# Onq-Gevrey Asymptotics for Singularly Perturbed q-Difference-Differential Problems with an Irregular Singularity

Authors: Alberto Lastra; Stéphane Malek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study aq-analog of a singularly perturbed Cauchy problem with irregular singularity
in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open q-spirals to the origin. By means of a q-Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the q-Gevrey asymptotic expansion (of certain type) of the actual solutions.


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## Body

## 1. Introduction

We study a family ofq-difference-differential equations of the following form:(1.1)ϵt∂zSX(ϵ,qt,z)+∂zSX(ϵ,t,z)=∑k=0S-1bk(ϵ,z)(tσq)m0,k(∂zkX)(ϵ,t,zq-m1,k),
where q∈ℂ such that |q|>1,m0,k,m1,k are positive integers, bk(ϵ,z) are polynomials in z with holomorphic coefficients in ϵ on some neighborhood of 0 in ℂ, and σq is the dilation operator given by (σqX)(ϵ,t,z)=X(ϵ,qt,z). As in previous works [1–3], the map (t,z)↦(qm0,kt,zq-m1,k) is assumed to be a volume shrinking map, meaning that the modulus of the Jacobian determinant |q|m0,k-m1,k is less than 1, for every 0≤k≤S-1.In [4], the second author studies a similar singularly perturbed Cauchy problem. In this previous work, the polynomial bk(ϵ,z)∶=∑s∈Ikbks(ϵ)zs is such that, for all 0≤k≤S-1,Ik is a finite subset of ℕ={0,1,…} and bks(ϵ) are bounded holomorphic functions on some disc D(0,r0) in ℂ which verify that the origin is a zero of order at least m0,k. The main point on these flatness conditions on the coefficients in bk(ϵ,z) is that the method used by Canalis-Durand et al. in [5] could be adapted so that the initial singularly perturbed problem turns into an auxiliary regularly perturbed q-difference-differential equation with an irregular singularity at t=0, preserving holomorphic coefficients bks (we refer to [4] for the details). These constricting conditions on the flatness of bk(ϵ,z) is now omitted, so that previous result is generalized. In the present work we will make use not only of the procedure considered in [5] but also of the methodology followed in [6]. In that work, the second author considers a family of singularly perturbed nonlinear partial differential equations such that the coefficients appearing possess poles with respect to ϵ at the origin after the change of variable t↦t/ϵ. This scenario fits our problem.In both the present work and [6], the procedure for locating actual solutions relies on the research of certain appropriate Banach spaces. The ones appearing here may be regarded as q-analogs of the ones in [6].In order to fix ideas we first settle a brief summary of the procedure followed. We consider a finite family of discreteq-spirals (UIq-ℕ)I∈ℐ in such a way that it provides a good covering at 0 (Definition 4.6).We depart from a finite family, with indices belonging to a setℐ, of perturbed Cauchy problems (4.22) and (4.23). Let I∈ℐ be fixed. Firstly, by means of a nondiscrete q-analog of Laplace transform introduced by Zhang in [7] (for details on classical Laplace transform we refer to [8, 9]), we are able to transform our initial problem into auxiliary equation (2.13) (or (3.8)).The transformed problem fits into a certain Cauchy auxiliary problem such as (2.13) and (2.14) which is considered in Section 2. Here, its solution is found in the space of formal power series in z with coefficients belonging to the space of holomorphic functions defined in the product of discrete q-spirals to the origin in the variable ϵ (this domain corresponds to UIq-ℕ in the auxiliary transformed problem) times a continuous q-spiral to infinity in the variable τ (VIqℝ+ for the auxiliary equation). Moreover, for any fixed ϵ and regarding our auxiliary equation, one can deduce that the coefficients, as functions in the variable τ, belong to the Banach space of holomorphic functions in VIqℝ+ subject to q-Gevrey bounds
(1.2)|WβI(ϵ,τ)|≤C1β!HβeMlog⁡2|τ/ϵ||τϵ|Cβ|q|-A1β2,τ∈VIqR+,
for positive constants C1,C,M,H,A1>0, where the index of the coefficient considered is β (see Theorem 2.4).Also, the transformed problem fits into the auxiliary problem (3.8) and (3.9), studied in detail in Section 3. In this case, the solution is found in the space of formal power series in z with coefficients belonging to the space of holomorphic functions defined in the product of a punctured disc at 0 in the variable ϵ times a punctured disc at the origin in τ. For a fixed ϵ, the coefficients belong to the Banach space of holomorphic functions in D(0,ρ0)∖{0} such that
(1.3)|WβI(ϵ,τ)|≤C1β!HβeMlog⁡2|τ/ϵ||ϵ|-Cβ|q|-A1β2,τ∈D(0,ρ0)∖{0},
for positive constants C1,C,M,H,A1>0 when β is the index of the coefficient considered (see Theorem 3.4).From these results, we get a sequence(WβI)β∈ℕ consisting of holomorphic functions in the variable τ so that the q-Laplace transform can be applied to its elements. In addition, the function(1.4)XI(ϵ,t,z)∶=∑β≥0Lq;1λIWβI(ϵ,ϵt)zββ!
turns out to be a holomorphic function defined in UIq-ℕ×𝒯×ℂ which is a solution of the initial problem. Here, 𝒯 is an adequate open half q-spiral to 0 and λI corresponds to certain q-directions for the q-Laplace transform (see Proposition 4.3). The way to proceed is also followed by the authors in [10, 11] when studying asymptotic properties of analytic solutions of q-difference equations with irregular singularities.It is worth pointing out that the choice of a continuous summation procedure unlike the discrete one in [4] is due to the requirement of the Cauchy theorem on the way.At this point we own a finite family(XI)I∈ℐ of solutions of (4.22) and (4.23). The main goal is to study its asymptotic behavior at the origin in some sense. Let ρ>0. One observes (Theorem 4.11) that whenever the intersection UI∩UI′ is not empty we have
(1.5)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤C1e-(1/A)log⁡2|ϵ|
for positive constants C1,A and for every (ϵ,t,z)∈(UIq-ℕ∩UI′q-ℕ)×𝒯×D(0,ρ). Equation (1.5) implies that the difference of two solutions of (4.22) and (4.23) admits q-Gevrey null expansion of type A>0 at 0 in UI∩UI′ as a function with values in the Banach space ℍ𝒯,ρ of holomorphic bounded functions defined in 𝒯×D(0,ρ) endowed with the supremum norm. Flatness condition (1.5) allows us to establish the main result of the present work (Theorem 6.3): the existence of a formal power series
(1.6)X̂(ϵ)=∑k≥0Xkk!ϵk∈HT,ρ[[ϵ]],
formal solution of (1.1), such that, for every I∈ℐ, each of the actual solutions (1.4) of the problem (4.22) and (4.23) admits X̂ as its q-Gevrey expansion of a certain type in the corresponding domain of definition.The main result heavily rests on a Malgrange-Sibuya-type theorem involvingq-Gevrey bounds, which generalizes a result in [4] where no precise bounds on the asymptotic appear. In this step, we make use of the Whitney-type extension results in the framework of ultradifferentiable functions. The Whitney-type extension theory is widely studied in literature under the framework of ultradifferentiable functions subject to bounds of their derivatives (see e.g., [12, 13]) and also it is a useful tool taken into account on the study of continuity of ultraholomorphic operators (see [14–16]). It is also worth saying that, although q-Gevrey bounds have been achieved in the present work, the type involved might be increased when applying an extension result for ultradifferentiable functions from [13].The paper is organized as follows.In Sections2 and 3, we introduce Banach spaces of formal power series and solve auxiliary Cauchy problems involving these spaces. In Section 2, this is done when the variables rely in a product of a discrete q-spiral to the origin times a q-spiral to infinity, while in Section 3 it is done when working on a product of a punctured disc at 0 times a disc at 0.In Section4 we first recall definitions and some properties related to q-Laplace transform appearing in [7], firstly developed by Zhang. In this section we also find actual solutions of the main Cauchy problem (4.22) and (4.23) and settle a flatness condition on the difference of two of them so that, when regarding the difference of two solutions in the variable ϵ, we are able to give some information on its asymptotic behavior at 0. Finally, in Section 6 we conclude with the existence of a formal power series in ϵ with coefficients in an adequate Banach space of functions which solves in a formal sense the problem considered. The procedure heavily rests on a q-Gevrey version of the Malgrange-Sibuya theorem, developed in Section 5.


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

M,A1,C>0 are fixed positive real numbers throughout the whole paper.LetU,V be nonempty bounded open sets in ℂ⋆:=ℂ∖{0}, and let q∈ℂ⋆ such that |q|>1. We define
(2.1)Uq-N={ϵq-n∈C:ϵ∈U,n∈N},VqR+={τql∈C:τ∈V,l∈R,l≥0}.
We assume there exists M1>0 such that |τ+1|>M1 for all τ∈Vqℝ+ and also that the distance from the set V to the origin is positive.Definition 2.1.
Letϵ∈Uq-ℕ and β∈ℕ.Eβ,ϵ,Vqℝ+ denotes the vector space of functions v∈𝒪(Vqℝ+) such that
(2.2)‖v(τ)‖β,ϵ,VqR+:=sup⁡τ∈VqR+{|v(τ)|eMlog⁡2|τ/ϵ||τϵ|-Cβ}|q|A1β2
is finite.
Letδ>0. H(ϵ,δ,Vqℝ+) denotes the complex vector space of all formal series v(τ,z)=∑β≥0vβ(τ)zβ/β! belonging to 𝒪(Vqℝ+)[[z]] such that
(2.3)‖v(τ,z)‖(ϵ,δ,VqR+)∶=∑β≥0‖vβ(τ)‖β,ϵ,VqR+δββ!<∞.
It is straightforward to check that the pair (H(ϵ,δ,Vqℝ+),∥·∥(ϵ,δ,Vqℝ+)) is a Banach space.We consider the formal integration operator∂z-1 defined on 𝒪(Vqℝ+)[[z]] by(2.4)∂z-1(v(τ,z))∶=∑β≥1vβ-1(τ)zββ!∈O(VqR+)[[z]].Lemma 2.2.
Lets,k,m1,m2∈ℕ,δ>0,ϵ∈Uq-ℕ. One assumes that the following conditions hold:
(2.5)m1≤C(k+s),m2≥2(k+s)A1.
Then, there exists a constantC1=C1(s,k,m1,m2,V,U,C,A1) (not depending on ϵ nor δ) such that
(2.6)‖zs(τϵ)m1∂z-kv(τ,zq-m2)‖(ϵ,δ,VqR+)≤C1δk+s‖v(τ,z)‖(ϵ,δ,VqR+),
for every v∈H(ϵ,δ,Vqℝ+).Proof.
Letv(τ,z)=∑β≥0vβ(τ)(zβ/β!)∈𝒪(Vqℝ+)[[z]]. We have that
(2.7)‖zs(τϵ)m1∂z-kv(τ,zq-m2)‖(ϵ,δ,VqR+)=‖∑β≥k+s(τϵ)m1vβ-(k+s)(τ)β!(β-s)!1qm2(β-s)zββ!‖(ϵ,δ,VqR+)=∑β≥k+s‖(τϵ)m1vβ-(k+s)(τ)β!(β-s)!1qm2(β-s)‖β,ϵ,VqR+δββ!.
Taking into account the definition of the norm ∥·∥β,ϵ,Vqℝ+, we get
(2.8)‖(τϵ)m1vβ-(k+s)(τ)β!(β-s)!1qm2(β-s)‖β,ϵ,VqR+=β!(β-s)!|q|A1(β-(k+s))2|q|p(β)sup⁡τ∈VqR+{|vβ-(k+s)(τ)|eMlog⁡2|τ/ϵ||τϵ|-C(β-(k+s))|ϵτ|C(k+s)-m1},
with p(β)=A1β2-A1(β-(k+s))2-m2(β-s). From (2.5) we derive |ϵ/τ|C(k+s)-m1≤(CU/CV)C(k+s)-m1 for every ϵ∈Uq-ℕ and τ∈Vqℝ+, where 0<CV∶=min{|τ|:τ∈V} and 0<CU∶=max{|ϵ|:ϵ∈U}. Moreover,
(2.9)p(β)=(2(k+s)A1-m2)β-(k+s)2A1+m2s,
for every β∈ℕ. Regarding condition (2.5) we obtain the existence of C1>0 such that
(2.10)|ϵτ|C(k+s)-m1|q|p(β)≤C1,
for every τ∈Vqℝ+ and β∈ℕ. Inequality (2.6) follows from (2.7), (2.8), and (2.10):
(2.11)‖zs(τϵ)m1∂z-kv(τ,zq-m2)‖(ϵ,δ,VqR+)≤C1∑β≥k+s‖vβ-(k+s)(τ)‖β-(k+s),ϵ,VqR+β!(β-s)!δββ!≤C1δk+s∑β≥k+s‖vβ-(k+s)(τ)‖β-(k+s),ϵ,VqR+δβ-(k+s)(β-(k+s))!.Lemma 2.3.
LetF(ϵ,τ) be a holomorphic and bounded function defined on  Uq-ℕ×Vqℝ+. Then, there exists a constant  C2=C2(F,U,V)>0 such that
(2.12)‖F(ϵ,τ)vϵ(τ,z)‖(ϵ,δ,VqR+)≤C2‖vϵ(τ,z)‖(ϵ,δ,VqR+)
for every ϵ∈Uq-ℕ, every δ>0, and all vϵ∈H(ϵ,δ,Vqℝ+).Proof.
Direct calculations regarding the definition of the elements inH(ϵ,δ,Vqℝ+) allow us to conclude when taking C2∶=max{|F(ϵ,τ)|:ϵ∈Uq-ℕ,τ∈Vqℝ+}.LetS≥1 be an integer. For all 0≤k≤S-1, let m0,k,m1,k be positive integers and bk(ϵ,z)=∑s∈Ikbks(ϵ)zs a polynomial in z, where Ik is a finite subset of ℕ and bks(ϵ) are holomorphic bounded functions on D(0,r0). We assume Uq-ℕ¯⊆D(0,r0).We consider the following functional equation:(2.13)∂zSW(ϵ,τ,z)=∑k=0S-1bk(ϵ,z)(τ+1)ϵm0,kτm0,k(∂zkW)(ϵ,τ,zq-m1,k)
with initial conditions
(2.14)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where the functions (ϵ,τ)↦Wj(ϵ,τ) belong to 𝒪(Uq-ℕ×Vqℝ+) for every 0≤j≤S-1.We make the following assumption.Assumption A.
For every0≤k≤S-1 and s∈Ik, we have
(2.15)m0,k≤C(S-k+s),m1,k≥2(S-k+s)A1.Theorem 2.4.
Let Assumption A be fulfilled. One also makes the following assumption on the initial conditions in (2.14): there exist a constant  Δ>0 and  0<M̃<M such that, for every  0≤j≤S-1(2.16)|Wj(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for all τ∈Vqℝ+,ϵ∈Uq-ℕ. Then, there exists W(ϵ,τ,z)∈𝒪(Uq-ℕ×Vqℝ+)[[z]], solution of (2.13) and (2.14), such that, if W(ϵ,τ,z)=∑β≥0Wβ(ϵ,τ)(zβ/β!), then there exist C2>0 and 0<δ<1 such that
(2.17)|Wβ(ϵ,τ)|≤C2β!(|q|2A1Sδ)β|τϵ|CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for every ϵ∈Uq-ℕ and τ∈Vqℝ+.Proof.
Letϵ∈Uq-ℕ. We define the map 𝒜ϵ from 𝒪(Vqℝ+)[[z]] into itself by
(2.18)Aϵ(W̃(τ,z))∶=∑k=0S-1bk(ϵ,z)(τ+1)ϵm0,kτm0,k[(∂zk-SW̃)(τ,zq-m1,k)+∂zkwϵ(τ,zq-m1,k)],
where wϵ(τ,z)∶=∑j=0S-1Wj(ϵ,τ)(zj/j!). In the following lemma, we show that the restriction of 𝒜ϵ to a neighborhood of the origin in H(ϵ,δ,Vqℝ+) is a Lipschitz shrinking map for an appropriate choice of δ>0.Lemma 2.5.
There existR>0 and  δ>0 (not depending on  ϵ) such that (1)
∥𝒜ϵ(W̃(τ,z))∥(ϵ,δ,Vqℝ+)≤R for every W̃(τ,z)∈B(0,R); B(0,R) denotes the closed ball centered at 0 with radius R in H(ϵ,δ,Vqℝ+);(2)
(2.19)‖Aϵ(W̃1(τ,z))-Aϵ(W̃2(τ,z))‖(ϵ,δ,VqR+)≤12‖W̃1(τ,z)-W̃2(τ,z)‖(ϵ,δ,VqR+)
for every W̃1,W̃2∈B(0,R).Proof.
LetR>0 and 0<δ<1.
For the first part we considerW̃(τ,z)∈B(0,R)⊆H(ϵ,δ,Vqℝ+). Lemmas 2.2 and 2.3 can be applied so that
(2.20)‖Aϵ(W̃(τ,z))‖(ϵ,δ,VqR+)≤∑k=0S-1∑s∈IkMksM1[C1δS-k+s‖W̃(τ,z)‖(ϵ,δ,VqR+)+‖zs(τϵ)m0,k∂zkwϵ(τ,zq-m1,k)‖(ϵ,δ,VqR+)],
with Mks=supϵ∈Uq-ℕ|bks(ϵ)|<∞,s∈Ik,0≤k≤S-1. Taking into account the definition of H(ϵ,δ,Vqℝ+) and (2.16) we have
(2.21)‖zs(τϵ)m0,k∂zkwϵ(τ,zq-m1,k)‖(ϵ,δ,VqR+)=‖∑j=0S-1-k(τϵ)m0,kWj+k(ϵ,τ)zj+sj!qm1,kj‖(ϵ,δ,VqR+)=∑j=0S-1-ksup⁡τ∈VqR+{|Wj+k(ϵ,τ)|eMlog⁡2|τ/ϵ||τϵ|m0,k-C(j+s)}|q|A1(j+s)2δj+sj!|q|m1,kj≤Δ∑j=0S-1-k|q|A1(j+s)2δj+sj!|q|m1,kjmax⁡{e-(M-M̃)log⁡2(x)xm0,k-C(j+s):x>0,0≤j+k≤S-1,s∈Ik}≤ΔC2′,
for a positive constant C2′.
We conclude this first part from an appropriate choice ofR and δ>0.
For the second part we takeW̃1,W̃2∈B(0,R)⊆H(ϵ,δ,Vqℝ+). Similar arguments as before yield
(2.22)‖Aϵ(W̃1)-Aϵ(W̃2)‖(ϵ,δ,VqR+)≤∑k=0S-1∑s∈IkMksM1C1δS-k+s‖W̃1-W̃2‖(ϵ,δ,VqR+).
An adequate choice for δ>0 allows us to conclude the proof.We choose constantsR,δ as in the previous lemma.From Lemma2.5 and taking into account the shrinking map theorem on complete metric spaces, we guarantee the existence of W̃ϵ(τ,z)∈H(ϵ,δ,Vqℝ+) which is a fixed point for 𝒜ϵ in B(0,R); it is to say, ∥W̃ϵ(τ,z)∥(ϵ,δ,Vqℝ+)≤R and 𝒜ϵ(W̃ϵ(τ,z))=W̃ϵ(τ,z).Let us define(2.23)Wϵ(τ,z)∶=∂z-SW̃ϵ(τ,z)+wϵ(τ,z).
If we write W̃ϵ(τ,z)=∑β≥0W̃β,ϵ(τ)(zβ/β!) and Wϵ(τ,z)=∑β≥0Wβ,ϵ(τ)(zβ/β!), then we have that Wβ+S,ϵ≡W̃β,ϵ for β≥0 and Wj,ϵ(τ)=Wj(ϵ,τ), 0≤j≤S-1.From∥W̃ϵ(τ,z)∥(ϵ,δ,Vqℝ+)≤R we arrive at ∥W̃β,ϵ∥β,ϵ,Vqℝ+≤Rβ!(1/δ)β for every β≥0. This implies
(2.24)|W̃β,ϵ(τ)|≤Rβ!(1δ)β|τϵ|CβeMlog⁡2|τ/ϵ||q|-A1β2,
for every β≥0 and τ∈Vqℝ+.This is valid for everyϵ∈Uq-ℕ. We define W(ϵ,τ,z)∶=Wϵ(τ,z) and Wβ(ϵ,τ)∶=Wβ,ϵ(τ) for every (ϵ,τ)∈Uq-ℕ×Vqℝ+,z∈ℂ and β≥S. From (2.23), it is straightforward to prove that W(ϵ,τ,z)=∑β≥0Wβ(ϵ,τ)(zβ/β!) is a solution of (2.13) and (2.14).Moreover, holomorphy ofWβ in Uq-ℕ×Vqℝ+ for every β≥0 can be deduced from the recursion formula verified by the coefficients:
(2.25)Wh+S(ϵ,τ)h!=∑k=0S-1∑h1+h2=h,h1∈Ikbkh1(ϵ)τm0,k(τ+1)ϵm0,kWh2+k(ϵ,τ)h2!qm1,kh2,h≥0.
This implies that Wβ(ϵ,τ) is holomorphic in Uq-ℕ×Vqℝ+ for every β∈ℕ.It only remiains to prove (2.17). Upper and lower bounds for the modulus of the elements in Uq-ℕ and Vqℝ+, respectively, and usual calculations lead us to assure the existence of a positive constant R1>0 such that
(2.26)|Wβ(ϵ,τ)|=|W̃β-S,ϵ(τ)|≤R1β!(|q|2A1Sδ)β|τϵ|CβeMlog⁡2|τ/ϵ||q|-A1β2,
for every β≥S, and for every ϵ∈Uq-ℕ and τ∈Vqℝ+. This concludes the proof for β≥S.Hypothesis (2.16) leads us to obtain (2.26) for 0≤k≤S-1.Remark 2.6.
Ifs>0 for every s∈Ik,0≤k≤S-1, then, for every R>0, there exists small enough δ>0 in such a way that Lemma 2.5 holds.


## 3. Second Cauchy Problem in a Weighted Banach Space of Taylor Series

This section is devoted to the study of the same equation as in the previous section when the initial conditions are of a different nature. Proofs will only be sketched not to repeat calculations.Let1<ρ0, and let U⊆ℂ⋆, a bounded and open set with positive distance to the origin. Ḋρ0 stands for D(0,ρ0)∖{0} in this section. M,A1,C remain the same positive constants as in the previous section.Definition 3.1.
Letr0>0,ϵ∈D(0,r0)∖{0}, and β∈ℕ. Eβ,ϵ,Ḋρ02 denotes the vector space of functions v∈𝒪(Ḋρ0) such that
(3.1)|v(τ)|β,ϵ,Ḋρ0∶=sup⁡τ∈Ḋρ0{|v(τ)||ϵ|CβeMlog⁡2|τ/ϵ|}|q|A1β2,
is finite. Let δ>0. H2(ϵ,δ,Ḋρ0) stands for the vector space of all formal series v(τ,z)=∑β≥0vβ(τ)zβ/β! belonging to 𝒪(Ḋρ0)[[z]] such that
(3.2)|v(τ,z)|(ϵ,δ,Ḋρ0)∶=∑β≥0|vβ(τ)|β,ϵ,Ḋρ0δββ!<∞.
It is straightforward to check that the pair (H2(ϵ,δ,Ḋρ0),|·|(ϵ,δ,Ḋρ0)) is a Banach space.Lemma 3.2.
Lets,k,m1,m2∈ℕ,δ>0 and  ϵ∈D(0,r0)∖{0}. One assumes that the following conditions hold:
(3.3)m1≤C(k+s),m2≥2(k+s)A1.
Then, there exists a constantC1=C1(s,k,m1,m2,Ḋρ0,U) (not depending on ϵ nor δ) such that
(3.4)|zs(τϵ)m1∂z-kv(τ,zq-m2)|(ϵ,δ,Ḋρ0)≤C1δk+s|v(τ,z)|(ϵ,δ,Ḋρ0),
for every v∈H2(ϵ,δ,Ḋρ0).Proof.
Letv(τ,z)∈𝒪(Ḋρ0)[[z]]. The proof follows similar steps to those in Lemma 2.2. We have
(3.5)|zs(τϵ)m1∂z-kv(τ,zq-m2)|(ϵ,δ,Ḋρ0)=∑β≥k+s|(τϵ)m1vβ-(k+s)(τ)β!(β-s)!1qm2(β-s)|β,ϵ,Ḋρ0δββ!.
From the definition of the norm |·|β,ϵ,Ḋρ0, we get
(3.6)|(τϵ)m1vβ-(k+s)(τ)β!(β-s)!1qm2(β-s)|β,ϵ,Ḋρ0≤β!(β-s)!|q|A1(β-(k+s))2|q|p(β)×sup⁡τ∈Ḋρ0{|vβ-(k+s)(τ)|eMlog⁡2|τ/ϵ||ϵ|C(β-(k+s))}ρ0m1|ϵ|C(k+s)-m1,
with p(β)=A1β2-A1(β-(k+s))2-m2(β-s). Identical arguments to those in Lemma 2.2 allow us to conclude.Lemma 3.3.
LetF(ϵ,τ) be a holomorphic and bounded function defined on  (D(0,r0)∖{0})×Ḋρ0. Then, there exists a constant  C2=C2(F)>0 such that
(3.7)|F(ϵ,τ)vϵ(τ,z)|(ϵ,δ,Ḋρ0)≤C2|vϵ(τ,z)|(ϵ,δ,Ḋρ0)
for every ϵ∈D(0,r0)∖{0}, every δ>0, and every vϵ∈H2(ϵ,δ,Ḋρ0).LetS,r0,m0,k,m1,k and bk, as in Section 2 and ρ0>0. One  considers the Cauchy problem
(3.8)∂zSW(ϵ,τ,z)=∑k=0S-1bk(ϵ,z)(τ+1)ϵm0,kτm0,k(∂zkW)(ϵ,τ,zq-m1,k)
with initial conditions
(3.9)(∂zjW)(ϵ,τ,0)=Wj(ϵ,τ),0≤j≤S-1,
where the functions (ϵ,τ)↦Wj(ϵ,τ) belong to 𝒪((D(0,r0)∖{0})×Ḋρ0) for every 0≤j≤S-1.Theorem 3.4.
Let Assumption A be fulfilled. One makes the following assumption on the initial conditions (3.9): there exist constants  Δ>0 and  0<M̃<M such that
(3.10)|Wj(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for every τ∈Ḋρ0,ϵ∈D(0,r0)∖{0} and 0≤j≤S-1. Then, there exists W(ϵ,τ,z)∈𝒪((D(0,r0)∖{0})×Ḋρ0)[[z]], solution of (3.8) and (3.9) such that, if W(ϵ,τ,z)=∑β≥0Wβ(ϵ,τ)(zβ/β!), then there exist C3>0 and 0<δ<1 such that
(3.11)|Wβ(ϵ,τ)|≤C3β!(|q|2A1Sδ)β|ϵ|-CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for every ϵ∈D(0,r0)∖{0} and τ∈Ḋρ0.Proof.
The proof of Theorem2.4 can be adapted here so details will be omitted.
Letϵ∈D(0,r0)∖{0} and 0<δ<1. We consider the map 𝒜ϵ from 𝒪(Ḋρ0)[[z]] into itself defined as in (2.18) and construct wϵ(τ,z) as above. From (3.10) we derive
(3.12)|zs(τϵ)m0,k∂zkwϵ(τ,zq-m1,k)|(ϵ,δ,Ḋρ0)=∑j=0S-1-ksup⁡τ∈Ḋρ0|Wj+k(ϵ,τ)||ϵ|C(j+s)eMlog⁡2|ϵ/τ||τϵ|m0,k|q|A1(j+s)2δj+sj!|q|m1,kj≤ΔC3′,
for a positive constant C3′ not depending on ϵ nor δ.
Lemmas3.2, 3.3, and (3.12) allow us to affirm that one can find R>0 and δ>0 such that the restriction of 𝒜ϵ to the disc D(0,R) in H2(ϵ,δ,Ḋρ0) is a Lipschitz shrinking map. Moreover, there exists W̃ϵ(τ,z)∈H2(ϵ,δ,Ḋρ0) which is a fixed point for 𝒜ϵ in B(0,R).
If we putW̃ϵ(τ,z)=∑β≥0W̃β,ϵ(τ)(zβ/β!), then one gets |W̃β,ϵ|β,ϵ,Ḋρ0≤Rβ!(1/δ)β for β≥0. This implies
(3.13)|W̃β,ϵ(τ)|≤Rβ!(1δ)β|ϵ|-CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,τ∈Ḋρ0.
The formal power series
(3.14)W(ϵ,τ,z)∶=∑β≥SW̃β-S,ϵ(τ)zββ!+wϵ(τ,z)∶=∑β≥0Wβ(ϵ,τ)zββ!
turns out to be a solution of (3.8) and (3.9) verifying that Wβ(ϵ,τ) is a holomorphic function in (D(0,r0)∖{0})×Ḋρ0 and the estimates (3.11) hold for β≥0.


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

### 4.1. Aq-Analog of the Laplace Transform and q-Asymptotic Expansion

In this subsection, we recall the definition and several results related to the Jacobi Theta function and also aq-analog of the Laplace transform which was firstly developed by Zhang in [7].Letq∈ℂ such that |q|>1.The Jacobi Theta function is defined inℂ⋆ by
(4.1)Θ(x)=∑n∈Zq-n(n-1)/2xn,x∈C⋆.
From the fact that the Jacobi Theta function satisfies the functional equation xqΘ(x)=Θ(qx), for x≠0, we have
(4.2)Θ(qmx)=qm(m+1)/2xmΘ(x),x∈C,x≠0
for every m∈ℤ. The following lower bounds for the Jacobi Theta function will be useful in the sequel.Lemma 4.1.
Letδ>0. There exists  C>0 (not depending on  δ) such that
(4.3)|Θ(x)|≥Cδe(log⁡2|x|)/2log⁡|q||x|1/2,
for every x∈ℂ⋆ such that |1+xqk|>δ for all k∈ℤ.Proof.
Letδ>0. From Lemma  5.1.6 in [17] we get the existence of a positive constant C1 such that |Θ(x)|≥C1δΘ|q|(|x|) for every x∈ℂ⋆ such that |1+xqk|>δ for all k∈ℤ. Now,
(4.4)Θ|q|(|x|)=∑n∈Z|q|-n(n-1)/2|x|n≥max⁡n∈Z|q|-n(n-1)/2|x|n.
Let us fix |x|. The function
(4.5)f(t)=exp⁡(-12t(t-1)log⁡|q|+tlog⁡|x|)
takes its maximum value at t0=log|x|/log|q|+1/2 with f(t0)=C2exp(log2|x|/2log|q|)|x|1/2, for certain C2>0. Taking into account that
(4.6)max⁡n∈Z|q|-n(n-1)/2|x|n≥f(⌊t0⌋)=f(t0)|q|-(⌊t0⌋-t0)2/2≥f(t0)|q|-1/2,
one can conclude the result. Here ⌊·⌋ stands for the entire part.Corollary 4.2.
Letδ>0. For any  ξ∈(0,1) there exists  Cξ=Cξ(δ)>0 such that
(4.7)|Θ(x)|≥Cξeξlog⁡2|x|/2log⁡|q|,
for every x∈ℂ⋆ such that |1+xqk|>δ, for all k∈ℤ.From now on,(ℍ,∥·∥ℍ) stands for a complex Banach space.For anyλ∈ℂ and δ>0(4.8)Rλ,q,δ∶={z∈C⋆:|1+λzqk|>δ,∀k∈R}.The following definition corresponds to aq-analog of the Laplace transform and can be found in [7] when working with sectors in the complex plane.Proposition 4.3.
Letδ>0 and  ρ0>0. One  fixes an open and bounded set  V in  ℂ⋆ such that  D(0,ρ0)∩V≠∅. Let  λ∈D(0,ρ0)∩V, and,  f be a holomorphic function defined in  Ḋρ0 with values in  ℍ such that let can be extended to a function  F defined in  Ḋρ0∪Vqℝ+ and
(4.9)‖F(x)‖H≤C1eM¯log⁡2|x|,x∈Ḋρ0∪VqR+,
for positive constants C1>0 and 0<M¯<1/2log|q|.
Letπq=log(q)∏n≥0(1-q-n-1)-1, and put
(4.10)Lq;1λF(z)=1πq∫0∞λF(ξ)Θ(ξ/z)dξξ,
where the path [0,∞λ] is given by t∈(-∞,∞)↦qtλ. Then, ℒq;1λF defines a holomorphic function in ℛλ,q,δ and it is known as the q-Laplace transform of f following direction [λ].Proof.
LetK⊆ℛλ,q,δ be a compact set and z∈K. From the parametrization of the path [0,∞λ] we have
(4.11)∫0∞λF(ξ)Θ(ξ/z)dξξ=log⁡(q)∫-∞∞F(qtλ)Θ(qtλ/z)dt.
Let 0<ξ1<1 such that 0<M¯<ξ1/2log|q|, and let t∈ℝ. We have that w=qtλ/z satisfies |1+qkw|>δ for every k∈ℤ. Corollary 4.2 and (4.9) yield
(4.12)∫-∞∞‖F(qtλ)Θ(qtλ/z)‖Hdt≤∫-∞∞C1eM¯log⁡2|qtλ|Cξ1e(ξ1/2log⁡|q|)log⁡2|qtλ/z|dt≤L1∫-∞∞|qtλ|ξ1log⁡|z|/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt,
for a positive constant L1. There exist 0<A<B such that A≤|z|≤B for every z∈K, so that the last term in the chain of inequalities above is upper bounded by
(4.13)L1∫-∞-log⁡|λ|/log⁡|q||qtλ|ξ1log⁡A/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt+L1∫-log⁡|λ|/log⁡|q|∞|qtλ|ξ1log⁡B/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt.
The result follows from this last expression.Remark 4.4.
If we letM¯=1/2log|q|, then ℒq;1λF will only remain holomorphic in ℛλ,q,δ∩D(0,r1) for certain r1>0.
In the next proposition, we recall a commutation formula for theq-Laplace transform and the multiplication by a polynomial.Proposition 4.5.
LetV be an open and bounded set in  ℂ⋆ and  D(0,ρ0) such that  V∩D(0,ρ0)≠∅. Let  ϕ be a holomorphic function on  Vqℝ+∪Ḋρ0 with values in the Banach space  (ℍ,∥·∥ℍ) which satisfies the following estimates: there exist  C1>0 and  0<M¯<1/2log|q| such that
(4.14)‖ϕ(x)‖H<C1eM¯log⁡2|x|,x∈Ḋρ0∪VqR+.
Then, the function mϕ(τ)=τϕ(τ) is holomorphic on Vqℝ+∪Ḋρ0 and satisfies estimates in the shape above. Let λ∈V∩D(0,ρ0) and δ>0. One  has the following equality:
(4.15)Lq;1λ(mϕ)(t)=tLq;1λϕ(qt)
for every t∈ℛλ,q,δ.Proof.
It is direct to prove thatmϕ is a holomorphic function in Vqℝ+∪Ḋρ0 and also that mϕ verifies bounds as in (4.14). From (4.2) we have Θ(x)=xΘ(x/q), x∈ℂ⋆, so
(4.16)Lq;1λ(mϕ)(t)=1πq∫0∞λ(mϕ)(ξ)Θ(ξ/t)dξξ=1πq∫0∞λϕ(ξ)Θ(ξ/t)dξ=1πq∫0∞λϕ(ξ)(ξ/t)Θ(ξ/qt)dξ=tLq;1λ(ϕ)(qt),
for every t∈ℛλ,q,δ.


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

The following definition of a good covering firstly appeared in [17], p. 36.Definition 4.6.
LetI=(I1,I2) be a pair of open intervals in ℝ each one of length smaller than 1/4, and let UI be the corresponding open bounded set in ℂ⋆ defined by
(4.17)UI={e2πuiqv∈C⋆:u∈I1,v∈I2}.
Let ℐ be a finite family of tuple I as above verifying(1)
⋃I∈ℐ(UIq-ℕ)=ν∖{0}, where ν is a neighborhood of 0 in ℂ,(2)
the open setsUIq-ℕ, I∈ℐ are four-by-four disjoint. Then, we say that (UIq-ℕ)I∈ℐ is a good covering.Definition 4.7.
Let(UIq-ℕ)I∈ℐ be a good covering. Let δ>0. We consider a family of open bounded sets {(VI)I∈ℐ,𝒯} in ℂ⋆ such that(1)
there exists1<ρ0 with VI∩D(0,ρ0)≠∅, for all I∈ℐ,(2)
for everyI∈ℐ and τ∈VIqℝ,|τ+1|>δ,(3)
for everyI∈ℐ,t∈𝒯,ϵu∈UI, and λv∈VI∩D(0,ρ0), we have
(4.18)|1+λvϵutqr|>δ,
for every r∈ℝ,(4)
|t|≤1 for every t∈𝒯. We say that the family {(VI)I∈ℐ,𝒯} is associated to the good covering (UIq-ℕ)I∈ℐ.
LetS≥1 be an integer. For every 0≤k≤S-1, let m0,k,m1,k be positive integers and bk(ϵ,z)=∑s∈Ikbks(ϵ)zs a polynomial in z, where Ik is a subset of ℕ and bks(ϵ) are bounded holomorphic functions on some disc D(0,r0) in ℂ,0<r0≤1. Let (UIq-ℕ)I∈ℐ be a good covering such that UIq-ℕ⊆D(0,r0) for every I∈ℐ.Assumption B.
We have(4.19)M<12log⁡|q|.Definition 4.8.
Letρ0>1such thatV∩D(0,ρ0)≠∅. LetΔ,M̃>0such thatM̃<M,and that(ϵ,τ)↦W(ϵ,τ)be a bounded holomorphic function on(D(0,r0)∖{0})×Ḋρ0verifying(4.20)|W(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for every (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0. Assume moreover that W(ϵ,τ) can be extended to an analytic function (ϵ,τ)↦WUV(ϵ,τ) on Uq-ℕ×(Vqℝ+∪Ḋρ0) and
(4.21)|WUV(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for every (ϵ,τ)∈Uq-ℕ×(Vqℝ+∪Ḋρ0). One says that the set {W,WUV,ρ0} is admissible.
Letℐ be a finite family of indices. For every I∈ℐ, we consider the following singularly perturbed Cauchy problem:
(4.22)ϵt∂zSXI(ϵ,qt,z)+∂zSXI(ϵ,t,z)=∑k=0S-1bk(ϵ,z)(tσq)m0,k(∂zkXI)(ϵ,t,zq-m1,k)
with bk as in (2.13), and with initial conditions
(4.23)(∂zjXI)(ϵ,t,0)=ϕI,j(ϵ,t),0≤j≤S-1,
where the functions ϕI,j(ϵ,t) are constructed as follows. Let {(VI)I∈ℐ,𝒯} be a family of open sets associated to the good covering (UIq-ℕ)I∈ℐ. For every 0≤j≤S-1 and I∈ℐ, let {Wj,WUI,VI,j,ρ0} be an admissible set. Let λI be a complex number in VI∩D(0,ρ0). We can assume that r0<1<|λI|. If not, we diminish r0 as desired. We put
(4.24)ϕI,j(ϵ,t)∶=Lq;1λI(τ⟼WUI,VI,j(ϵ,τ))(ϵ,ϵt).Lemma 4.9.
The function(ϵ,t)↦ϕI,j(ϵ,t), constructed as above, turns out to be holomorphic and bounded on  UIq-ℕ×𝒯 for every  I∈ℐ and all  0≤j≤S-1.Proof.
LetI∈ℐ and 0≤j≤S-1. From (4.21), one has
(4.25)|WUI,VI,j(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|=ΔeM̃log⁡2|ϵ||τ|-2M̃log⁡|ϵ|eM̃log⁡2|τ|,
for every (ϵ,τ)∈UIq-ℕ×(VIqℝ+∪Ḋρ0). Let ϵ∈UIq-ℕ and M̃<M̃2<1/2log|q|. Then, (4.25) can be upper bounded by Δ̃exp(M̃2log2|τ|), for some Δ̃=Δ̃(ϵ)>0. Estimates in (4.9) hold so that Proposition 4.3 can be applied here. The third item in Definition 4.7 derives the holomorphy of ϕI,j on UIq-ℕ×𝒯.
We now prove the boundness ofϕI,j in its domain of definition. One has
(4.26)|ϕI,j(ϵ,t)|=|Lq;1λIWUI,VI,j(ϵ,ϵt)|≤|Lq;1,+λIWUI,VI,j(ϵ,ϵt)|+|Lq;1,-λIWUI,VI,j(ϵ,ϵt)|,
for every (ϵ,t)∈UIq-ℕ×𝒯, where
(4.27)Lq;1,+λIWUI,VI,j(ϵ,ϵt)=log⁡(q)πq∫0∞WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)ds,Lq;1λIWUI,VI,j,-(ϵ,ϵt)=log⁡(q)πq∫-∞0WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)ds.
We only give bounds for the first integral. The estimates for the second one can be deduced following a similar procedure.
Let0<ξ<1 such that M̃<ξ/2log|q|. From Corollary 4.2 and (4.21) we deduce
(4.28)|Lq;1,+λIWUI,VI,j(ϵ,ϵt)|≤|log⁡(q)||πq|∫0∞|WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)|ds≤|log⁡(q)|Δ|πq|Cξ∫0∞eM̃log⁡2|qsλI/ϵ|eξlog⁡2|qsλI/ϵt|/2log⁡|q|ds≤|log⁡(q)|Δ|πq|Cξe(M̃-ξ/2log⁡|q|)log⁡2|λI/ϵ|e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|×∫0∞e2(M̃-(ξ/2log⁡|q|))log⁡2|q|s2e(M̃-ξ/2log⁡|q|)log⁡|q|log⁡|λI/ϵ|seξlog⁡|t|sds≤Cj,
for some Cj>0 which does not depend on ϵ nor t.The following assumption is related to technical reasons appearing in the proof of Lemma4.9 and Theorem 4.11.Assumption C.
There exista1,a2>0,0<ξ,ξ¯<1 such that(C.1)
M<ξ/2log|q|,(C.2)
ξ/2-Mlog|q|-Ca1/2a2>0,(C.3)
Ca2/2a1+C2/4ξ¯log|q|(ξ/2log|q|-M)<A1.
The next remark clarifies the availability of these constants for a posed problem.Remark 4.10.
Assumptions A, B, and C strongly depend on the choice ofq whose modulus must rest near 1. For example, these assumptions on the constants are verified when taking log|q|=1/16,M=1,A1=5,C=1,ξ=1/2,ξ¯=1/2,a1=1,a2=4. Then, the next theorem provides a solution for the equation
(4.29)ϵt∂z2XI(ϵ,qt,z)+∂z2XI(ϵ,t,z)=(b00(ϵ)+b01(ϵ)z)t2XI(ϵ,q2t,zq-30)+b10(ϵ)t∂zXI(ϵ,qt,zq-10),
with b00,b01,b10 being holomorphic functions near the origin.Theorem 4.11.
Let Assumption A be fulfilled by the integersm0,k,m1,k, for  0≤k≤S-1 and also Assumptions B and C for  M,A1,C. One considers the problem (4.22) and (4.23) where the initial conditions are constructed as above. Then, for every  I∈ℐ, the problem (4.22) and (4.23) has a solution  XI(ϵ,t,z) which is holomorphic and bounded in  UIq-ℕ×𝒯×ℂ.
Moreover, for everyρ>0, if  I,I′∈ℐ are such that  UIq-ℕ∩UI′q-ℕ≠∅, then there exists a positive constant  C1=C1(ρ)>0 such that
(4.30)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤C1e-(1/A)log⁡2|ϵ|,(ϵ,t,z)∈(UIq-N∩UI′q-N)×T×D(0,ρ),
with 1/A=(1-ξ¯)(ξ/2log|q|-M) with ξ,ξ¯ chosen as in Assumption C.Proof.
Letδ>0 and I∈ℐ. We consider the Cauchy problem (3.8) with initial conditions (∂zjW)(ϵ,τ,0)=Wj(ϵ,τ) for 0≤j≤S-1. From Theorem 3.4 we obtain the existence of a unique formal solution W(ϵ,τ,z)=∑β≥0Wβ(ϵ,τ)(zβ/β)∈𝒪((D(0,r0)∖{0})×Ḋρ0)[[z]] and positive constants C3>0 and 0<δ1<1 such that
(4.31)|Wβ(ϵ,τ)|≤C3β!(|q|2A1Sδ1)β|ϵ|-CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0.
Moreover, from Theorem2.4 we get that the coefficients Wβ(ϵ,τ) can be extended to holomorphic functions defined in UIq-ℕ×VIqℝ+ and also the existence of positive constants C2 and 0<δ2<1 such that
(4.32)|Wβ(ϵ,τ)|≤C2β!(|q|2A1Sδ2)β|τϵ|CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for (ϵ,τ)∈UIq-ℕ×VIqℝ+.
We chooseλI∈VI∩D(0,ρ0). In the following estimates we will make use of the fact that |ϵ|≤|λI| for every ϵ∈D(0,r0∖{0}). Proposition 4.3 allows us to calculate the q-Laplace transform of Wβ with respect to τ for every β≥0,ℒq;1λI(Wβ)(ϵ,τ). It defines a holomorphic function in UIq-ℕ×ℛλI,q,δ. From the fact that {(VI)I∈ℐ,𝒯} is chosen to be a family associated to the good covering (UIq-ℕ)I∈ℐ we derive that the function
(4.33)(ϵ,t)⟼Lq;1λI(Wβ)(ϵ,ϵt)
is a holomorphic and bounded function defined in UIq-ℕ×𝒯. We can define, at least formally,
(4.34)XI(ϵ,t,z)∶=∑β≥0Lq;1λI(Wβ)(ϵ,ϵt)zββ!,
in 𝒪(UIq-ℕ×𝒯)[[z]]. If XI(ϵ,t,z) were a holomorphic function in UIq-ℕ×𝒯×ℂ, then Proposition 4.5 would allow us to affirm that (4.34) is an actual solution of (4.22) and (4.23). In order to end the first part of the proof it remian to demonstrate that (4.34) defines in fact a bounded holomorphic function in UIq-ℕ×𝒯×ℂ. Let (ϵ,t)∈UIq-ℕ×𝒯 and β≥0. We have
(4.35)|Lq;1λIWβ(ϵ,ϵt)|≤|Lq;1,+λIWβ(ϵ,ϵt)|+|Lq;1,-λIWβ(ϵ,ϵt)|,
where
(4.36)Lq;1,+λIWβ(ϵ,ϵt)=log⁡(q)πq∫0∞Wβ(ϵ,qsλI)Θ(qsλI/ϵt)ds,Lq;1,-λIWβ(ϵ,ϵt)=log⁡(q)πq∫-∞0Wβ(ϵ,qsλI)Θ(qsλI/ϵt)ds.
We now establish bounds for both integrals:
(4.37)|Lq;1,+λIWβ(ϵ,ϵt)|≤|log⁡q||πq|∫0∞|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds.
Let0<ξ<1 as in Assumption C. From (4.32) and (4.7), the previous integral is bounded by
(4.38)|log⁡q||πq|∫0∞C2β!(|q|2A1S/δ2)β|qsλI/ϵ|CβeMlog⁡2|qsλI/ϵ||q|-A1β2Cξexp⁡(ξlog⁡2|qsλI/ϵt|/2log⁡|q|)ds≤|log⁡q||πq|C2Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2∫0∞|q|CsβeMlog⁡2|qsλI/ϵ|exp⁡(ξlog⁡2|qsλI/ϵt|/2log⁡|q|)ds.
Leta1,a2 be as in Assumptions (C.2) and (C.3).
From(a1s-a2β)2≥0 and (2.5) in Definition 4.7, the previous inequality is upper bounded by
(4.39)A∫0∞|q|-Bs2e(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|e((2Mlog⁡|q|-ξ)log⁡|λI/ϵ|+ξlog⁡|t|)sds,
where 0<B=ξ/2-Mlog|q|-Ca1/2a2 and
(4.40)A=|log⁡q||πq|C2Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2+Ca2β2/2a1e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|.
The previous integral is uniformly bounded forϵ∈D(0,r0)∖{0} and t∈𝒯 from the hypotheses made on these sets. The expression in (4.39) can be bounded by
(4.41)|log⁡q||πq|C2′Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβe(M-ξ/2log⁡|q|)log⁡2|λI/ϵ||q|-A1β2+Ca2β2/2a1×e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|,
for an appropriate constant C2′>0.
The functions↦sγβe-αlog2(s) takes its maximum at s=eγβ/(2α) so each element in the image set is bounded by e(γβ)2/(4α). Taking this to the expression above we get
(4.42)|Lq;1,+λIWβ(ϵ,ϵt)|≤|log⁡q||πq|C2′′Cξβ!(|q|2A1Sδ2)β|q|-A1β2+Ca2β2/2a1+C2β2/4log⁡|q|(ξ/(2log⁡|q|)-M),
for certain C2′′>0.
Assumption (C.3) applied to the last term in the previous expression allows us to deduce that the sum(4.43)∑β≥0|Lq;1,+λIWβ(ϵ,ϵt)||z|ββ!
converges in the variable z uniformly in the compact sets of ℂ.
We now studyℒq;1,-λIWβ(ϵ,ϵt). We have
(4.44)|Lq;1,-λIWβ(ϵ,ϵt)|≤|log⁡q||πq|∫-∞0|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds.
From (3.11) and (4.7) the previous integral is bounded by
(4.45)|log⁡q||πq|∫-∞0C3β!(|q|2A1S/δ1)β|ϵ|-CβeMlog⁡2|qsλI/ϵ||q|-A1β2Cξeξlog⁡2|qsλI/ϵt|/2log⁡|q|ds.
Similar calculations to those in the first part of the proof resting on Assumption C can be made so that the series(4.46)∑β≥0Lq;1,-λIWβ(ϵ,ϵt)zββ!
is uniformly convergent with respect to the variable z in the compact sets of ℂ, for (ϵ,t)∈UIq-ℕ×𝒯. We will not go into detail not to repeat calculations.
The estimates (4.43) and (4.46) imply the convergence of the series in (4.34) for every z∈ℂ. The boundness of the q-Laplace transform with respect to ϵ is guaranteed so the first part of the result is achieved.
LetI,I′∈ℐ such that UIq-ℕ∩UI′q-ℕ≠∅ and ρ>0. For every (ϵ,t,z)∈(UIq-ℕ∩UI′q-ℕ)×𝒯×D(0,ρ) we have
(4.47)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤∑β≥0|Lq;1λIWβ(ϵ,ϵt)-Lq;1λI′Wβ(ϵ,ϵt)|ρββ!.
We can write(4.48)Lq;1λIWβ(ϵ,ϵt)-Lq;1λI′Wβ(ϵ,ϵt)=1πq(∫γ1Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ-∫γ2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ+∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ),
where the path γ1 is given by s∈(0,∞)↦qsλI, γ2 by s∈(0,∞)↦qsλI′, γ3 by s∈(-∞,0)↦qsλI, and γ4 by s∈(-∞,0)↦qsλI′.
Without loss of generality, we can assume that|λI|=|λI′|.
For the first integral we deduce(4.49)|∫1Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ|≤|log⁡(q)|∫0∞|Wβ(ϵ,qsλI)||Θ(qsλI/ϵt)|ds.
Similar estimates as in the first part of the proof lead us to bound the right part of the previous inequality by
(4.50)C2′′′Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2+(Ca2/2a1)β2e(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|,
for certain C2′′′>0. For any ξ¯∈(0,1) we have
(4.51)|λIϵ|Cβeξ¯(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|≤eC2β2/4ξ¯(ξ/2log⁡|q|-M),β≥0.
This yields
(4.52)∫γ1|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds≤C2′′′Cξβ!(|q|2A1Sδ2)β|q|(-A1+(Ca2/2a1)+C2/4ξ¯log⁡|q|((ξ/2log⁡|q|)-M))β2e(1-ξ¯)(M-(ss))log⁡2|λI/ϵ|.
We choose ξ¯ as in Assumption C.
The integral corresponding to the pathγ2 can be bounded following identical steps.
We now give estimates concerningγ3-γ4. It is worth saying that the function in the integrand is well defined for (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0 and does not depend on the index I∈ℐ. This fact and the Cauchy theorem allow us to write for any n∈ℕ(4.53)∫ΓnWβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=0,
where Γn=γn,1+γ5-γn,2-γn,3 is the closed path defined in the following way: s∈[-n,0]↦γn,1(s)=λIqs,γ5 is the arc of circumference from λI to λI′, s∈[-n,0]↦γn,2(s)=λI′qs, and γn,3 is the arc of circumference from λIq-n to λI′q-n. Taking n→∞ we derive
(4.54)0=lim⁡n→∞∫ΓnWβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=lim⁡n→∞∫γn,1+γ5-γn,2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ-lim⁡n→∞∫γn,3Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ.
Usual estimates lead us to prove that
(4.55)lim⁡n→∞∫γn,3Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=0.
Moreover,
(4.56)lim⁡n→∞∫γn,1+γ5-γn,2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫γ3+γ5-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ.
From (4.54), (4.55), and (4.56) we obtain
(4.57)∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫-γ5Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫θI′θIWβ(ϵ,|λI|eiθ)Θ(|λI|eiθ/ϵt)dθ,
where θI=arg(λI),θI′=arg(λI′). Taking into account Definition 4.7 and (4.31) we derive that the modulus of the last term in the previous equality is bounded by
(4.58)length(γ5)C3Cξβ!(|q|2A1Sδ1)β|ϵ|-CβeMlog⁡2|λI/ϵ|e(ξ/2log⁡|q|)log⁡2|λI/ϵt||q|-A1β2≤C3′β!(|q|2A1Sδ1)β|ϵ|-Cβe(M-ξ/2log⁡|q|)log⁡2|λI/ϵ||q|-A1β2≤C3′β!(|q|2A1Sδ1)β|ϵ|-Cβeξ¯(M-ξ/2log⁡|q|)log⁡2|ϵ||q|-A1β2e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|
for adequate positive constants C3,C3′. From the standard estimates we achieve
(4.59)|∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ|≤C3′β!(|q|2A1Sδ1)β|q|-A1β2eC2/4ξ¯(ξ/2log⁡|q|-M)β2e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|.
From (4.47), (4.48), (4.52), (4.59) and Assumption (C.3) we conclude the existence of a positive constant C1′>0 such that
(4.60)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤C1′∑β≥0β!(|q|2A1Sδ0)β|q|(-A1+(Ca2/2a1)+C2/4ξ¯log⁡|q|(ξ/2log⁡|q|-M))β2×e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|ρββ!≤C1e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|,
for every (ϵ,t,z)∈(UIq-ℕ∩UI′q-ℕ)×𝒯×D(0,ρ), with δ0=min{δ1,δ2}.


## 4.1. Aq-Analog of the Laplace Transform and q-Asymptotic Expansion

In this subsection, we recall the definition and several results related to the Jacobi Theta function and also aq-analog of the Laplace transform which was firstly developed by Zhang in [7].Letq∈ℂ such that |q|>1.The Jacobi Theta function is defined inℂ⋆ by
(4.1)Θ(x)=∑n∈Zq-n(n-1)/2xn,x∈C⋆.
From the fact that the Jacobi Theta function satisfies the functional equation xqΘ(x)=Θ(qx), for x≠0, we have
(4.2)Θ(qmx)=qm(m+1)/2xmΘ(x),x∈C,x≠0
for every m∈ℤ. The following lower bounds for the Jacobi Theta function will be useful in the sequel.Lemma 4.1.
Letδ>0. There exists  C>0 (not depending on  δ) such that
(4.3)|Θ(x)|≥Cδe(log⁡2|x|)/2log⁡|q||x|1/2,
for every x∈ℂ⋆ such that |1+xqk|>δ for all k∈ℤ.Proof.
Letδ>0. From Lemma  5.1.6 in [17] we get the existence of a positive constant C1 such that |Θ(x)|≥C1δΘ|q|(|x|) for every x∈ℂ⋆ such that |1+xqk|>δ for all k∈ℤ. Now,
(4.4)Θ|q|(|x|)=∑n∈Z|q|-n(n-1)/2|x|n≥max⁡n∈Z|q|-n(n-1)/2|x|n.
Let us fix |x|. The function
(4.5)f(t)=exp⁡(-12t(t-1)log⁡|q|+tlog⁡|x|)
takes its maximum value at t0=log|x|/log|q|+1/2 with f(t0)=C2exp(log2|x|/2log|q|)|x|1/2, for certain C2>0. Taking into account that
(4.6)max⁡n∈Z|q|-n(n-1)/2|x|n≥f(⌊t0⌋)=f(t0)|q|-(⌊t0⌋-t0)2/2≥f(t0)|q|-1/2,
one can conclude the result. Here ⌊·⌋ stands for the entire part.Corollary 4.2.
Letδ>0. For any  ξ∈(0,1) there exists  Cξ=Cξ(δ)>0 such that
(4.7)|Θ(x)|≥Cξeξlog⁡2|x|/2log⁡|q|,
for every x∈ℂ⋆ such that |1+xqk|>δ, for all k∈ℤ.From now on,(ℍ,∥·∥ℍ) stands for a complex Banach space.For anyλ∈ℂ and δ>0(4.8)Rλ,q,δ∶={z∈C⋆:|1+λzqk|>δ,∀k∈R}.The following definition corresponds to aq-analog of the Laplace transform and can be found in [7] when working with sectors in the complex plane.Proposition 4.3.
Letδ>0 and  ρ0>0. One  fixes an open and bounded set  V in  ℂ⋆ such that  D(0,ρ0)∩V≠∅. Let  λ∈D(0,ρ0)∩V, and,  f be a holomorphic function defined in  Ḋρ0 with values in  ℍ such that let can be extended to a function  F defined in  Ḋρ0∪Vqℝ+ and
(4.9)‖F(x)‖H≤C1eM¯log⁡2|x|,x∈Ḋρ0∪VqR+,
for positive constants C1>0 and 0<M¯<1/2log|q|.
Letπq=log(q)∏n≥0(1-q-n-1)-1, and put
(4.10)Lq;1λF(z)=1πq∫0∞λF(ξ)Θ(ξ/z)dξξ,
where the path [0,∞λ] is given by t∈(-∞,∞)↦qtλ. Then, ℒq;1λF defines a holomorphic function in ℛλ,q,δ and it is known as the q-Laplace transform of f following direction [λ].Proof.
LetK⊆ℛλ,q,δ be a compact set and z∈K. From the parametrization of the path [0,∞λ] we have
(4.11)∫0∞λF(ξ)Θ(ξ/z)dξξ=log⁡(q)∫-∞∞F(qtλ)Θ(qtλ/z)dt.
Let 0<ξ1<1 such that 0<M¯<ξ1/2log|q|, and let t∈ℝ. We have that w=qtλ/z satisfies |1+qkw|>δ for every k∈ℤ. Corollary 4.2 and (4.9) yield
(4.12)∫-∞∞‖F(qtλ)Θ(qtλ/z)‖Hdt≤∫-∞∞C1eM¯log⁡2|qtλ|Cξ1e(ξ1/2log⁡|q|)log⁡2|qtλ/z|dt≤L1∫-∞∞|qtλ|ξ1log⁡|z|/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt,
for a positive constant L1. There exist 0<A<B such that A≤|z|≤B for every z∈K, so that the last term in the chain of inequalities above is upper bounded by
(4.13)L1∫-∞-log⁡|λ|/log⁡|q||qtλ|ξ1log⁡A/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt+L1∫-log⁡|λ|/log⁡|q|∞|qtλ|ξ1log⁡B/log⁡|q|e(M¯-ξ1/2log⁡|q|)log⁡2|qtλ|dt.
The result follows from this last expression.Remark 4.4.
If we letM¯=1/2log|q|, then ℒq;1λF will only remain holomorphic in ℛλ,q,δ∩D(0,r1) for certain r1>0.
In the next proposition, we recall a commutation formula for theq-Laplace transform and the multiplication by a polynomial.Proposition 4.5.
LetV be an open and bounded set in  ℂ⋆ and  D(0,ρ0) such that  V∩D(0,ρ0)≠∅. Let  ϕ be a holomorphic function on  Vqℝ+∪Ḋρ0 with values in the Banach space  (ℍ,∥·∥ℍ) which satisfies the following estimates: there exist  C1>0 and  0<M¯<1/2log|q| such that
(4.14)‖ϕ(x)‖H<C1eM¯log⁡2|x|,x∈Ḋρ0∪VqR+.
Then, the function mϕ(τ)=τϕ(τ) is holomorphic on Vqℝ+∪Ḋρ0 and satisfies estimates in the shape above. Let λ∈V∩D(0,ρ0) and δ>0. One  has the following equality:
(4.15)Lq;1λ(mϕ)(t)=tLq;1λϕ(qt)
for every t∈ℛλ,q,δ.Proof.
It is direct to prove thatmϕ is a holomorphic function in Vqℝ+∪Ḋρ0 and also that mϕ verifies bounds as in (4.14). From (4.2) we have Θ(x)=xΘ(x/q), x∈ℂ⋆, so
(4.16)Lq;1λ(mϕ)(t)=1πq∫0∞λ(mϕ)(ξ)Θ(ξ/t)dξξ=1πq∫0∞λϕ(ξ)Θ(ξ/t)dξ=1πq∫0∞λϕ(ξ)(ξ/t)Θ(ξ/qt)dξ=tLq;1λ(ϕ)(qt),
for every t∈ℛλ,q,δ.


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

The following definition of a good covering firstly appeared in [17], p. 36.Definition 4.6.
LetI=(I1,I2) be a pair of open intervals in ℝ each one of length smaller than 1/4, and let UI be the corresponding open bounded set in ℂ⋆ defined by
(4.17)UI={e2πuiqv∈C⋆:u∈I1,v∈I2}.
Let ℐ be a finite family of tuple I as above verifying(1)
⋃I∈ℐ(UIq-ℕ)=ν∖{0}, where ν is a neighborhood of 0 in ℂ,(2)
the open setsUIq-ℕ, I∈ℐ are four-by-four disjoint. Then, we say that (UIq-ℕ)I∈ℐ is a good covering.Definition 4.7.
Let(UIq-ℕ)I∈ℐ be a good covering. Let δ>0. We consider a family of open bounded sets {(VI)I∈ℐ,𝒯} in ℂ⋆ such that(1)
there exists1<ρ0 with VI∩D(0,ρ0)≠∅, for all I∈ℐ,(2)
for everyI∈ℐ and τ∈VIqℝ,|τ+1|>δ,(3)
for everyI∈ℐ,t∈𝒯,ϵu∈UI, and λv∈VI∩D(0,ρ0), we have
(4.18)|1+λvϵutqr|>δ,
for every r∈ℝ,(4)
|t|≤1 for every t∈𝒯. We say that the family {(VI)I∈ℐ,𝒯} is associated to the good covering (UIq-ℕ)I∈ℐ.
LetS≥1 be an integer. For every 0≤k≤S-1, let m0,k,m1,k be positive integers and bk(ϵ,z)=∑s∈Ikbks(ϵ)zs a polynomial in z, where Ik is a subset of ℕ and bks(ϵ) are bounded holomorphic functions on some disc D(0,r0) in ℂ,0<r0≤1. Let (UIq-ℕ)I∈ℐ be a good covering such that UIq-ℕ⊆D(0,r0) for every I∈ℐ.Assumption B.
We have(4.19)M<12log⁡|q|.Definition 4.8.
Letρ0>1such thatV∩D(0,ρ0)≠∅. LetΔ,M̃>0such thatM̃<M,and that(ϵ,τ)↦W(ϵ,τ)be a bounded holomorphic function on(D(0,r0)∖{0})×Ḋρ0verifying(4.20)|W(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for every (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0. Assume moreover that W(ϵ,τ) can be extended to an analytic function (ϵ,τ)↦WUV(ϵ,τ) on Uq-ℕ×(Vqℝ+∪Ḋρ0) and
(4.21)|WUV(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|,
for every (ϵ,τ)∈Uq-ℕ×(Vqℝ+∪Ḋρ0). One says that the set {W,WUV,ρ0} is admissible.
Letℐ be a finite family of indices. For every I∈ℐ, we consider the following singularly perturbed Cauchy problem:
(4.22)ϵt∂zSXI(ϵ,qt,z)+∂zSXI(ϵ,t,z)=∑k=0S-1bk(ϵ,z)(tσq)m0,k(∂zkXI)(ϵ,t,zq-m1,k)
with bk as in (2.13), and with initial conditions
(4.23)(∂zjXI)(ϵ,t,0)=ϕI,j(ϵ,t),0≤j≤S-1,
where the functions ϕI,j(ϵ,t) are constructed as follows. Let {(VI)I∈ℐ,𝒯} be a family of open sets associated to the good covering (UIq-ℕ)I∈ℐ. For every 0≤j≤S-1 and I∈ℐ, let {Wj,WUI,VI,j,ρ0} be an admissible set. Let λI be a complex number in VI∩D(0,ρ0). We can assume that r0<1<|λI|. If not, we diminish r0 as desired. We put
(4.24)ϕI,j(ϵ,t)∶=Lq;1λI(τ⟼WUI,VI,j(ϵ,τ))(ϵ,ϵt).Lemma 4.9.
The function(ϵ,t)↦ϕI,j(ϵ,t), constructed as above, turns out to be holomorphic and bounded on  UIq-ℕ×𝒯 for every  I∈ℐ and all  0≤j≤S-1.Proof.
LetI∈ℐ and 0≤j≤S-1. From (4.21), one has
(4.25)|WUI,VI,j(ϵ,τ)|≤ΔeM̃log⁡2|τ/ϵ|=ΔeM̃log⁡2|ϵ||τ|-2M̃log⁡|ϵ|eM̃log⁡2|τ|,
for every (ϵ,τ)∈UIq-ℕ×(VIqℝ+∪Ḋρ0). Let ϵ∈UIq-ℕ and M̃<M̃2<1/2log|q|. Then, (4.25) can be upper bounded by Δ̃exp(M̃2log2|τ|), for some Δ̃=Δ̃(ϵ)>0. Estimates in (4.9) hold so that Proposition 4.3 can be applied here. The third item in Definition 4.7 derives the holomorphy of ϕI,j on UIq-ℕ×𝒯.
We now prove the boundness ofϕI,j in its domain of definition. One has
(4.26)|ϕI,j(ϵ,t)|=|Lq;1λIWUI,VI,j(ϵ,ϵt)|≤|Lq;1,+λIWUI,VI,j(ϵ,ϵt)|+|Lq;1,-λIWUI,VI,j(ϵ,ϵt)|,
for every (ϵ,t)∈UIq-ℕ×𝒯, where
(4.27)Lq;1,+λIWUI,VI,j(ϵ,ϵt)=log⁡(q)πq∫0∞WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)ds,Lq;1λIWUI,VI,j,-(ϵ,ϵt)=log⁡(q)πq∫-∞0WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)ds.
We only give bounds for the first integral. The estimates for the second one can be deduced following a similar procedure.
Let0<ξ<1 such that M̃<ξ/2log|q|. From Corollary 4.2 and (4.21) we deduce
(4.28)|Lq;1,+λIWUI,VI,j(ϵ,ϵt)|≤|log⁡(q)||πq|∫0∞|WUI,VI,j(ϵ,qsλI)Θ(qsλI/ϵt)|ds≤|log⁡(q)|Δ|πq|Cξ∫0∞eM̃log⁡2|qsλI/ϵ|eξlog⁡2|qsλI/ϵt|/2log⁡|q|ds≤|log⁡(q)|Δ|πq|Cξe(M̃-ξ/2log⁡|q|)log⁡2|λI/ϵ|e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|×∫0∞e2(M̃-(ξ/2log⁡|q|))log⁡2|q|s2e(M̃-ξ/2log⁡|q|)log⁡|q|log⁡|λI/ϵ|seξlog⁡|t|sds≤Cj,
for some Cj>0 which does not depend on ϵ nor t.The following assumption is related to technical reasons appearing in the proof of Lemma4.9 and Theorem 4.11.Assumption C.
There exista1,a2>0,0<ξ,ξ¯<1 such that(C.1)
M<ξ/2log|q|,(C.2)
ξ/2-Mlog|q|-Ca1/2a2>0,(C.3)
Ca2/2a1+C2/4ξ¯log|q|(ξ/2log|q|-M)<A1.
The next remark clarifies the availability of these constants for a posed problem.Remark 4.10.
Assumptions A, B, and C strongly depend on the choice ofq whose modulus must rest near 1. For example, these assumptions on the constants are verified when taking log|q|=1/16,M=1,A1=5,C=1,ξ=1/2,ξ¯=1/2,a1=1,a2=4. Then, the next theorem provides a solution for the equation
(4.29)ϵt∂z2XI(ϵ,qt,z)+∂z2XI(ϵ,t,z)=(b00(ϵ)+b01(ϵ)z)t2XI(ϵ,q2t,zq-30)+b10(ϵ)t∂zXI(ϵ,qt,zq-10),
with b00,b01,b10 being holomorphic functions near the origin.Theorem 4.11.
Let Assumption A be fulfilled by the integersm0,k,m1,k, for  0≤k≤S-1 and also Assumptions B and C for  M,A1,C. One considers the problem (4.22) and (4.23) where the initial conditions are constructed as above. Then, for every  I∈ℐ, the problem (4.22) and (4.23) has a solution  XI(ϵ,t,z) which is holomorphic and bounded in  UIq-ℕ×𝒯×ℂ.
Moreover, for everyρ>0, if  I,I′∈ℐ are such that  UIq-ℕ∩UI′q-ℕ≠∅, then there exists a positive constant  C1=C1(ρ)>0 such that
(4.30)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤C1e-(1/A)log⁡2|ϵ|,(ϵ,t,z)∈(UIq-N∩UI′q-N)×T×D(0,ρ),
with 1/A=(1-ξ¯)(ξ/2log|q|-M) with ξ,ξ¯ chosen as in Assumption C.Proof.
Letδ>0 and I∈ℐ. We consider the Cauchy problem (3.8) with initial conditions (∂zjW)(ϵ,τ,0)=Wj(ϵ,τ) for 0≤j≤S-1. From Theorem 3.4 we obtain the existence of a unique formal solution W(ϵ,τ,z)=∑β≥0Wβ(ϵ,τ)(zβ/β)∈𝒪((D(0,r0)∖{0})×Ḋρ0)[[z]] and positive constants C3>0 and 0<δ1<1 such that
(4.31)|Wβ(ϵ,τ)|≤C3β!(|q|2A1Sδ1)β|ϵ|-CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0.
Moreover, from Theorem2.4 we get that the coefficients Wβ(ϵ,τ) can be extended to holomorphic functions defined in UIq-ℕ×VIqℝ+ and also the existence of positive constants C2 and 0<δ2<1 such that
(4.32)|Wβ(ϵ,τ)|≤C2β!(|q|2A1Sδ2)β|τϵ|CβeMlog⁡2|τ/ϵ||q|-A1β2,β≥0,
for (ϵ,τ)∈UIq-ℕ×VIqℝ+.
We chooseλI∈VI∩D(0,ρ0). In the following estimates we will make use of the fact that |ϵ|≤|λI| for every ϵ∈D(0,r0∖{0}). Proposition 4.3 allows us to calculate the q-Laplace transform of Wβ with respect to τ for every β≥0,ℒq;1λI(Wβ)(ϵ,τ). It defines a holomorphic function in UIq-ℕ×ℛλI,q,δ. From the fact that {(VI)I∈ℐ,𝒯} is chosen to be a family associated to the good covering (UIq-ℕ)I∈ℐ we derive that the function
(4.33)(ϵ,t)⟼Lq;1λI(Wβ)(ϵ,ϵt)
is a holomorphic and bounded function defined in UIq-ℕ×𝒯. We can define, at least formally,
(4.34)XI(ϵ,t,z)∶=∑β≥0Lq;1λI(Wβ)(ϵ,ϵt)zββ!,
in 𝒪(UIq-ℕ×𝒯)[[z]]. If XI(ϵ,t,z) were a holomorphic function in UIq-ℕ×𝒯×ℂ, then Proposition 4.5 would allow us to affirm that (4.34) is an actual solution of (4.22) and (4.23). In order to end the first part of the proof it remian to demonstrate that (4.34) defines in fact a bounded holomorphic function in UIq-ℕ×𝒯×ℂ. Let (ϵ,t)∈UIq-ℕ×𝒯 and β≥0. We have
(4.35)|Lq;1λIWβ(ϵ,ϵt)|≤|Lq;1,+λIWβ(ϵ,ϵt)|+|Lq;1,-λIWβ(ϵ,ϵt)|,
where
(4.36)Lq;1,+λIWβ(ϵ,ϵt)=log⁡(q)πq∫0∞Wβ(ϵ,qsλI)Θ(qsλI/ϵt)ds,Lq;1,-λIWβ(ϵ,ϵt)=log⁡(q)πq∫-∞0Wβ(ϵ,qsλI)Θ(qsλI/ϵt)ds.
We now establish bounds for both integrals:
(4.37)|Lq;1,+λIWβ(ϵ,ϵt)|≤|log⁡q||πq|∫0∞|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds.
Let0<ξ<1 as in Assumption C. From (4.32) and (4.7), the previous integral is bounded by
(4.38)|log⁡q||πq|∫0∞C2β!(|q|2A1S/δ2)β|qsλI/ϵ|CβeMlog⁡2|qsλI/ϵ||q|-A1β2Cξexp⁡(ξlog⁡2|qsλI/ϵt|/2log⁡|q|)ds≤|log⁡q||πq|C2Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2∫0∞|q|CsβeMlog⁡2|qsλI/ϵ|exp⁡(ξlog⁡2|qsλI/ϵt|/2log⁡|q|)ds.
Leta1,a2 be as in Assumptions (C.2) and (C.3).
From(a1s-a2β)2≥0 and (2.5) in Definition 4.7, the previous inequality is upper bounded by
(4.39)A∫0∞|q|-Bs2e(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|e((2Mlog⁡|q|-ξ)log⁡|λI/ϵ|+ξlog⁡|t|)sds,
where 0<B=ξ/2-Mlog|q|-Ca1/2a2 and
(4.40)A=|log⁡q||πq|C2Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2+Ca2β2/2a1e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|.
The previous integral is uniformly bounded forϵ∈D(0,r0)∖{0} and t∈𝒯 from the hypotheses made on these sets. The expression in (4.39) can be bounded by
(4.41)|log⁡q||πq|C2′Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβe(M-ξ/2log⁡|q|)log⁡2|λI/ϵ||q|-A1β2+Ca2β2/2a1×e-ξlog⁡2|t|/2log⁡|q|eξlog⁡|λI/ϵ|log⁡|t|/log⁡|q|,
for an appropriate constant C2′>0.
The functions↦sγβe-αlog2(s) takes its maximum at s=eγβ/(2α) so each element in the image set is bounded by e(γβ)2/(4α). Taking this to the expression above we get
(4.42)|Lq;1,+λIWβ(ϵ,ϵt)|≤|log⁡q||πq|C2′′Cξβ!(|q|2A1Sδ2)β|q|-A1β2+Ca2β2/2a1+C2β2/4log⁡|q|(ξ/(2log⁡|q|)-M),
for certain C2′′>0.
Assumption (C.3) applied to the last term in the previous expression allows us to deduce that the sum(4.43)∑β≥0|Lq;1,+λIWβ(ϵ,ϵt)||z|ββ!
converges in the variable z uniformly in the compact sets of ℂ.
We now studyℒq;1,-λIWβ(ϵ,ϵt). We have
(4.44)|Lq;1,-λIWβ(ϵ,ϵt)|≤|log⁡q||πq|∫-∞0|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds.
From (3.11) and (4.7) the previous integral is bounded by
(4.45)|log⁡q||πq|∫-∞0C3β!(|q|2A1S/δ1)β|ϵ|-CβeMlog⁡2|qsλI/ϵ||q|-A1β2Cξeξlog⁡2|qsλI/ϵt|/2log⁡|q|ds.
Similar calculations to those in the first part of the proof resting on Assumption C can be made so that the series(4.46)∑β≥0Lq;1,-λIWβ(ϵ,ϵt)zββ!
is uniformly convergent with respect to the variable z in the compact sets of ℂ, for (ϵ,t)∈UIq-ℕ×𝒯. We will not go into detail not to repeat calculations.
The estimates (4.43) and (4.46) imply the convergence of the series in (4.34) for every z∈ℂ. The boundness of the q-Laplace transform with respect to ϵ is guaranteed so the first part of the result is achieved.
LetI,I′∈ℐ such that UIq-ℕ∩UI′q-ℕ≠∅ and ρ>0. For every (ϵ,t,z)∈(UIq-ℕ∩UI′q-ℕ)×𝒯×D(0,ρ) we have
(4.47)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤∑β≥0|Lq;1λIWβ(ϵ,ϵt)-Lq;1λI′Wβ(ϵ,ϵt)|ρββ!.
We can write(4.48)Lq;1λIWβ(ϵ,ϵt)-Lq;1λI′Wβ(ϵ,ϵt)=1πq(∫γ1Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ-∫γ2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ+∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ),
where the path γ1 is given by s∈(0,∞)↦qsλI, γ2 by s∈(0,∞)↦qsλI′, γ3 by s∈(-∞,0)↦qsλI, and γ4 by s∈(-∞,0)↦qsλI′.
Without loss of generality, we can assume that|λI|=|λI′|.
For the first integral we deduce(4.49)|∫1Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ|≤|log⁡(q)|∫0∞|Wβ(ϵ,qsλI)||Θ(qsλI/ϵt)|ds.
Similar estimates as in the first part of the proof lead us to bound the right part of the previous inequality by
(4.50)C2′′′Cξβ!(|q|2A1Sδ2)β|λIϵ|Cβ|q|-A1β2+(Ca2/2a1)β2e(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|,
for certain C2′′′>0. For any ξ¯∈(0,1) we have
(4.51)|λIϵ|Cβeξ¯(M-ξ/2log⁡|q|)log⁡2|λI/ϵ|≤eC2β2/4ξ¯(ξ/2log⁡|q|-M),β≥0.
This yields
(4.52)∫γ1|Wβ(ϵ,qsλI)Θ(qsλI/ϵt)|ds≤C2′′′Cξβ!(|q|2A1Sδ2)β|q|(-A1+(Ca2/2a1)+C2/4ξ¯log⁡|q|((ξ/2log⁡|q|)-M))β2e(1-ξ¯)(M-(ss))log⁡2|λI/ϵ|.
We choose ξ¯ as in Assumption C.
The integral corresponding to the pathγ2 can be bounded following identical steps.
We now give estimates concerningγ3-γ4. It is worth saying that the function in the integrand is well defined for (ϵ,τ)∈(D(0,r0)∖{0})×Ḋρ0 and does not depend on the index I∈ℐ. This fact and the Cauchy theorem allow us to write for any n∈ℕ(4.53)∫ΓnWβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=0,
where Γn=γn,1+γ5-γn,2-γn,3 is the closed path defined in the following way: s∈[-n,0]↦γn,1(s)=λIqs,γ5 is the arc of circumference from λI to λI′, s∈[-n,0]↦γn,2(s)=λI′qs, and γn,3 is the arc of circumference from λIq-n to λI′q-n. Taking n→∞ we derive
(4.54)0=lim⁡n→∞∫ΓnWβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=lim⁡n→∞∫γn,1+γ5-γn,2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ-lim⁡n→∞∫γn,3Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ.
Usual estimates lead us to prove that
(4.55)lim⁡n→∞∫γn,3Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=0.
Moreover,
(4.56)lim⁡n→∞∫γn,1+γ5-γn,2Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫γ3+γ5-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ.
From (4.54), (4.55), and (4.56) we obtain
(4.57)∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫-γ5Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ=∫θI′θIWβ(ϵ,|λI|eiθ)Θ(|λI|eiθ/ϵt)dθ,
where θI=arg(λI),θI′=arg(λI′). Taking into account Definition 4.7 and (4.31) we derive that the modulus of the last term in the previous equality is bounded by
(4.58)length(γ5)C3Cξβ!(|q|2A1Sδ1)β|ϵ|-CβeMlog⁡2|λI/ϵ|e(ξ/2log⁡|q|)log⁡2|λI/ϵt||q|-A1β2≤C3′β!(|q|2A1Sδ1)β|ϵ|-Cβe(M-ξ/2log⁡|q|)log⁡2|λI/ϵ||q|-A1β2≤C3′β!(|q|2A1Sδ1)β|ϵ|-Cβeξ¯(M-ξ/2log⁡|q|)log⁡2|ϵ||q|-A1β2e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|
for adequate positive constants C3,C3′. From the standard estimates we achieve
(4.59)|∫γ3-γ4Wβ(ϵ,ξ)Θ(ξ/ϵt)dξξ|≤C3′β!(|q|2A1Sδ1)β|q|-A1β2eC2/4ξ¯(ξ/2log⁡|q|-M)β2e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|.
From (4.47), (4.48), (4.52), (4.59) and Assumption (C.3) we conclude the existence of a positive constant C1′>0 such that
(4.60)|XI(ϵ,t,z)-XI′(ϵ,t,z)|≤C1′∑β≥0β!(|q|2A1Sδ0)β|q|(-A1+(Ca2/2a1)+C2/4ξ¯log⁡|q|(ξ/2log⁡|q|-M))β2×e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|ρββ!≤C1e(1-ξ¯)(M-ξ/2log⁡|q|)log⁡2|ϵ|,
for every (ϵ,t,z)∈(UIq-ℕ∩UI′q-ℕ)×𝒯×D(0,ρ), with δ0=min{δ1,δ2}.


## 5. Aq-Gevrey Malgrange-Sibuya-Type Theorem

In this section we obtain aq-Gevrey version of the so-called Malgrange-Sibuya theorem which allows us to reach our final main achievement: the existence of a formal series solution of problem (4.22) and (4.23) which asymptotically represents the actual solutions obtained in Theorem 4.11, meaning that, for every I∈ℐ,XI admits this formal solution as its q-Gevrey asymptotic expansion in the variable ϵ.In [4], a Malgrange-Sibuya-type theorem appears with similar aims as in this work. We complete the information there giving bounds on the estimates appearing for the q-asymptotic expansion. This mentioned work heavily rests on the theory developed by Ramis et al. in [17].In the present work, althoughq-Gevrey bounds are achieved, the q-Gevrey type involved will not be preserved, suffering an increase on the way.The nature of the proof relies on the one concerning the classical Malgrange-Sibuya theorem for Gevrey asymptotics which can be found in [18].Letℍ be a complex Banach space.Definition 5.1.
LetU be a bounded open set in ℂ⋆ and A>0. One says a holomorphic function f:Uq-ℕ→ℍ admits f̂=∑n≥0fnϵn∈ℍ[[ϵ]] as its q-Gevrey asymptotic expansion of type A in Uq-ℕ if for every compact set K⊆U there exist C1,H>0 such that
(5.1)‖f(ϵ)-∑n=0Nfnϵn‖H≤C1HN|q|A(N2/2)|ϵ|N+1(N+1)!,N≥0,
for every ϵ∈Kq-ℕ.
The following proposition can be found, under slight modifications in Section  4 of [17].Proposition 5.2.
LetA>0 and  U⊆ℂ⋆ be an open and bounded set. Let  f:Uq-ℕ→ℍ be a holomorphic function that admits a formal power series  f̂∈ℍ[[ϵ]] as its  q-Gevrey asymptotic expansion of type  A in  Uq-ℕ. Then, if  f̂(k) stands for the  kth formal derivative of  f̂ for every  k∈ℕ, one has that  f(k) admits  f̂(k) as its  q-Gevrey asymptotic expansion of type  A in  Uq-ℕ.Proposition 5.3.
LetA>0, and let  f:Uq-ℕ→ℍ a holomorphic function in  Uq-ℕ. Then, (i)
iff admits 0̂ as its q-Gevrey expansion of type A, then for every compact set K⊆U there exists C1>0 with
(5.2)‖f(ϵ)‖H≤C1e-(1/ã)(1/2log⁡|q|)log⁡2|ϵ|,
for every ϵ∈Kq-ℕ and every ã>A;(ii)
if for every compact setK⊆U there exists C1>0 with
(5.3)‖f(ϵ)‖H≤C1e-(1/A)(1/2log⁡|q|)log⁡2|ϵ|,
for every ϵ∈Kq-ℕ, then f admits 0̂ as its q-Gevrey asymptotic expansion of type ã in Uq-ℕ, for every ã>A.Proof.
LetC1,H,A>0 and ϵ∈ℂ⋆. The function
(5.4)G(x)=C1exp⁡(log⁡(H)x+log⁡|q|A2x2+(x+1)log⁡|ϵ|)
reaches its minimum for x>0 at x0=(-log(H)-log|ϵ|)/Alog|q|. We deduce both results from standard calculations.Definition 5.4.
Let(UIq-ℕ)I∈ℐ be a good covering at 0 (see Definition 4.6) and gI,I′:UIq-ℕ∩UI′q-ℕ→ℍ a holomorphic function in UIq-ℕ∩UI′q-ℕ for I,I′∈ℐ when the intersection is not empty. The family (gI,I′)(I,I′)∈ℐ2 is a q-Gevrey ℍ-cocycle of type A>0 attached to a good covering (UIq-ℕ)I∈ℐ if the following properties are satisfied.(1)
gI,I′ admits 0̂ as its q-Gevrey asymptotic expansion of type A>0 on UIq-ℕ∩UI′q-ℕ for every (I,I′)∈ℐ.(2)
gI,I′(ϵ)=-gI′,I(ϵ) for every (I,I′)∈ℐ, and ϵ∈UIq-ℕ∩UI′q-ℕ.(3)
One hasgI,I′′(ϵ)=gI,I′(ϵ)+gI′,I′′(ϵ) for all ϵ∈UIq-ℕ∩UI′q-ℕ∩UI′′q-ℕ, I,I′,I′′∈ℐ.Letρ>0 and 𝒯⊆ℂ⋆ be an open and bounded set. ℍ𝒯,ρ stands for the Banach space of holomorphic and bounded functions in 𝒯×D(0,ρ) with the supremum norm.Proposition 5.5.
Letρ>0. We consider the family  (XI(ϵ,t,z))I∈ℐ constructed in Theorem 4.11. Then, the set of functions  (gI,I′(ϵ))(I,I′)∈ℐ2 defined by
(5.5)gI,I′(ϵ)∶=(t,z)∈T×D(0,ρ)⟼XI′(ϵ,t,z)-XI(ϵ,t,z)
for I,I′∈ℐ is a q-Gevrey ℍ𝒯,ρ-cocycle of type Ã for every
(5.6)Ã>A∶=1(1-ξ¯)((ξ/2log⁡|q|)-M)2log⁡|q|=1(1-ξ¯)(ξ-2Mlog⁡|q|),
attached to the good covering (UIq-ℕ)I∈ℐ.Proof.
The first property in Definition5.4 directly comes from Theorem 4.11 and Proposition 5.3. The other two are verified by the construction of the cocycle.We recall several definitions and an extension result from [13] which will be crucial in our work.Definition 5.6.
A continuous increasing functionw:[0,∞)→[0,∞) is a weight function if it satisfies the following:(α)
there existsk≥1 with w(2t)≤k(w(t)+1) for all t≥0,(β)
∫0∞w(t)/(1+t2)dt<∞,(γ)
limt→∞(logt)/w(t)=0,(δ)
ϕ:t↦w(et) is convex. The Young conjugate associated to ϕ,ϕ⋆:[0,∞)→ℝ is defined by
(5.7)ϕ⋆(y)∶=sup⁡{xy-ϕ(x):x≥0}.Definition 5.7.
LetK be a nonempty compact set in ℝ2. A jet on K is a family F=(fα)α∈ℕ2 where fα:K→ℂ is a continuous function on K for each α∈ℕ2.
Letw be a weight function. A jet F=(fα)α∈ℕ2 on K is said to be a w-Whitney jet (of Roumieu type) on K if there exist m>0 and M>0 such that
(5.8)‖f‖K,1/m∶=sup⁡x∈K,α∈N2|fα(x)|exp⁡(-1mϕ⋆(m|α|))≤M,
and for every l∈ℕ,α∈ℕ2 with |α|≤l and x,y∈K one has
(5.9)|(RxlF)α(y)|≤M|x-y|l+1-|α|(l+1-|α|)!exp⁡(1mϕ⋆(m(l+1))),
where (RxlF)α(y)∶=fα(y)-∑|α+β|≤l(1/β!)fα+β(x)(y-x)β.ℰ{w}(K) denotes the linear space of w-Whitney jets on K.Definition 5.8.
LetK⊆ℝ2 be a nonempty compact set and w a weight function in K. A continuous function f:K→ℂ is w-𝒞∞ in the sense of Whitney in K if there exists a w-Whitney jet on K, (fα)α∈ℕ2 such that f(0,0)=f.
For an open setΩ∈ℝ2 we defineℰ{w}(Ω)∶={f∈𝒞∞(Ω):∀K⊆Ω,Kcompact,∃m>0,∥f∥K,1/m<∞}.The following result establishes conditions on a weight function so that a jet inℰ{w}(K) can be extended to an element in ℰ{w}(ℝ2).Theorem 5.9 (Corollary  3.10, [13]).
For a given weight functionw, the following statements are equivalent.(1)
For every nonempty closed setK in ℝ2 the restriction map sending a function f∈ℰ{w}(ℝ2) to the family of derivatives of f in K, (f(α)∣K)α∈ℕ2∈ℰ{w}(K), is a surjective map.(2)
w is a strong weight function, that is to say,(5.10)lim⁡ϵ→0+lim⁡t→∞̅ϵw(t)w(ϵt)=0.Letk1=1/4log|q|. One considers the weight function defined by w0(t)=k1log2(t) for t≥1 and w0(t)=0 for 0≤t≤1. As the authors write in [13], the value of a weight function near the origin is not relevant for the space of functions generated in the sequel.The following lemma can be easily verified.Lemma 5.10.
w0 is a weight function.Under this definition ofw0 one has(5.11)ϕw0⋆(y)=sup⁡{xy-ϕw0(x):x≥0}=sup⁡{xy-x24log⁡|q|:x≥0}=log⁡|q|y2,y≥0.The spaces appearing in Definition5.7 concerning this weight function are the following: for any nonempty compact set K⊆ℝ2,ℰ{w0}(K) is the set of w0-Whitney jets on K, which consists of every jet F=(fα)α∈ℕ2 on K such that there exist m∈ℕ,M>0 with(5.12)|fα(x)|≤M|q|m|α|2,x∈K,α∈N2,and such that for everyl∈ℕ and α∈ℕ2 with |α|≤l one has(5.13)|(RxlF)α(y)|≤M|x-y|l+1-|α|(l+1-|α|)!|q|m(l+1)2,x,y∈K.We derive thatℰ{w0}(K) consists of the Whitney jets on K such that there exist C1,H>0 with
(5.14)|fα(x)|≤C1H|α||q|A(|α|2/2),x∈K,α∈N2,
and for every x,y∈K and all l∈ℕ,α∈ℕ2 with |α|≤l,(5.15)|(RxlF)α(y)|≤C1Hl|q|A(l2/2)|x-y|l+1-|α|(l+1-|α|)!.Theorem 5.11.
w0 is a strong weight function so that Theorem 5.9 holds.Proof.
We have(5.16)lim⁡ϵ→0+lim⁡t→∞ϵw(t)w(ϵt)=lim⁡ϵ→0+lim⁡t→∞ϵk1log⁡2(t)k1log⁡2(ϵt)=lim⁡ϵ→0+ϵ=0.Remark 5.12.
A continuous functionf which is w0-𝒞∞ in the sense of Whitney on a compact set K is indeed 𝒞∞ in the usual sense in Int(K) and verifies q-Gevrey bounds of the same type. Moreover, we have
(5.17)fk(x,y)=∂xk1∂yk2f(x,y),
for every k=(k1,k2)∈ℕ2 and (x,y)∈Int(K).
The next result is an adaptation of Lemma  4.1.2 in [17]. Here, we need to determine bounds in order to achieve a q-Gevrey-type result.Lemma 5.13.
LetU be an open set in  ℂ⋆ and  f:Uq-ℕ→ℍ a holomorphic function with  f̂=∑h≥0ahϵh∈ℍ[[ϵ]] being its  q-Gevrey asymptotic expansion of type  A>0 in  Uq-ℕ. Then, for any  n∈ℕ, the family  ∂ϵnf(ϵ) of  n-complex derivatives of  f satisfies that,  for every compact set  K⊆U and  k,m∈ℕ with  k≤m, there exist  C1,H>0 such that
(5.18)‖∂ϵkf(ϵa)-∑h=0m-k∂ϵk+hf(ϵb)h!(ϵa-ϵb)h‖H≤C1Hm|q|A(m2/2)|ϵa-ϵb|m+1-k(m+1-k)!,
for every ϵa,ϵb∈Kq-ℕ∪{0}. Here, one writes ∂ϵlf(0)=l!al  for l∈ℕ.Proof.
We will first state the result whenϵb=0. Indeed, we prove in this first step that the family of functions with q-Gevrey asymptotic expansion of type A>0 in a fixed q-spiral is closed under derivation. Proposition 5.2 turns out to be a particular case of this result.
Letm∈ℕ, K be a compact set in U, and consider another compact set K1 such that K⊆K1⊆U. We define
(5.19)Rm(ϵ)∶=ϵ-m-1(f(ϵ)-∑h=0m∂ϵhf(0)h!ϵh),ϵ∈Kq-N,
where ∂ϵhf(0) denotes the limit of ∂ϵhf(ϵ) for ϵ∈Kq-ℕ tending to 0. Then, we have that
(5.20)∂ϵf(ϵ)=∑h=1m∂ϵhf(0)h!hϵh-1+(∂ϵRm(ϵ))ϵm+1+(m+1)Rm(ϵ)ϵm.
Moreover, from Definition 5.1, there exist C,H>0 such that ∥Rm(ϵ)∥≤CHm|q|A(m2/2)/(m+1)! for every ϵ∈K1q-ℕ.
Lemma 5.14 (Lemma  4.4.1 [17]).
There existsρ>0 such that D¯(ϵ,ρ|ϵ|)⊆K1q-ℕ for every ϵ∈Kq-ℕ.
The Cauchy’s integral formula andq-Gevrey expansion of f guarantee the existence of a positive constant C2>0 such that
(5.21)‖∂ϵRm(ϵ)‖H≤C2Hm|q|A(m2/2)(m+1)!1ρ|ϵ|,ϵ∈Kq-N,
This yields the existence of C3>0 such that
(5.22)‖ϵ-m(∂ϵf(ϵ)-∑h=0m-1∂ϵh+1f(0)h!ϵh)‖H≤‖∂ϵRm(ϵ)‖H|ϵ|+(m+1)‖Rm(ϵ)‖H≤C2A1m|q|A(m2/2)m!,ϵ∈Kq-N.
An induction reasoning is sufficient to conclude the proof for every m≥0.
We now study the case whereϵb≠0 and only give details for k=0. For k≥1 one only has to take into account that the derivatives of f also admit q-Gevrey asymptotic expansion of type A and consider the function ∂ϵkf.
Ifϵb≠0 we treat two cases:
if|ϵa-ϵb|≤ρ|ϵb|, then [ϵa,ϵb] is contained in K1q-ℕ and we conclude from the Cauchy integral formula.
If|ϵa-ϵb|>ρ|ϵb|, then we bear in mind that the result is obvious when f is a  polynomial and write f(ϵ)=ϵm+1Rm(ϵ)+p(ϵ) where p(ϵ)=∑h=0m(∂ϵhf(0)/h!ϵh). So, it is sufficient to prove (5.18) when f(ϵ)∶=ϵm+1Rm(ϵ). The result follows from q-Gevrey bounds for ∥∂ϵkRm∥ℍ, k=0,…,n and usual estimates.The following lemma generalizes Lemma  6 in [4].Lemma 5.15.
Letf:Uq-ℕ→ℍ be a holomorphic function having  f̂(ϵ)=∑h≥0ahϵh∈ℍ[[ϵ]] as its  q-Gevrey asymptotic expansion of type  A>0 on  Uq-ℕ. Let  K⊆U be a compact set. Then, the function  (ϵ1,ϵ2)↦ϕ(ϵ1+iϵ2)=f(ϵ1,ϵ2) is a  w0-𝒞∞ function in the sense of Whitney on the compact set
(5.23)K′={(ϵ1,ϵ2)∈R2:ϵ1+iϵ2∈Kq-N∪{0}}.Proof.
We consider the set of functions(ϕ(k1,k2))(k1,k2)∈ℕ2 defined by
(5.24)ϕ(k1,k2)∶=ik2∂ϵk1+k2f(ϵ),(k1,k2)∈N2,(ϵ1,ϵ2)∈K′.
From Lemma 5.13, function f satisfies bounds as in (5.18). Written in terms of the elements in (ϕ(k1,k2))(k1,k2)∈ℕ2 we have the existence of C1,H>0 such that for, every (k1,k2)∈ℕ2, m≥0,
(5.25)‖1ik2ϕ(k1,k2)(x1,y1)-∑p=0m-|(k1,k2)|∑h1+h2=pϕ(k1+h1,k2+h2)(x2,y2)ik2+h2p!×p!h1!h2!(x1-x2)h1ih2(y1-y2)h2∑p=0m-|k1,k2|‖H≤C1Hm|q|A(m2/2)‖(x1-x2,y1-y2)‖R2m+1-|(k1,k2)|(m+1-|(k1,k2)|)!
for (x1,y1),(x2,y2)∈K′. Expression (5.14) can be directly checked from (5.24) and (5.18) for ϵb=0 and m=k. This yields that the set (ϕ(k1,k2))(k1,k2)∈ℕ2 is an element in ℰ{w0}(K′).The next result allows us to glue together a finite number of jets inℰ{w0}(K), for a given compact set K.Theorem 5.16 ([19]. Theorem II.1.3).
LetK1,K2 be compact sets in  ℝ2. The following statements are equivalent. (i)
The sequence(5.26)0⟶E{w0}(K1∪K2)⟶πE{w0}(K1)⊕E{w0}(K2)⟶δE{w0}(K1∩K2)⟶0
is exact. π(f)=(f∣K1,f∣K2) and δ(f,g)=f∣K1∩K2-g∣K1∩K2.(ii)
Letf1∈ℰ{w0}(K1) and f2∈ℰ{w0}(K2) be such that f1(x)=f2(x) for every x∈K1∩K2. The function f defined by f(x)=f1(x) if x∈K1 and f(x)=f2(x) if x∈K2 belongs to ℰ{w0}(K1∪K2).(iii)
IfK1∩K2≠∅, then there exist A3,A4>0 such that
(5.27)M¯(A3dist⁡(x,K1∩K2))≤A4M¯(dist⁡(x,K2)),
for every x∈K1. Here, M¯ denotes the function given by M¯(0)=0 and M¯(t)=infn∈ℕtnMn for t>0. dist(x,K) stands for the distance from x to the set K.Corollary 5.17 ([17], Lemma  4.3.6).
GivenK̃1,K̃2 nonempty compact sets in  ℂ⋆, if one puts  Kj:={(ϵ1,ϵ2)∈ℝ2:ϵ1+iϵ2∈K̃jq-ℕ∪{0}},j=1,2, then the previous theorem holds for  K1 and  K2.As the authors remark in [17], condition (iii) in the previous result is known as the transversality condition which is more constricting than Łojasiewicz’s condition (see [20]).The next proposition is devoted to show that the cocycle constructed in Proposition5.5 splits in the space of w0–𝒞∞ functions in the sense of Whitney. Whitney-type extension results on ℰ{w0}(K) (Theorems 5.9 and 5.11) will play an important role in the following step.Proposition 5.18.
Let(UIq-ℕ)I∈ℐ be a good covering, and let  (gI,I′(ϵ))(I,I′)∈ℐ2 be the  q-Gevrey  ℍ𝒯,ρ-cocycle of type  Ã constructed in Proposition 5.5. One chooses a family of compact sets  KI⊆UI for  I∈ℐ, with  Int(KI)≠∅, in such a way that  ⋃I∈ℐ(KIq-ℕ) is  𝒰∖{0}, where  𝒰 is a neighborhood of 0 in  ℂ.
Then, for allI∈ℐ, there exists a w0-𝒞∞ function fI(ϵ1,ϵ2) in the sense of Whitney on the compact set AI={(ϵ1,ϵ2)∈ℝ2:ϵ1+iϵ2∈KIq-ℕ∪{0}}, with values in the Banach space ℍ𝒯,ρ, such that
(5.28)gI,I′(ϵ1+iϵ2)=fI′(ϵ1,ϵ2)-fI(ϵ1,ϵ2)
for all I,I′∈ℐ such that AI∩AI′≠∅ and, for every (ϵ1,ϵ2)∈AI∩AI′.Proof.
The proof follows similar arguments to those Lemma  3.12 in [17] and it is an adaptation of Proposition  5 in [4] under q-Gevrey settings.
LetI,I′∈ℐ such that AI∩AI′≠∅. From Lemma 5.15, we have that the function (ϵ1,ϵ2)↦gI,I′(ϵ1+iϵ2) is a w0-𝒞∞ function in the sense of Whitney on AI∩AI′. In the following we provide the construction of fI for I∈ℐ verifying (5.28).
Let us fix anyI∈ℐ. We consider any w0-𝒞∞ function in the sense of Whitney on AI. By definition of the good covering (UIq-ℕ)I∈ℐ the following cases are possible.
Case 1.
If there is at least oneI′∈ℐ,I≠I′, such that AI∩AI′≠∅ but AI∩AI′∩AI′′=∅ for every I′′∈ℐ with I′′≠I′≠I, then we define eI,I′(ϵ1,ϵ2)=fI(ϵ1,ϵ2)+gI,I′(ϵ1+iϵ2) for every (ϵ1,ϵ2)∈AI∩AI′. eI,I′ is a w0-𝒞∞ function in the sense of Whitney in AI∩AI′. From Theorems 5.9 and 5.11, we can extend eI,I′ to a w0-𝒞∞ function in the sense of Whitney on AI′. This extension is called fI′. We have
(5.29)gI,I′(ϵ1+iϵ2)=fI′(ϵ1,ϵ2)-fI(ϵ1,ϵ2),(ϵ1,ϵ2)∈AI∩AI′.Case 2.
There exist two differentI′,I′′∈ℐ with I′≠I≠I′′ such that AI∩AI′∩AI′′≠∅. We first construct a w0-𝒞∞ function in the sense of Whitney on AI′, fI′(ϵ1,ϵ2), verifying
(5.30)gI,I′(ϵ1+iϵ2)=fI′(ϵ1,ϵ2)-fI(ϵ1,ϵ2),(ϵ1,ϵ2)∈AI∩AI′.
We define eI,I′′(ϵ1,ϵ2)=fI(ϵ1,ϵ2)+gI,I′′(ϵ1+iϵ2) for every (ϵ1,ϵ2)∈AI∩AI′′ and eI′,I′′(ϵ1,ϵ2)=fI′(ϵ1,ϵ2)+gI′,I′′(ϵ1+iϵ2) whenever (ϵ1,ϵ2)∈AI′∩AI′′. From (5.30), we have eI,I′′(ϵ1,ϵ2)=eI′,I′′(ϵ1,ϵ2) for every (ϵ1,ϵ2)∈AI∩AI′∩AI′′. From this, we can define
(5.31)eI′′(ϵ1,ϵ2)∶={eI,I′′(ϵ1,ϵ2)if(ϵ1,ϵ2)∈AI∩AI′′,eI′,I′′(ϵ1,ϵ2)if(ϵ1,ϵ2)∈AI′∩AI′′.
From Theorem 5.16 and Corollary 5.17 we deduce that eI′′(ϵ1,ϵ2) can be extended to a w0-𝒞∞ function in the sense of Whitney in AI′′,fI′′(ϵ1,ϵ2). It is straightforward to check, from the way fI′′ was constructed, that fI′′(ϵ1,ϵ2)=fI(ϵ1,ϵ2)+gI,I′′(ϵ1+iϵ2) when (ϵ1,ϵ2)∈AI∩AI′′ and also fI′′(ϵ1,ϵ2)=fI′(ϵ1,ϵ2)+gI′,I′′(ϵ1+iϵ2) for (ϵ1,ϵ2)∈AI′∩AI′′.These two cases solve completely the problem since nonempty intersection of four different compacts in (AI)I∈ℐ is not allowed when working with a good covering. The functions in (fI)I∈ℐ satisfy (5.28).


## 6. Existence of Formal Series Solutions andq-Gevrey Expansions

In the current section we set the main result in this work. We establish the existence of a formal power series with coefficients belonging toℍ𝒯,ρ which asymptotically represents the actual solutions found in Theorem 4.11 for the problem (4.22) and (4.23). Moreover, each actual solution turns out to admit this formal power series as q-Gevrey expansion of a certain type in the q-spiral where the solution is defined.The following lemma will be useful in the following. We only sketch its proof. For more details we refer to [21].Lemma 6.1.
LetU be an open and bounded set in  ℝ2. We consider  h∈𝒞∞(U) (in the classical sense) verifying bounds as in (5.14) and (5.15) for every  (ϵ1,ϵ2)∈U. Let  g be the solution of the equation
(6.1)∂ϵ¯g(ϵ1,ϵ2)∶=12(∂ϵ1+i∂ϵ2)g(ϵ1+iϵ2)=h(ϵ1,ϵ2),(ϵ1,ϵ2)∈U.
Then, g also verifies bounds such as those in (5.14) and (5.15) for (ϵ1,ϵ2)∈U.Proof.
Leth1 be any extension of the function h to ℝ2 with compact support which preserves bounds in (5.14) and (5.15) in ℝ2. We have that
(6.2)g(ϵ1,ϵ2)∶=-1π∫R2h1(x)x-ϵdξdη,(ϵ1,ϵ2)∈U,
solves (6.1). Here, ϵ=(ϵ1,ϵ2),x=(ξ,η), and dξdη stands for the Lebesgue measure in x-plane. Bounds in (5.14) for the function g come out from
(6.3)∂α1+α2g∂ϵ1α1∂ϵ2α2(ϵ1,ϵ2)=-1π∫R2∂α1+α2h1∂ϵ1α1∂ϵ2α2(x)1x-ϵdξdη,
for every α=(α1,α2)∈ℕ2 and (ϵ1,ϵ2)∈U and from the fact that the function x=(x1,x2)↦1/|x| is Lebesgue integrable in any compact set containing 0.
On the other hand,g satisfies the estimates in (5.15) from the Taylor formula with integral remainder.We now give a decomposition result of the functionsXI constructed in Theorem 4.11. The procedure is adapted from [4] under q-Gevrey settings. For every I∈ℐ, we write XI(ϵ):UIq-ℕ→ℍ𝒯,ρ for the holomorphic function given by XI(ϵ)∶=(t,z)↦XI(ϵ,t,z).Proposition 6.2.
There exist aw0-𝒞∞ function  u(ϵ1,ϵ2) and a holomorphic function  a(ϵ1+iϵ2) defined on the neighborhood  Int(⋃I∈ℐAI) of 0 such that
(6.4)XI(ϵ1+iϵ2)=fI(ϵ1,ϵ2)+u(ϵ1,ϵ2)+a(ϵ1+iϵ2),(ϵ1,ϵ2)∈Int(AI),
for every I∈ℐ.Proof.
From the definition of the cocycle(gI,I′)(I,I′)∈ℐ2 in Proposition 5.5 and from Proposition 5.18 we derive
(6.5)XI(ϵ1+iϵ2)-fI(ϵ1,ϵ2)=XI′(ϵ1+iϵ2)-fI′(ϵ1,ϵ2),(ϵ1,ϵ2)∈AI∩AI′∖{(0,0)},
whenever (I,I′)∈ℐ2 and AI∩AI′≠∅. The function X-f given by
(6.6)(X-f)(ϵ1,ϵ2)∶=XI(ϵ1+iϵ2)-fI(ϵ1,ϵ2),(ϵ1,ϵ2)∈AI∖{(0,0)},
is well defined on W∖{(0,0)}, where W=⋃I∈ℐAI is a closed neighborhood of (0,0).
For everyI∈ℐ,XI is a holomorphic function on UIq-ℕ so that the Cauchy-Riemann equations hold:
(6.7)∂ϵ¯(XI)(ϵ1+iϵ2)=0,(ϵ1,ϵ2)∈AI∖{(0,0)}.
This yields ∂ϵ¯(X-f)(ϵ1,ϵ2)=-∂ϵ¯fI(ϵ1,ϵ2) for every I∈ℐ and (ϵ1,ϵ2)∈Int(AI).
We have that-∂ϵ¯fI(ϵ1,ϵ2) can be extended to a w0-𝒞∞ function in the sense of Whitney on AI. This yields that fI is w0-𝒞∞ in the sense of Whitney on AI. In fact, their q-Gevrey types coincide.
From this, we deduce that∂ϵ¯(X-f) is a w0-𝒞∞ function in the sense of Whitney on AI for every I∈ℐ and also that ∂ϵ¯fI(ϵ1,ϵ2)=∂ϵ¯fI′(ϵ1,ϵ2) for every (ϵ1,ϵ2)∈Int(AI∩AI′) and every I,I′∈ℐ because gI,I′(ϵ) is a holomorphic function on UIq-ℕ∩UI′q-ℕ. The previous equality is also true for (ϵ1,ϵ2)∈AI∩AI′ from the fact that fI is w0-𝒞∞ in the sense of Whitney on AI.
From Theorem5.16 and Corollary 5.17 we derive that ∂ϵ¯(X-f) is a w0-𝒞∞ function in the sense of Whitney on ⋃I∈ℐAI.
Taking into account Lemma6.1 we derive the existence of a 𝒞∞ function u(ϵ1,ϵ2) in the usual sense, defined in Int(W) and verifying q-Gevrey bounds of a certain positive type, such that
(6.8)∂ϵ¯u(ϵ1,ϵ2)=∂ϵ¯(X-f)(ϵ1,ϵ2),(ϵ1,ϵ2)∈Int(W).
From this last expression we have that u(ϵ1,ϵ2)-(X-f)(ϵ1,ϵ2) defines a holomorphic function on Int(W)∖{(0,0)}.
For everyI∈ℐ, XI is a bounded ℍ𝒯,ρ-function in Int(W)∖{(0,0)}, and so it is the function u(ϵ1,ϵ2)-(X-f)(ϵ1,ϵ2). The origin turns out to be a removable singularity so the function u(ϵ1,ϵ2)-(X-f)(ϵ1,ϵ2) can be extended to a holomorphic function defined on Int(W). The result follows from here.We are under conditions to enunciate the main result in the present work.Theorem 6.3.
Under the same hypotheses as in Theorem4.11, there exists a formal power series
(6.9)X̂(ϵ,t,z)=∑k≥0Xk(t,z)k!ϵk∈HT,ρ[[ϵ]],
formal solution of
(6.10)ϵt∂zSX̂(ϵ,qt,z)+∂zSX̂(ϵ,t,z)=∑k=0S-1bk(ϵ,z)(tσq)m0,k(∂zkX̂)(ϵ,t,zq-m1,k).
Moreover, let I∈ℐ, and let K̃I be any compact subset of Int(KI). There exists B>0 such that the function XI(ϵ,t,z) constructed in Theorem 4.11 admits X̂(ϵ,t,z) as its q-Gevrey asymptotic expansion of type B in K̃Iq-ℕ.Proof.
LetI∈ℐ, and let K̃I be any compact subset of Int(KI).
From Proposition6.2 we can extend XI(ϵ1+iϵ2) to a w0-𝒞∞ function in the sense of Whitney on ÃI={(ϵ1,ϵ2)∈ℝ2:ϵ1+iϵ2∈K̃Iq-ℕ∪{0}}⊆Int(AI)∪{(0,0)}. Let us fix I∈ℐ. We consider the family (X(h1,h2)(ϵ1,ϵ2))(h1,h2)∈ℕ2 associated to XI by Definition 5.7. We have
(6.11)XI(h1,h2)(ϵ1,ϵ2)=∂ϵ1h1∂ϵ2h2XI(ϵ1+iϵ2)=ih2∂ϵh1+h2XI(ϵ),(ϵ1,ϵ2)∈ÃI∖{(0,0)},
because XI(ϵ) is holomorphic on Int(KI)q-ℕ.
We have thatXI(h1,h2)(ϵ1,ϵ2) is continuous at (0,0) for every (h1,h2)∈ℕ2 so we can define k≥0(6.12)Xk,I∶=XI(h1,h2)(0,0)ih2∈HT,ρ,
whenever h1+h2=k. The estimates held by any w0-𝒞∞ function in the sense of Whitney (see Definition 5.7 for α=(0,0)) lead us to the existence of positive constants C1,H,B>0 such that
(6.13)‖XI(ϵ1+iϵ2)-∑p=0mXp,Ip!(ϵ1+iϵ2)p‖HT,ρ≤C1Hm|q|B(m2/2)|ϵ1+iϵ2|m+1(m+1)!,
for every m≥0 and ϵ1+iϵ2∈K̃Iq-ℕ. As a matter of fact, this shows that XI admits X̂I(ϵ)=∑k≥0(Xk/k!)ϵk as its q-Gevrey expansion of type B>0 in K̃Iq-ℕ.
The formal power seriesX̂I does not depend on I∈ℐ. Indeed, from Theorem 4.11 we have that XI(ϵ)-XI′(ϵ) admits both 0̂ and X̂I′-X̂I as q-asymptotic expansion on K̃Iq-ℕ∩K̃I′q-ℕ whenever this intersection is not empty. We put X̂∶=X̂I for any I∈ℐ. The function Xk,I=Xk,I(t,z)∈ℍ𝒯,ρ does not depend on I for every k≥0. We write Xk∶=Xk,I for k≥0. XI admits X̂=∑k≥0(Xk/k!)ϵk as its q-Gevrey asymptotic expansion of type B>0 in K̃Iq-ℕ for all I∈ℐ.
In order to achieve the result, it only remains to prove thatX̂(ϵ,t,z) is a formal solution of (6.10). Let l≥1. If we derive l times with respect to ϵ in (6.10), we get that ∂ϵlXI(ϵ,t,z) is a solution of
(6.14)ϵt∂zS∂ϵlXI(ϵ,qt,z)+t∂zSl∂ϵl-1XI(ϵ,qt,z)+∂zS∂ϵlXI(ϵ,t,z)=∑k=0S-1∑l1+l2=ll!l1!l2!∂ϵl1bk(ϵ,z)∂ϵl2(((tσq)m0,k)∂zkXI)(ϵ,t,zq-m1,k).
for every l≥1,(t,z)∈𝒯×D(0,ρ) and ϵ∈K̃Iq-ℕ. Letting ϵ tend to 0 in (6.14), we obtain
(6.15)t∂zSXl-1(qt,z)(l-1)!+∂zSXl(t,z)l!=∑k=0S-1∑l1+l2=l∂ϵl1bk(ϵ,z)∣ϵ=0l1!((tσq)m0,k∂zkXl2)(t,zq-m1,k)l2!
for every l≥1,(t,z)∈𝒯×D(0,ρ). The holomorphy of bk(ϵ,z) with respect to ϵ at 0 implies
(6.16)bk(ϵ,z)=∑l≥0∂ϵlbk(ϵ,z)|ϵ=0l!ϵl,
for ϵ near 0 and for every z∈ℂ. Statements (6.14) and (6.15) conclude that X̂(ϵ,t,z)=∑k≥0Xk(t,z)(ϵk/k!) is a formal solution of (6.10).



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*Source: 860716-2012-02-06.xml*