjW)(ϵ,τ,0)=Wℰi,S~i,j,j(ϵ,τ),0≤j≤S-1.
Theorem 5 shows that the problem (31)+(80) has a formal solution W(ϵ,τ,z)=∑β≥0‍Wβ(ϵ,τ)(zβ/β!), with Wβ∈𝒪(ℰi×S~i,β) for every β≥0. Moreover, for every β≥0 one has
(81)|Wβ(ϵ,τ)|≤C13β!(C14δ)β×eMlog2(|τ|/|ϵ|+δ1)|τϵ|CβqA1β2,
for every (ϵ,τ)∈ℰi×S~i,β, where C13,C14, and δ are positive constants provided in the proof of Theorem 5. In a parallel direction, one can consider the same Cauchy problem with initial conditions given by
(82)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where Wj∈𝒪((D(0,r0)∖{0})×D˙R^j) are as previously shown.
From Theorem15, one concludes that the formal power series W(ϵ,τ,z) is such that Wβ can be extended to a holomorphic function defined in (D(0,r0)∖{0})×D˙R^β, for every β≥0. We preserve notation for these extensions. Moreover, for every β≥0 one has
(83)|Wβ(ϵ,τ)|≤C23β!(C24δ)β|ϵ|-CβeMlog2(|τ|+δ1)qA1β2,
for every (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^β, and some positive constants C23 and C24 determined in the proof of Theorem 15.
We putXi(ϵ,τ,z)=∑β≥0‍Xi,β(t,ϵ)(zβ/β!), where
(84)Xi,β(ϵ,t):=∫Lγi‍Wβ(τ,ϵ)e-τt/ϵdτ.
We fist check thatXi is, at least formally, a solution of (72)+(73). From (67), one can check by inserting the formal power series Xi in (72), that it turns out to be a formal solution in the variable z of (72)+(73) if and only if W(ϵ,τ,z) is a formal solution of (28)+(29) and (56)+(57).
Bearing in mind thatWβ verifies (82) and (83), one derives Xi,β is well defined in ℰi×𝒯, for every β≥0. We now state a proof for the fact that (ϵ,τ,z)↦Xi(ϵ,τ,z) is indeed a holomorphic solution of (72)+(73) in ℰi×𝒯×ℂ. Let ϵ∈ℰi, t∈𝒯, and β∈ℕ. One has
(85)|∫Lγi‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤|∫Lγi,1‍Wβ(τ,ϵ)e-tτ/ϵdτ|+|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|+|∫Lγi,3‍Wβ(τ,ϵ)e-tτ/ϵdτ|,
where Lγi,1:=Lγi∩DR^β, Lγi,2:=Lγi∩S~i,β, and Lγi,3:=Lγi∩D˙R^β∩S~i,β. 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 (85) can be upper bounded by means of (83) and the choice of direction γi.
Consider(86)|∫Lγi,1‍Wβ(τ,ϵ)e-τ/ϵtdτ|≤∫0R^β‍|Wβ(se-1γi,ϵ)|e-(s|t|/|ϵ|)cos(γi-arg(t/ϵ))ds≤C23β!(C24δ)βqA1β2×∫0R^β‍|ϵ|-CβeMlog2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|ds.
One has(87)|ϵ|-CβeMlog2(s/|ϵ|+δ1)e-sδ2r𝒯/|ϵ|≤C~23|ϵ|-Cβe-sδ2r𝒯/2|ϵ|,
for some C~23>0. Now, the function x↦x-Cβe-sδ2r𝒯/|ϵ| attains its maximum at x=Cβ2/sδ2r𝒯. One can reduce r0, if necessary, to conclude that this function is increasing for x∈[0,ϵ0]. The expression in (86) is upper bounded by
(88)C23β!(C24δr0C)βqA1β2∫0∞‍e-sδ2r𝒯/2r0ds.
This yields(89)|∫Lγi,1‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C31β!C32βqA1β2,
for some constants C31,C32>0 only depending on δ, r0, R0, q, S, A1, C, δ1, M, r𝒯, and δ2. We now consider the second integral appearing on the right-hand side of (85). From (81) and similar estimates as before we get
(90)|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C13β!(C14δ)βqA1β2×∫Rβ∞‍eMlog2((s/|ϵ|)+δ1)(s|ϵ|)Cβe-(sδ2r𝒯)/|ϵ|ds.
The functionx↦g1(x)=eMlog2(x+δ1)xCβ, x≥0 is such that g1(x)≤g2(x) for all x≥0, where g2(x)=C~13eMlog2(x)xCβ, for some positive constant C~13, not depending on β. g2 attains its maximum value at x0=exp(-Cβ/2M) so that g(x)≤g(x0)=exp(-C2β2/4M), for every x>0. This implies
(91)eMlog2(s/|ϵ|+δ1)(s|ϵ|)Cβe-sδ2r𝒯/|ϵ|≤e-(C2/4M)β2e-sδ2r𝒯/|ϵ|.
From (90) we derive
(92)|∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C13∫0∞‍e-sδ2r𝒯/2r0dsβ!qA1β2e-(C2/4M)β2e-Rβδ2r𝒯/2r0=C~14β!q(A1-C2/4Mlog(q))β2,
for some C~13>0.
From (89) and (92), we lead to the existence of positive constants C41,C42, not depending on β, such that
(93)|∑β≥0‍Xi,β(t,ϵ)zββ!|≤C41∑β≥0‍C42βqA1β2|z|β,
for every z∈ℂ. This allows us to conclude the first part of the proof.
Let0≤i≤ν-1 and ρ>0. For every (ϵ,t,z)∈(ℰi∩ℰi+1)×𝒯×D(0,ρ) we have
(94)|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤∑β≥0‍|Xi+1,β(ϵ,t)-Xi,β(ϵ,t)|ρββ!.
We can write(95)Xi+1,β(ϵ,t)-Xi,β(ϵ,t)=∫Lγi+1,2‍Wβ(τ,ϵ)e-tτ/ϵdτ-∫Lγi,2‍Wβ(τ,ϵ)e-tτ/ϵdτ+∫Lγi+1,4-Lγi,4‍Wβ(τ,ϵ)e-tτ/ϵdτ,
where Lγi+1,4-Lγi,4 stands for the path consisting of two parts: the first one going from Rβe-1γi+1 to 0 along the segment [0,Rβe-1γi+1] and the path going from 0 to Rβe-1γi following direction γi.
This integral has already been estimated in (92), for the first part of the proof, so we omit the details. We also omit the details on the integral concerning the path Lγi+1,2 which is analogous.
In order to estimate the integral along the pathLγi+1,4-Lγi,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 (ϵ,τ)∈(D(0,r0)∖{0})×D˙R^β. One can apply Cauchy Theorem to derive
(96)∫L‍Wβ(τ,ϵ)e-tτ/ϵdτ=0,
where L=Lγi+1,4-Lγi,4-L1 is the closed path with s∈[γi,γi+1]→L1(s)=Rβe-1s. Moreover, |∫Lγi+1,4-Lγi,4‍Wβ(τ,ϵ)e-tτ/ϵdτ| equals
(97)|∫L1‍Wβ(τ,ϵ)e-tτ/ϵdτ|≤C23β!(C24δ)β|ϵ|-CβeMlog2(Rβ+δ1)qA1β2Rβ×∫γiγi+1‍e-Rβ|t|cos(θ-arg(t/ϵ))/|ϵ|dθ≤C^23β!(C24δ)β|ϵ|-CβqA1β2Rβ×∫γiγi+1‍e-Rβr𝒯δ2/2|ϵ|ds≤C˘23β!(C24δ)βqA1β2|ϵ|-Cβ×e-Rβr𝒯δ2/2|ϵ|e-Rβr𝒯δ2/2|ϵ|,
for some C^23,C˘23>0. It only rests to take into account that the function x∈(0,r0)↦x-Cβe-Rβr𝒯δ2/2x is monotonely increasing in (0,r0), so that |ϵ|-Cβe-Rβr𝒯δ2/2|ϵ| can be included in the constants C˘23 and C24.
From (92) and (97) one gets the existence of positive constants C6andC7 such that
(98)|Xi+1,β(ϵ,t)-Xi,β(ϵ,t)|≤C6β!C7βqA1β2e-(d1δ2r𝒯/2)(qd2β/|ϵ|),
for every (ϵ,t)∈(ℰi∩ℰi+1)×𝒯. Taking this last estimate into the expression of Xi+1-Xi one can conclude that
(99)|Xi+1(ϵ,t,z)-Xi(ϵ,t,z)|≤C6∑β≥0‍(C7ρ)βqA1β2e-(d1δ2r𝒯/2)(qd2β/|ϵ|),
for every (ϵ,t,z)∈(ℰi∩ℰi+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 [2], Lemma 16, when dealing with Gevrey asymptotic expansions.Lemma 22.
LetA1, D1, D2, and letd2 be positive constants, with D2>1. Then, for every Δ>1 there exist E1>0 and δ>0 such that
(100)∑β≥0‍D1βqA1β2e-D2(qd2β/ϵ)≤E1e-(A1/d22Δ)(1/(-log(q))2)log2ϵ,
for every ϵ∈(0,δ].The proof of Lemma22 heavily rests on the q-Gevrey version of some preliminary results which are classical in Gevrey case (see [2] and the references therein). Their proofs do not differ from the classical ones, so we omit them.Lemma 23.
Letb>0 and f:[0,b]→ℂ a continuous function having the formal expansion ∑n≥0‍antn∈ℂ[[t]] as its q-asymptotic expansion of type A1>0 at 0, meaning there exist C,H>0 such that
(101)|f(t)-∑n=0N-1‍antn|≤CHNq-A1N2/2|t|NN!,
for every N≥1 and t∈[0,δ], for some 0<δ<b.
Then, the function(102)I(x)=∫0b‍f(s)e-s/xds
admits the formal power series ∑n≥0‍ann!ϵn+1∈ℂ[[ϵ]] as its q-Gevrey asymptotic expansion of type A1 at 0. It is to say, there exist C~,H~>0 such that
(103)|I(x)-∑n=0N-1‍ann!xn+1|≤C~H~N+1q-A1(N+1)2/2|x|N+1(N+1)!,
for every N≥0 and x∈[0,δ′] for some 0<δ′<b.One can adapt the proof of Proposition  4 in [5] in our framework.Lemma 24.
LetA1,δ>0, and let ψ:[0,δ]→ℂ be a continuous function. The following holds. (1)
If there existC,H>0 such that |ψ(x)|≤CHnq-A1n2/2|x|n/n!, for every n∈ℕ, n≥0 and x∈[0,δ], then for every A~1>A1 there exists C~>0 such that
(104)|ψ(x)|≤C~e-((1/A~1)(1/(-log(q))2))log2|x|,
for every x∈(0,δ].(2)
If there existsC>0 such that |ψ(x)|≤Ce-((1/A1)(1/(-log(q))2))log2|x|, for every n∈ℕ, and x∈[0,δ], then for every A~1>A1 there exists C~,H~>0 such that
(105)|ψ(x)|≤C~H~nq-A~1n2/2|x|nn!,
for every n∈ℕ and for every x∈(0,δ].Proof of Lemma22.
Letf:[0,+∞)→ℝ be a 𝒞1 function. For every n∈ℕ, one can apply the Euler-Mac-Laurin formula
(106)∑κ=0n‍f(κ)=12(f(0)+f(n))+∫0n‍f(t)dt+∫0n‍B1(t-⌊t⌋)f′(t)dt,
where B1(s)=s-1/2 is the Bernoulli polynomial and ⌊·⌋ stands for the floor function to f(s)=D1sqA1s2e-D2(qd2s/ϵ). One has
(107)∑κ=0n‍D1κqA1κ2e-D2(qd2κ/ϵ)=12(e-D2/ϵ+D1nqA1n2e-D2(qd2n/ϵ))+∫0n‍D1tqA1t2e-D2(qd2t/ϵ)dt+∫0n‍B1(t-⌊t⌋)D1tqA1t2e-D2(qd2t/ϵ)×(log(D1)+log(q)A12t-D2log(q)d2ϵ)dt.
Taking the limit whenn tends to infinity in the previous expression we arrive at the following equality for a convergent series:
(108)∑κ=0∞‍D1κqA1κ2e-D2(qd2κ/ϵ)=12e-D2/ϵ+∫0∞‍D1tqA1t2e-D2(qd2t/ϵ)dt+∫0∞‍B1(t-⌊t⌋)D1tqA1t2e-D2(qd2t/ϵ)×(log(q)d2ϵlog(D1)+log(q)A12t-D2log(q)d2ϵ)dt.
LetI1:=∫0∞‍f(t)dt and I2:=∫0∞‍B1(t-⌊t⌋)(log(D1)+log(q)A12t-D2(log(q)d2/ϵ)qD2t/ϵ)f(t)dt.
From the fact thatB1(t-⌊t⌋)≤1/2 for every t≥0 and the change of variable D2qd2t=u, one gets
(109)I1=∫0D2‍f1(u)e-u/ϵdu,wheref1(u)≔D1log(u/D2)/log(q)d2qA1log2(u/D2)/log2(q)d22×1d2(-log(q))u,I2≤12∫0D2‍(f2(u)+f3(u)+f4(u))e-u/ϵdu,
with f2(u):=log(D1)f1(u), f3(u):=(2A1/d2)log(u/D2)f1(u), and f4:=((-log(q))d2/ϵ)eD2log(u/D2)/d2ϵf1(u), for every u∈(0,D2]. Bearing in mind that f3(u)<0 for u∈(0,D2] and from usual estimates we derive I2≤C1(1+1/ϵ)∫0D2‍f1(u)e-u/ϵdu for some C1>0. The proof is complete if one can estimate e-D2/ϵ, I1, and 1/ϵI1 appropriately. The first expression is clearly upper bounded according to (100).
From usual estimates we arrive at(110)I1≤C3∫0D2‍e-(A1/(-log(q))d22)log2(u/D2)e-u/ϵdu=C3∫0D3‍f~(u)e-u/ϵ,
for some C3>0. From Lemma 24, the function g~ defined by u∈[0,1]↦f~(D2u) (extended by continuity to u=0) is such that for every Δ~>1 there exist C~,H~>0 with
(111)|g~(u)|≤C~H~nq-((d22Δ~2)/A1)n2|u|nn!,
for every u∈[0,D2] and for every n≥0. From Lemma 23, the functions I1(ϵ) and (1/ϵ)I1(ϵ) admit the series with null coefficients as q-asymptotic expansion of type (d22Δ~)/A1. Again, from Lemma 24, one can conclude that for every Δ>Δ~ there exists C4>0 such that both I1 and (1/ϵ)I1 are upper bounded by C4e-((A1/d22Δ)(1/(-log(q))2))log2(ϵ), for every ϵ∈(0,ϵ1], for some ϵ1>0.


## 4.3. Existence of Formal Series Solutions in the Complex Parameter

In this last subsection we obtain aq-Gevrey version of a Malgrange-Sibuya type theorem. A result in this direction has already been obtained by the authors in [5] 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.Givenq∈ℂ with 0<|q|<1 and a nonempty open subset U⊂ℂ⋆, the discrete q-spiral associated with U and q consists of the products of an element in U and qm, 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 theq-Gevrey version of Malgrange-Sibuya theorem in [5] is based on the use of extension results on ultradifferential spaces of weighted functions which preserve the information of q-Gevrey bounds but causes the q-Gevrey type involved in the q-Gevrey asymptotic to suffer 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 [6], 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 25 (q-MS).
Let(𝔼,∥·∥𝔼) be a Banach space over ℂ and let {ℰi}0≤i≤ν-1 be a good covering in ℂ*. For all 0≤i≤ν-1, let Gi be a holomorphic function from ℰi into the Banach space (𝔼,∥·∥𝔼) and let the cocycle Δi(ϵ)=Gi+1(ϵ)-Gi(ϵ) be a holomorphic function from the sector Zi=ℰi+1∩ℰi into 𝔼 (with the convention that ℰν=ℰ0 and Gν=G0). We make the following assumptions.(1)
The functionsGi(ϵ) are bounded as ϵ∈ℰi tends to the origin in ℂ, for all 0≤i≤ν-1.(2)
Δi has a q-exponential decreasing of some type L>0, for every 0≤i≤ν-1, meaning there exists Ci>0 such that
(112)∥Δi(ϵ)∥𝔼≤Cie-((1/L)(1/(-log(q))2))log2|ϵ|,Then, there exists a formal power seriesG^(ϵ)∈𝔼[[ϵ]] such that Gi(ϵ) admits G^(ϵ) as its q-Gevrey asymptotic expansion of type L on ℰi, for every 0≤i≤ν-1.Proof.
We first state an auxiliary result.Lemma 26.
For all0≤i≤ν-1, there exist bounded holomorphic functions Ψi:ℰi→ℂ such that
(113)Δi(ϵ)=Ψi+1(ϵ)-Ψi(ϵ),
for all ϵ∈Zi, where by convention Ψν(ϵ)=Ψ0(ϵ). Moreover, there exist φm∈𝔼, m≥0, such that for each 0≤l≤ν-1 any L^>L and every 𝒲≺ℰl, there exist K^l,M^l>0 with
(114)∥Ψl(ϵ)-∑m=0M-1‍φmϵm∥𝔼≤K^l(M^l)Mq-L^((M-1)2/2)|ϵ|MM!
for all ϵ∈𝒲 and all M≥2.Proof.
We follow analogous arguments as in Lemma XI-2-6 from [34] with appropriate modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform.
For all0≤l≤ν-1, we choose a segment
(115)𝒞l={te-1θl,t∈[0,r]}⊂ℰl∩ℰl+1.
Theseν segments divide the open punctured disc D(0,r)∖{0} into ν open sectors ℰ~0,…,ℰ~ν-1, where
(116)ℰ~l={ϵ∈ℂ*θl-1<arg(ϵ)<θl,|ϵ|<r},0≤l≤ν-1,
where by convention θ-1=θν-1. Let
(117)Ψl(ϵ)=-12π-1∑h=0ν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ
for all ϵ∈ℰ~l, for 0≤l≤ν-1, be defined as a sum of Cauchy-Heine transforms of the functions Δh(ϵ). By deformation of the paths Cl-1 and Cl without moving their endpoints and letting the other paths Ch, h≠l-1,andl untouched (with the convention that C-1=Cν-1), one can continue analytically the function Ψl onto ℰl. Therefore, Ψl defines a holomorphic function on ℰl, for all 0≤l≤ν-1.
Now, takeϵ∈ℰl∩ℰl+1. In order to compute Ψl+1(ϵ)-Ψl(ϵ), we write
(118)Ψl(ϵ)=-12π-1∫C^l‍Δl(ξ)ξ-ϵdξ+-12π-1×∑h=0,h≠lν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,Ψl+1(ϵ)=-12π-1∫C˘l‍Δl(ξ)ξ-ϵdξ+-12π-1×∑h=0,h≠lν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,
where the paths C^l and C˘l are obtained by deforming the same path Cl without moving its endpoints in such a way that(a)
C^l⊂ℰl∩ℰl+1 and C˘l⊂ℰl∩ℰl+1,(b)
Γl,l+1:=-C˘l+C^l is a simple closed curve with positive orientation whose interior contains ϵ.
Therefore, due to the residue formula, we can write(119)Ψl+1(ϵ)-Ψl(ϵ)=12π-1∫Γl,l+1‍Δl(ξ)ξ-ϵdξ=Δl(ϵ),
for all ϵ∈ℰl∩ℰl+1 and for all 0≤l≤ν-1 (with the convention that Ψν=Ψ0).
In a second step, we derive asymptotic properties ofΨl. We fix an 0≤l≤ν-1 and a proper closed sector 𝒲 contained in ℰl. Let C~l (resp., C~l-1) be a path obtained by deforming Cl (resp., Cl-1) without moving the endpoints so that 𝒲 is contained in the interior of the simple closed curve C~l-1+γl-C~l (which is itself contained in ℰl), where γl is a circular arc joining the two points re-1θl-1 and re-1θl. We get the representation
(120)Ψl(ϵ)=-12π-1∫C~l‍Δl(ξ)ξ-ϵdξ+-12π-1×∫C~l-1‍Δl-1(ξ)ξ-ϵdξ+-12π-1∑h=0,h≠l,l-1ν-1‍∫Ch‍Δh(ξ)ξ-ϵdξ,
for all ϵ∈𝒲. One assumes that the path C~l is given as the union of a segment Ll={te-1wl:t∈[0,r1]}, where r1<r and wl>θl and a curve Γl={μl(τ):τ∈[0,1]} such that μl(0)=r1e-1wl, μl(1)=re-1θl, and r1≤|μl(τ)|<r for all τ∈[0,1). We also assume that there exists a positive number σ<1 with |ϵ|≤σr1 for all ϵ∈𝒲. By construction of the path Γl, we get that the function ϵ↦(1/2π)-1∫Γl‍(Δl(ξ)/(ξ-ϵ))dξ defines an analytic function on the open disc D(0,r1).
It remains to give estimates for the integral(1/2π-1)∫Ll‍(Δl(ξ)/(ξ-ϵ))dξ. Let M≥0 be an integer. From the usual geometric series expansion, one can write
(121)12π-1∫Ll‍Δl(ξ)ξ-ϵdξ=∑m=0M‍αl,mϵm+ϵM+1El,M+1(ϵ),
where
(122)αl,m=12π-1∫Ll‍Δl(ξ)ξm+1dξ,El,M+1(ϵ)=12π-1∫Ll‍Δl(ξ)ξM+1(ξ-ϵ)dξ,
for all ϵ∈𝒲.
Gathering (112) and (122), we get
(123)∥αl,m∥𝔼≤Kl2π∫0r1‍e-((1/L)(1/(-log(q))2))log2ττm+1dτ.
The changes of variable log(τ)=s first, and s=2L(-log(q))t afterwards, transform the right-hand side of (123) into
(124)Kl2π∫-∞log(r1)‍e-((1/L)(1/(-log(q))2))s2esmds=Kl2L(-log(q))2π×∫-∞log(r1)/2L(-log(q))‍exp(2L(-log(q))t-t2-m2L(-log(q))t)dt≤Kl2L(-log(q))2π×∫-∞∞‍exp(-t2-m2L(-log(q))t)dt.
The application of(125)ea2/4π=∫-∞∞‍e-x2-axdx,
for every a∈ℝ, which can be found in [35] (Chapter 10, page 498), leads us to
(126)∥αl,m∥𝔼≤2L(-log(q))Kl2πq-L(m2/2).
Moreover, as previously described, one can choose a positive numberη>0 (depending on 𝒲) such that |ξ-ϵ|≥|ξ|sin(η) for all ξ∈Ll and all ϵ∈𝒲. Again by (112) and (122) and following analogous calculations as before, we obtain
(127)∥El,M+1(ϵ)∥𝔼≤Kl2πsin(η)∫0r1‍e-((1/L)(1/2(-log(q))))log2ττM+2dτ≤2L(-log(q))Kl2sin(ν)πq-L((M+1)2/2),
for all ϵ∈𝒲. Using comparable arguments, one can give analogous estimates when estimating the other integrals
(128)-12π-1∫C~l-1‍Δl-1(ξ)ξ-ϵdξ,-12π-1∫Ch‍Δh(ξ)ξ-ϵdξ,
for all h≠l,l-1.
As a consequence, for any0≤l≤ν-1, there exist φl,m∈𝔼, for all m≥0 and a constant K^l>0 such that
(129)∥Ψl(ϵ)-∑m=0M-1‍φl,mϵm∥𝔼≤K^lq-L(M2/2)|ϵ|M,
for all M≥2 and all ϵ∈𝒲.
Taking into account Proposition18, we deduce that for every ℰ^i,i+1≺ℰl∩ℰl+1 and for every L^>L, the function Ψl+1(ϵ)-Ψl(ϵ) has the formal series 0^ as q-Gevrey asymptotic expansion of type L^ in ℰ^i,i+1. From the unicity of the asymptotic expansions on sectors, we deduce that all the formal series ∑m≥0‍φl,mϵm, 0≤l≤ν-1, are equal to some formal series denoted G^(ϵ)=∑m≥0‍φmϵm∈𝔼[[ϵ]].We consider now the bounded holomorphic functions(130)ai(ϵ)=Gi(ϵ)-Ψi(ϵ),
for all 0≤i≤ν-1 and all ϵ∈ℰi. By definition, for any i∈{0,…,ν-1}, we have that
(131)ai+1(ϵ)-ai(ϵ)=Gi+1(ϵ)-Gi(ϵ)-Δi(ϵ)=0,
for all ϵ∈Zi. Therefore, each ai(ϵ) is the restriction on ℰi of a holomorphic function a(ϵ) on D(0,r)∖{0}. Since a(ϵ) is bounded on D(0,r)∖{0}, the origin turns out to be a removable singularity for a(ϵ) which, as a consequence, defines a convergent power series on D(0,r).Finally, one can write(132)Gi(ϵ)=a(ϵ)+Ψi(ϵ),
for all ϵ∈ℰi, all 0≤i≤ν-1. Moreover, a(ϵ) is a convergent power series, and for every L^>L, Ψi(ϵ) has the series G^(ϵ)=∑m≥0‍φmϵm as q-Gevrey asymptotic expansion of type L^ on ℰi, for all 0≤i≤ν-1.We are under conditions to enunciate the main result in the present work.Theorem 27.
Letρ>0.Under the same hypotheses as in Theorem 21, we denote ℍ𝒯,ρ the Banach space of holomorphic and bounded functions in 𝒯×D(0,ρ) with the supremum norm. Then, there exists a formal power series
(133)X^(ϵ,t,z)=∑k≥0‍Xk(t,z)k!ϵk∈ℍ𝒯,ρ[[ϵ]],
which is formal solution of
(134)ϵ∂t∂zSX^(ϵ,t,z)+a∂zSX^(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍bκ_(ϵ,z)(∂tκ0∂zκ1X^)×(ϵ,qmκ_,1t,qmκ_,2z).
Moreover, for every 0≤i≤ν-1 and every L2>d22/A1, the function Xi(ϵ,t,z) constructed in Theorem 21 admits X^(ϵ,t,z) as its q-Gevrey asymptotic expansion of type L2 in ℰi, meaning that for every 0≤i≤ν-1 and ℰ~i≺ℰi, there exist L0,L1>0 such that
(135)supz∈D(0,ρ)t∈𝒯|X(ϵ,t,z)-∑k=0N‍Xk(t,z)k!ϵk|≤L0L1Nq-L2(N2/2)|ϵ|N+1(N+1)!,
for every N≥0 and all ϵ∈ℰ~i.Proof.
Let us consider the family(Xi(ϵ,t,z))0≤i≤ν-1 constructed in Theorem 21. For every 0≤i≤ν-1, we define the function ϵ∈ℰi↦Gi(ϵ):=Xi(ϵ,t,z), which belongs to the space 𝔼𝒯,ρ. From (79), we derive the cocycle Δi:=Gi+1(ϵ)-Gi(ϵ) verifies (112) in Theorem 25, with 1/L:=A1/d22Δ for some fixed 0<Δ<1. Theorem 25 guarantees the existence of a formal power series X^(ϵ)∈ℍ𝒯,ρ[[ϵ]], such that for every L2>L, Gi(ϵ) admits G^(ϵ) as its q-Gevrey asymptotic expansion of type L2 on ℰi. This is valid for every 0≤i≤ν-1. This concludes the second part of the result.
It only rests to verify thatX^ is a formal solution of (72)+(73).
If we writeX^(ϵ,t,z):=∑k≥0‍(Xk(t,z)/k!)ϵk, we have
(136)limϵ∈ℰ~iϵ→0sup(t,z)∈𝒯×D(0,ρ)|∂ϵℓXi(t,z,ϵ)-Xℓ(t,z)|=0,
for every 0≤i≤ν-1 and all ℓ≥0.
Let0≤i≤ν-1. By construction, Xi satisfies (72)+(73). We differentiate in the equality (72) ℓ≥1 times with respect to ϵ. By Leibniz’s rule, we deduce that ∂ϵℓXi(ϵ,t,z) satisfies
(137)ϵ∂ϵℓ∂t∂zSXi(ϵ,t,z)+ℓ∂ϵℓ-1∂t∂zSXi(ϵ,t,z)+a∂zSXi(ϵ,t,z)=∑κ_=(κ0,κ1)∈𝒩‍∑ℓ0+ℓ1=ℓ‍ℓ!ℓ0!ℓ1!∂ϵℓ1bκ_(ϵ,z)×(∂ϵℓ1∂tκ0∂zκ1Xi)(ϵ,qmκ_,1t,qmκ_,2z),
for every (ϵ,t,z)∈ℰi×𝒯×D(0,ρ). Let ϵ→0 in the previous expression. From (136) we obtain
(138)∂t∂zS(Xℓ-1(t,z)(ℓ-1)!)+a∂zS(Xℓ(t,z)ℓ!)=∑κ_=(κ0,κ1)∈𝒩‍∑ℓ0+ℓ1=ℓ‍(∂ϵℓ0bκ_(0,z)ℓ0!)×(∂tκ0∂zκ1Xℓ1(qmκ_,1t,qmκ_,2z)ℓ1!).bκ_(z,ϵ) is holomorphic wih respect to ϵ for every κ_∈𝒩. This entails bκ_(ϵ,z)=∑h≥0‍(∂ϵhbκ_(0,z)/h!)ϵh, for every (ϵ,z) in a neighborhood of the origin in ℂ2. From this and (138), we deduce X^(ϵ,t,z)=∑k≥0‍(Xk(t,z)/k!)ϵk∈ℍ𝒯,ρ[[ϵ]] is a formal solution of (72)+(73).



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