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

Abstract

We study a 𝑞-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.

1. Introduction

We study a family of 𝑞-difference-differential equations of the following form:𝜖𝑡𝜕^𝑆_𝑧𝑋𝜖,𝑞𝑡,𝑧+𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)𝑡𝜎_𝑞^𝑚0,𝑘𝜕^𝑘_𝑧𝑋𝜖,𝑡,𝑧𝑞^{−𝑚1,𝑘}, (1.1) where 𝑞∈ℂ such that |𝑞|>1,𝑚_0,𝑘,𝑚_1,𝑘 are positive integers, 𝑏_𝑘(𝜖,𝑧) are polynomials in 𝑧 with holomorphic coefficients in 𝜖 on some neighborhood of 0 in ℂ, and 𝜎_𝑞 is the dilation operator given by (𝜎_𝑞𝑋)(𝜖,𝑡,𝑧)=𝑋(𝜖,𝑞𝑡,𝑧). As in previous works [1–3], the map (𝑡,𝑧)↦(𝑞^𝑚0,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 𝑏_𝑘(𝜖,𝑧)∶=∑_{𝑠∈𝐼𝑘}𝑏_𝑘𝑠(𝜖)𝑧^𝑠 is such that, for all 0≤𝑘≤𝑆−1,𝐼_𝑘 is a finite subset of ℕ={0,1,…} and 𝑏_𝑘𝑠(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 𝑚_0,𝑘. The main point on these flatness conditions on the coefficients in 𝑏_𝑘(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 𝑏_𝑘𝑠 (we refer to [4] for the details). These constricting conditions on the flatness of 𝑏_𝑘(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 discrete 𝑞-spirals (𝑈_𝐼𝑞^−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 set ℐ, 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 𝑈_𝐼𝑞^−ℕ in the auxiliary transformed problem) times a continuous 𝑞-spiral to infinity in the variable 𝜏 (𝑉_𝐼𝑞^ℝ+ 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 𝑉_𝐼𝑞^ℝ+ subject to 𝑞-Gevrey bounds 𝑊^𝐼_𝛽(𝜖,𝜏)≤𝐶₁𝛽!𝐻^𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝜏𝜖^𝐶𝛽𝑞^{−𝐴1𝛽2},𝜏∈𝑉_𝐼𝑞^ℝ+, (1.2) 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 𝑊^𝐼_𝛽(𝜖,𝜏)≤𝐶₁𝛽!𝐻^𝛽𝑒^{𝑀log2|𝜏/𝜖|}|𝜖|^−𝐶𝛽𝑞^{−𝐴1𝛽2},𝜏∈𝐷0,𝜌₀⧵{0}, (1.3) 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 (𝑊^𝐼_𝛽)_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 function𝑋_𝐼(𝜖,𝑡,𝑧)∶=_𝛽≥0ℒ^𝜆𝐼_𝑞;1𝑊^𝐼_𝛽()𝑧𝜖,𝜖𝑡^𝛽𝛽! (1.4) turns out to be a holomorphic function defined in 𝑈_𝐼𝑞^−ℕ×𝒯×ℂ which is a solution of the initial problem. Here, 𝒯 is an adequate open half 𝑞-spiral to 0 and 𝜆_𝐼 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 (𝑋_𝐼)_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 𝑈_𝐼∩𝑈_𝐼′ is not empty we have 𝑋_𝐼(𝜖,𝑡,𝑧)−𝑋_𝐼′(𝜖,𝑡,𝑧)≤𝐶₁𝑒^{−(1/𝐴)log2|𝜖|} (1.5) for positive constants 𝐶₁,𝐴 and for every (𝜖,𝑡,𝑧)∈(𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′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 𝑈_𝐼∩𝑈_𝐼′ as a function with values in the Banach space ℍ_𝒯,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 𝑋(𝜖)=_𝑘≥0𝑋_𝑘𝜖𝑘!^𝑘∈ℍ_𝒯,𝜌[[𝜖]], (1.6) 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 𝑋 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 involving 𝑞-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 Sections 2 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 Section 4 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.

Let 𝑈,𝑉 be nonempty bounded open sets in ℂ^⋆∶=ℂ⧵{0}, and let 𝑞∈ℂ^⋆ such that |𝑞|>1. We define 𝑈𝑞^−ℕ=𝜖𝑞^−𝑛∈ℂ∶𝜖∈𝑈,𝑛∈ℕ,𝑉𝑞^ℝ+=𝜏𝑞^𝑙∈ℂ∶𝜏∈𝑉,𝑙∈ℝ,𝑙≥0. (2.1) We assume there exists 𝑀₁>0 such that |𝜏+1|>𝑀₁ for all 𝜏∈𝑉𝑞^ℝ+ and also that the distance from the set 𝑉 to the origin is positive.

Definition 2.1. Let 𝜖∈𝑈𝑞^−ℕ and 𝛽∈ℕ.𝐸_{𝛽,𝜖,𝑉𝑞ℝ+} denotes the vector space of functions 𝑣∈𝒪(𝑉𝑞^ℝ+) such that ‖𝑣(𝜏)‖_{𝛽,𝜖,𝑉𝑞ℝ+}∶=sup_{𝜏∈𝑉𝑞ℝ+}|𝑣(𝜏)|𝑒^{𝑀log2|𝜏/𝜖|}𝜏𝜖^−𝐶𝛽𝑞^𝐴1𝛽2 (2.2) is finite.
Let 𝛿>0. 𝐻(𝜖,𝛿,𝑉𝑞^ℝ+) denotes the complex vector space of all formal series 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)𝑧^𝛽/𝛽! belonging to 𝒪(𝑉𝑞^ℝ+)[[𝑧]] such that ‖𝑣(𝜏,𝑧)‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}∶=_𝛽≥0𝑣_𝛽(𝜏)_{𝛽,𝜖,𝑉𝑞ℝ+}𝛿^𝛽𝛽!<∞. (2.3) It is straightforward to check that the pair (𝐻(𝜖,𝛿,𝑉𝑞^ℝ+),‖⋅‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}) is a Banach space.

We consider the formal integration operator 𝜕_𝑧⁻¹ defined on 𝒪(𝑉𝑞^ℝ+)[[𝑧]] by𝜕_𝑧⁻¹(𝑣(𝜏,𝑧))∶=_𝛽≥1𝑣_𝛽−1(𝜏)𝑧^𝛽𝛽!∈𝒪𝑉𝑞^ℝ+[[𝑧]]. (2.4)

Lemma 2.2. Let 𝑠,𝑘,𝑚₁,𝑚₂∈ℕ,𝛿>0,𝜖∈𝑈𝑞^−ℕ. One assumes that the following conditions hold: 𝑚₁≤𝐶(𝑘+𝑠),𝑚₂≥2(𝑘+𝑠)𝐴₁. (2.5)
Then, there exists a constant 𝐶₁=𝐶₁(𝑠,𝑘,𝑚₁,𝑚₂,𝑉,𝑈,𝐶,𝐴₁) (not depending on 𝜖 nor 𝛿) such that 𝑧^𝑠𝜏𝜖^𝑚1𝜕_𝑧^−𝑘𝑣𝜏,𝑧𝑞^−𝑚2_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤𝐶₁𝛿^𝑘+𝑠‖𝑣(𝜏,𝑧)‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}, (2.6) for every 𝑣∈𝐻(𝜖,𝛿,𝑉𝑞^ℝ+).

Proof. Let 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)(𝑧^𝛽/𝛽!)∈𝒪(𝑉𝑞^ℝ+)[[𝑧]]. We have that 𝑧^𝑠𝜏𝜖^𝑚1𝜕_𝑧^−𝑘𝑣𝜏,𝑧𝑞^−𝑚2_{(𝜖,𝛿,𝑉𝑞ℝ+)}=_{𝛽≥𝑘+𝑠}𝜏𝜖^𝑚1𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝛽!!1𝛽−𝑠𝑞^{𝑚2(𝛽−𝑠)}𝑧^𝛽𝛽!_{(𝜖,𝛿,𝑉𝑞ℝ+)}=_{𝛽≥𝑘+𝑠}𝜏𝜖^𝑚1𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝛽!!1𝛽−𝑠𝑞^{𝑚2(𝛽−𝑠)}_{𝛽,𝜖,𝑉𝑞ℝ+}𝛿^𝛽.𝛽! (2.7) Taking into account the definition of the norm ‖⋅‖_{𝛽,𝜖,𝑉𝑞ℝ+}, we get 𝜏𝜖^𝑚1𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝛽!!1𝛽−𝑠𝑞^{𝑚2(𝛽−𝑠)}_{𝛽,𝜖,𝑉𝑞ℝ+}=𝛽!!𝑞𝛽−𝑠^{𝐴1(𝛽−(𝑘+𝑠))2}𝑞^𝑝(𝛽)sup_{𝜏∈𝑉𝑞ℝ+}𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝑒^{𝑀log2|𝜏/𝜖|}𝜏𝜖^{−𝐶(𝛽−(𝑘+𝑠))}𝜖𝜏^{𝐶(𝑘+𝑠)−𝑚1}, (2.8) with 𝑝(𝛽)=𝐴₁𝛽²−𝐴₁(𝛽−(𝑘+𝑠))²−𝑚₂(𝛽−𝑠). From (2.5) we derive |𝜖/𝜏|^{𝐶(𝑘+𝑠)−𝑚1}≤(𝐶_𝑈/𝐶_𝑉)^{𝐶(𝑘+𝑠)−𝑚1} for every 𝜖∈𝑈𝑞^−ℕ and 𝜏∈𝑉𝑞^ℝ+, where 0<𝐶_𝑉∶=min{|𝜏|∶𝜏∈𝑉} and 0<𝐶_𝑈∶=max{|𝜖|∶𝜖∈𝑈}. Moreover, 𝑝𝛽=2(𝑘+𝑠)𝐴₁−𝑚₂𝛽−(𝑘+𝑠)²𝐴₁+𝑚₂𝑠, (2.9) for every 𝛽∈ℕ. Regarding condition (2.5) we obtain the existence of 𝐶₁>0 such that 𝜖𝜏^{𝐶(𝑘+𝑠)−𝑚1}𝑞^𝑝(𝛽)≤𝐶₁, (2.10) for every 𝜏∈𝑉𝑞^ℝ+ and 𝛽∈ℕ. Inequality (2.6) follows from (2.7), (2.8), and (2.10): 𝑧^𝑠𝜏𝜖^𝑚1𝜕_𝑧^−𝑘𝑣𝜏,𝑧𝑞^−𝑚2_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤𝐶_{1𝛽≥𝑘+𝑠}𝑣_{𝛽−(𝑘+𝑠)}(𝜏)_{𝛽−(𝑘+𝑠),𝜖,𝑉𝑞ℝ+}𝛽!!𝛿𝛽−𝑠𝛽^𝛽!≤𝐶₁𝛿^𝑘+𝑠_{𝛽≥𝑘+𝑠}𝑣_{𝛽−(𝑘+𝑠)}(𝜏)_{𝛽−(𝑘+𝑠),𝜖,𝑉𝑞ℝ+}𝛿^{𝛽−(𝑘+𝑠)}()!.𝛽−𝑘+𝑠 (2.11)

Lemma 2.3. Let 𝐹(𝜖,𝜏) be a holomorphic and bounded function defined on 𝑈𝑞^−ℕ×𝑉𝑞^ℝ+. Then, there exists a constant 𝐶₂=𝐶₂(𝐹,𝑈,𝑉)>0 such that 𝐹(𝜖,𝜏)𝑣_𝜖(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤𝐶₂𝑣_𝜖(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)} (2.12) for every 𝜖∈𝑈𝑞^−ℕ, every 𝛿>0, and all 𝑣_𝜖∈𝐻(𝜖,𝛿,𝑉𝑞^ℝ+).

Proof. Direct calculations regarding the definition of the elements in 𝐻(𝜖,𝛿,𝑉𝑞^ℝ+) allow us to conclude when taking 𝐶₂∶=max{|𝐹(𝜖,𝜏)|∶𝜖∈𝑈𝑞^−ℕ,𝜏∈𝑉𝑞^ℝ+}.

Let 𝑆≥1 be an integer. For all 0≤𝑘≤𝑆−1, let 𝑚_0,𝑘,𝑚_1,𝑘 be positive integers and 𝑏_𝑘(𝜖,𝑧)=∑_{𝑠∈𝐼𝑘}𝑏_𝑘𝑠(𝜖)𝑧^𝑠 a polynomial in 𝑧, where 𝐼_𝑘 is a finite subset of ℕ and 𝑏_𝑘𝑠(𝜖) are holomorphic bounded functions on 𝐷(0,𝑟₀). We assume 𝑈𝑞^−ℕ⊆𝐷(0,𝑟₀).

We consider the following functional equation: 𝜕^𝑆_𝑧𝑊(𝜖,𝜏,𝑧)=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)(𝜏+1)𝜖^𝑚0,𝑘𝜏^𝑚0,𝑘𝜕^𝑘_𝑧𝑊𝜖,𝜏,𝑧𝑞^{−𝑚1,𝑘} (2.13) with initial conditions 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏),0≤𝑗≤𝑆−1, (2.14) where the functions (𝜖,𝜏)↦𝑊_𝑗(𝜖,𝜏) belong to 𝒪(𝑈𝑞^−ℕ×𝑉𝑞^ℝ+) for every 0≤𝑗≤𝑆−1.

We make the following assumption.

Assumption A. For every 0≤𝑘≤𝑆−1 and 𝑠∈𝐼_𝑘, we have 𝑚_0,𝑘≤𝐶(𝑆−𝑘+𝑠),𝑚_1,𝑘≥2(𝑆−𝑘+𝑠)𝐴₁. (2.15)

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<𝑀<𝑀 such that, for every 0≤𝑗≤𝑆−1𝑊_𝑗(𝜖,𝜏)≤Δ𝑒𝑀_log²_|𝜏/𝜖|, (2.16) for all 𝜏∈𝑉𝑞^ℝ+,𝜖∈𝑈𝑞^−ℕ. Then, there exists 𝑊(𝜖,𝜏,𝑧)∈𝒪(𝑈𝑞^−ℕ×𝑉𝑞^ℝ+)[[𝑧]], solution of (2.13) and (2.14), such that, if 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), then there exist 𝐶₂>0 and 0<𝛿<1 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₂𝑞𝛽!^2𝐴1𝑆𝛿^𝛽𝜏𝜖^𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2},𝛽≥0, (2.17) for every 𝜖∈𝑈𝑞^−ℕ and 𝜏∈𝑉𝑞^ℝ+.

Proof. Let 𝜖∈𝑈𝑞^−ℕ. We define the map 𝒜_𝜖 from 𝒪(𝑉𝑞^ℝ+)[[𝑧]] into itself by 𝒜_𝜖𝑊(𝜏,𝑧)∶=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)(𝜏+1)𝜖^𝑚0,𝑘𝜏^𝑚0,𝑘𝜕_𝑧^𝑘−𝑆𝑊𝜏,𝑧𝑞^{−𝑚1,𝑘}+𝜕^𝑘_𝑧𝑤_𝜖𝜏,𝑧𝑞^{−𝑚1,𝑘}, (2.18) where 𝑤_𝜖(𝜏,𝑧)∶=∑^𝑆−1_𝑗=0𝑊_𝑗(𝜖,𝜏)(𝑧^𝑗/𝑗!). In the following lemma, we show that the restriction of 𝒜_𝜖 to a neighborhood of the origin in 𝐻(𝜖,𝛿,𝑉𝑞^ℝ+) is a Lipschitz shrinking map for an appropriate choice of 𝛿>0.

Lemma 2.5. There exist 𝑅>0 and 𝛿>0 (not depending on 𝜖) such that (1)‖𝒜_𝜖(𝑊(𝜏,𝑧))‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤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,𝑉𝑞^ℝ+);(2)𝒜_𝜖(𝑊₁(𝜏,𝑧))−𝒜_𝜖𝑊₂(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤12𝑊₁(𝜏,𝑧)−𝑊₂(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)} (2.19) for every 𝑊₁,𝑊₂∈𝐵(0,𝑅).

Proof. Let 𝑅>0 and 0<𝛿<1.
For the first part we consider 𝑊(𝜏,𝑧)∈𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝑉𝑞^ℝ+). Lemmas 2.2 and 2.3 can be applied so that 𝒜_𝜖𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤^𝑆−1_{𝑘=0𝑠∈𝐼𝑘}𝑀_𝑘𝑠𝑀₁𝐶₁𝛿^{𝑆−𝑘+𝑠}𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝑉𝑞ℝ+)}+𝑧^𝑠𝜏𝜖^𝑚0,𝑘𝜕^𝑘_𝑧𝑤_𝜖𝜏,𝑧𝑞^{−𝑚1,𝑘}_{(𝜖,𝛿,𝑉𝑞ℝ+)}, (2.20) with 𝑀_𝑘𝑠=sup_{𝜖∈𝑈𝑞−ℕ}|𝑏_𝑘𝑠(𝜖)|<∞,𝑠∈𝐼_𝑘,0≤𝑘≤𝑆−1. Taking into account the definition of 𝐻(𝜖,𝛿,𝑉𝑞^ℝ+) and (2.16) we have 𝑧^𝑠𝜏𝜖^𝑚0,𝑘𝜕^𝑘_𝑧𝑤_𝜖𝜏,𝑧𝑞^{−𝑚1,𝑘}_{(𝜖,𝛿,𝑉𝑞ℝ+)}=^{𝑆−1−𝑘}_𝑗=0𝜏𝜖^𝑚0,𝑘𝑊_𝑗+𝑘()𝑧𝜖,𝜏^𝑗+𝑠𝑗!𝑞^{𝑚1,𝑘𝑗}_{(𝜖,𝛿,𝑉𝑞ℝ+)}=^{𝑆−1−𝑘}_𝑗=0sup_{𝜏∈𝑉𝑞ℝ+}𝑊_𝑗+𝑘(𝜖,𝜏)𝑒^{𝑀log2|𝜏/𝜖|}𝜏𝜖^{𝑚0,𝑘−𝐶(𝑗+𝑠)}𝑞^{𝐴1(𝑗+𝑠)2}𝛿^𝑗+𝑠𝑗!𝑞^{𝑚1,𝑘𝑗}≤Δ^{𝑆−1−𝑘}_𝑗=0𝑞^{𝐴1(𝑗+𝑠)2}𝛿^𝑗+𝑠𝑗!𝑞^{𝑚1,𝑘𝑗}max𝑒^−(𝑀−𝑀_)log²_(𝑥)𝑥^{𝑚0,𝑘−𝐶(𝑗+𝑠)}∶𝑥>0,0≤𝑗+𝑘≤𝑆−1,𝑠∈𝐼_𝑘≤Δ𝐶^′₂, (2.21) for a positive constant 𝐶^′₂.
We conclude this first part from an appropriate choice of 𝑅 and 𝛿>0.
For the second part we take 𝑊₁,𝑊₂∈𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝑉𝑞^ℝ+). Similar arguments as before yield 𝒜_𝜖𝑊₁−𝒜_𝜖𝑊_{2(𝜖,𝛿,𝑉𝑞ℝ+)}≤^𝑆−1_{𝑘=0𝑠∈𝐼𝑘}𝑀_𝑘𝑠𝑀₁𝐶₁𝛿^{𝑆−𝑘+𝑠}𝑊₁−𝑊_{2(𝜖,𝛿,𝑉𝑞ℝ+)}. (2.22) An adequate choice for 𝛿>0 allows us to conclude the proof.

We choose constants 𝑅,𝛿 as in the previous lemma.

From Lemma 2.5 and taking into account the shrinking map theorem on complete metric spaces, we guarantee the existence of 𝑊_𝜖(𝜏,𝑧)∈𝐻(𝜖,𝛿,𝑉𝑞^ℝ+) which is a fixed point for 𝒜_𝜖 in 𝐵(0,𝑅); it is to say, ‖𝑊_𝜖(𝜏,𝑧)‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤𝑅 and 𝒜_𝜖(𝑊_𝜖(𝜏,𝑧))=𝑊_𝜖(𝜏,𝑧).

Let us define 𝑊_𝜖(𝜏,𝑧)∶=𝜕_𝑧^−𝑆𝑊_𝜖(𝜏,𝑧)+𝑤_𝜖(𝜏,𝑧). (2.23) If we write 𝑊_𝜖(𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽,𝜖(𝜏)(𝑧^𝛽/𝛽!) and 𝑊_𝜖(𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽,𝜖(𝜏)(𝑧^𝛽/𝛽!), then we have that 𝑊_{𝛽+𝑆,𝜖}≡𝑊_𝛽,𝜖 for 𝛽≥0 and 𝑊_𝑗,𝜖(𝜏)=𝑊_𝑗(𝜖,𝜏), 0≤𝑗≤𝑆−1.

From ‖𝑊_𝜖(𝜏,𝑧)‖_{(𝜖,𝛿,𝑉𝑞ℝ+)}≤𝑅 we arrive at ‖𝑊_𝛽,𝜖‖_{𝛽,𝜖,𝑉𝑞ℝ+}≤𝑅𝛽!(1/𝛿)^𝛽 for every 𝛽≥0. This implies 𝑊_𝛽,𝜖(𝜏1)≤𝑅𝛽!𝛿^𝛽𝜏𝜖^𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2}, (2.24) for every 𝛽≥0 and 𝜏∈𝑉𝑞^ℝ+.

This is valid for every 𝜖∈𝑈𝑞^−ℕ. We define 𝑊(𝜖,𝜏,𝑧)∶=𝑊_𝜖(𝜏,𝑧) and 𝑊_𝛽(𝜖,𝜏)∶=𝑊_𝛽,𝜖(𝜏) for every (𝜖,𝜏)∈𝑈𝑞^−ℕ×𝑉𝑞^ℝ+,𝑧∈ℂ and 𝛽≥𝑆. From (2.23), it is straightforward to prove that 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!) is a solution of (2.13) and (2.14).

Moreover, holomorphy of 𝑊_𝛽 in 𝑈𝑞^−ℕ×𝑉𝑞^ℝ+ for every 𝛽≥0 can be deduced from the recursion formula verified by the coefficients: 𝑊_ℎ+𝑆(𝜖,𝜏)=ℎ!^𝑆−1_{𝑘=0ℎ1+ℎ2=ℎ,ℎ1∈𝐼𝑘}𝑏_𝑘ℎ1(𝜖)𝜏^𝑚0,𝑘(𝜏+1)𝜖^𝑚0,𝑘𝑊_ℎ2+𝑘(𝜖,𝜏)ℎ₂!𝑞^{𝑚1,𝑘ℎ2},ℎ≥0. (2.25) This implies that 𝑊_𝛽(𝜖,𝜏) is holomorphic in 𝑈𝑞^−ℕ×𝑉𝑞^ℝ+ 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 𝑉𝑞^ℝ+, respectively, and usual calculations lead us to assure the existence of a positive constant 𝑅₁>0 such that 𝑊_𝛽(𝜖,𝜏)=𝑊_{𝛽−𝑆,𝜖}(𝜏)≤𝑅₁𝑞𝛽!^2𝐴1𝑆𝛿^𝛽𝜏𝜖^𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2}, (2.26) for every 𝛽≥𝑆, and for every 𝜖∈𝑈𝑞^−ℕ and 𝜏∈𝑉𝑞^ℝ+. This concludes the proof for 𝛽≥𝑆.

Hypothesis (2.16) leads us to obtain (2.26) for 0≤𝑘≤𝑆−1.

Remark 2.6. If 𝑠>0 for every 𝑠∈𝐼_𝑘,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.

Let 1<𝜌₀, and let 𝑈⊆ℂ^⋆, a bounded and open set with positive distance to the origin. 𝐷̇_𝜌0 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. Let 𝑟₀>0,𝜖∈𝐷(0,𝑟₀)⧵{0}, and 𝛽∈ℕ. 𝐸²_{̇𝐷𝛽,𝜖,𝜌0} denotes the vector space of functions 𝑣∈𝒪(𝐷̇_𝜌0) such that |𝑣(𝜏)|_{̇𝐷𝛽,𝜖,𝜌0}∶=sup_{̇𝐷𝜏∈𝜌0}𝑣(𝜏|𝜖||)|^𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^𝐴1𝛽2, (3.1) is finite. Let 𝛿>0. 𝐻₂(𝜖,𝛿,𝐷̇_𝜌0) stands for the vector space of all formal series 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)𝑧^𝛽/𝛽! belonging to 𝒪(𝐷̇_𝜌0)[[𝑧]] such that |𝑣(𝜏,𝑧)|_{̇𝐷(𝜖,𝛿,𝜌0)}∶=_𝛽≥0𝑣_𝛽(𝜏)_{̇𝐷𝛽,𝜖,𝜌0}𝛿^𝛽𝛽!<∞. (3.2) It is straightforward to check that the pair (𝐻₂(𝜖,𝛿,𝐷̇_𝜌0),|⋅|_{̇𝐷(𝜖,𝛿,𝜌0)}) is a Banach space.

Lemma 3.2. Let 𝑠,𝑘,𝑚₁,𝑚₂∈ℕ,𝛿>0 and 𝜖∈𝐷(0,𝑟₀)⧵{0}. One assumes that the following conditions hold: 𝑚₁≤𝐶(𝑘+𝑠),𝑚₂≥2(𝑘+𝑠)𝐴₁. (3.3)
Then, there exists a constant 𝐶₁=𝐶₁(𝑠,𝑘,𝑚₁,𝑚₂,𝐷̇_𝜌0,𝑈) (not depending on 𝜖 nor 𝛿) such that 𝑧^𝑠𝜏𝜖^𝑚1𝜕_𝑧^−𝑘𝑣𝜏,𝑧𝑞^−𝑚2_{̇𝐷(𝜖,𝛿,𝜌0)}≤𝐶₁𝛿^𝑘+𝑠|𝑣(𝜏,𝑧)|_{̇𝐷(𝜖,𝛿,𝜌0)}, (3.4) for every 𝑣∈𝐻₂(𝜖,𝛿,𝐷̇_𝜌0).

Proof. Let 𝑣(𝜏,𝑧)∈𝒪(𝐷̇_𝜌0)[[𝑧]]. The proof follows similar steps to those in Lemma 2.2. We have 𝑧^𝑠𝜏𝜖^𝑚1𝜕_𝑧^−𝑘𝑣𝜏,𝑧𝑞^−𝑚2_{̇𝐷(𝜖,𝛿,𝜌0)}=_{𝛽≥𝑘+𝑠}𝜏𝜖^𝑚1𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝛽!!1𝛽−𝑠𝑞^{𝑚2(𝛽−𝑠)}_{̇𝐷𝛽,𝜖,𝜌0}𝛿^𝛽.𝛽! (3.5) From the definition of the norm |⋅|_{̇𝐷𝛽,𝜖,𝜌0}, we get 𝜏𝜖^𝑚1𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝛽!!1𝛽−𝑠𝑞^{𝑚2(𝛽−𝑠)}_{̇𝐷𝛽,𝜖,𝜌0}≤𝛽!!𝑞𝛽−𝑠^{𝐴1(𝛽−(𝑘+𝑠))2}𝑞^𝑝(𝛽)×sup_{̇𝐷𝜏∈𝜌0}𝑣_{𝛽−(𝑘+𝑠)}(𝜏)𝑒^{𝑀log2|𝜏/𝜖|}|𝜖|^{𝐶(𝛽−(𝑘+𝑠))}𝜌^𝑚1₀|𝜖|^{𝐶(𝑘+𝑠)−𝑚1}, (3.6) 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. Let 𝐹(𝜖,𝜏) be a holomorphic and bounded function defined on (𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0. Then, there exists a constant 𝐶₂=𝐶₂(𝐹)>0 such that 𝐹(𝜖,𝜏)𝑣_𝜖(𝜏,𝑧)_{̇𝐷(𝜖,𝛿,𝜌0)}≤𝐶₂𝑣_𝜖(𝜏,𝑧)_{̇𝐷(𝜖,𝛿,𝜌0)} (3.7) for every 𝜖∈𝐷(0,𝑟₀)⧵{0}, every 𝛿>0, and every 𝑣_𝜖∈𝐻₂(𝜖,𝛿,𝐷̇_𝜌0).

Let 𝑆,𝑟₀,𝑚_0,𝑘,𝑚_1,𝑘 and 𝑏_𝑘, as in Section 2 and 𝜌₀>0. One considers the Cauchy problem 𝜕^𝑆_𝑧𝑊(𝜖,𝜏,𝑧)=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)(𝜏+1)𝜖^𝑚0,𝑘𝜏^𝑚0,𝑘𝜕^𝑘_𝑧𝑊𝜖,𝜏,𝑧𝑞^{−𝑚1,𝑘} (3.8) with initial conditions 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏),0≤𝑗≤𝑆−1, (3.9) where the functions (𝜖,𝜏)↦𝑊_𝑗(𝜖,𝜏) belong to 𝒪((𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0) 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<𝑀<𝑀 such that 𝑊_𝑗(𝜖,𝜏)≤Δ𝑒𝑀_log²_|𝜏/𝜖|, (3.10) for every 𝜏∈𝐷̇_𝜌0,𝜖∈𝐷(0,𝑟₀)⧵{0} and 0≤𝑗≤𝑆−1. Then, there exists 𝑊(𝜖,𝜏,𝑧)∈𝒪((𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0)[[𝑧]], solution of (3.8) and (3.9) such that, if 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), then there exist 𝐶₃>0 and 0<𝛿<1 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₃𝑞𝛽!^2𝐴1𝑆𝛿^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2},𝛽≥0, (3.11) for every 𝜖∈𝐷(0,𝑟₀)⧵{0} and 𝜏∈𝐷̇_𝜌0.

Proof. The proof of Theorem 2.4 can be adapted here so details will be omitted.
Let 𝜖∈𝐷(0,𝑟₀)⧵{0} and 0<𝛿<1. We consider the map 𝒜_𝜖 from 𝒪(𝐷̇_𝜌0)[[𝑧]] into itself defined as in (2.18) and construct 𝑤_𝜖(𝜏,𝑧) as above. From (3.10) we derive 𝑧^𝑠𝜏𝜖^𝑚0,𝑘𝜕^𝑘_𝑧𝑤_𝜖𝜏,𝑧𝑞^{−𝑚1,𝑘}_{̇𝐷(𝜖,𝛿,𝜌0)}=^{𝑆−1−𝑘}_𝑗=0sup_{̇𝐷𝜏∈𝜌0}𝑊_𝑗+𝑘(|𝜖|𝜖,𝜏)^{𝐶(𝑗+𝑠)}𝑒^{𝑀log2|𝜖/𝜏|}𝜏𝜖^𝑚0,𝑘𝑞^{𝐴1(𝑗+𝑠)2}𝛿^𝑗+𝑠𝑗!𝑞^{𝑚1,𝑘𝑗}≤Δ𝐶^′₃, (3.12) for a positive constant 𝐶^′₃ not depending on 𝜖 nor 𝛿.
Lemmas 3.2, 3.3, and (3.12) allow us to affirm that one can find 𝑅>0 and 𝛿>0 such that the restriction of 𝒜_𝜖 to the disc 𝐷(0,𝑅) in 𝐻₂(𝜖,𝛿,𝐷̇_𝜌0) is a Lipschitz shrinking map. Moreover, there exists 𝑊_𝜖(𝜏,𝑧)∈𝐻₂(𝜖,𝛿,𝐷̇_𝜌0) which is a fixed point for 𝒜_𝜖 in 𝐵(0,𝑅).
If we put 𝑊_𝜖(𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽,𝜖(𝜏)(𝑧^𝛽/𝛽!), then one gets |𝑊_𝛽,𝜖|_{̇𝐷𝛽,𝜖,𝜌0}≤𝑅𝛽!(1/𝛿)^𝛽 for 𝛽≥0. This implies 𝑊_𝛽,𝜖(𝜏1)≤𝑅𝛽!𝛿^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2},𝛽≥0,𝜏∈𝐷̇_𝜌0. (3.13) The formal power series 𝑊(𝜖,𝜏,𝑧)∶=_𝛽≥𝑆𝑊_{𝛽−𝑆,𝜖}(𝜏)𝑧^𝛽𝛽!+𝑤_𝜖(𝜏,𝑧)∶=_𝛽≥0𝑊_𝛽()𝑧𝜖,𝜏^𝛽𝛽! (3.14) turns out to be a solution of (3.8) and (3.9) verifying that 𝑊_𝛽(𝜖,𝜏) is a holomorphic function in (𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0 and the estimates (3.11) hold for 𝛽≥0.

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

4.1. A 𝑞-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 a 𝑞-analog of the Laplace transform which was firstly developed by Zhang in [7].

Let 𝑞∈ℂ such that |𝑞|>1.

The Jacobi Theta function is defined in ℂ^⋆ by Θ(𝑥)=_𝑛∈ℤ𝑞^{−𝑛(𝑛−1)/2}𝑥^𝑛,𝑥∈ℂ^⋆. (4.1) From the fact that the Jacobi Theta function satisfies the functional equation 𝑥𝑞Θ(𝑥)=Θ(𝑞𝑥), for 𝑥≠0, we have Θ𝑞^𝑚𝑥=𝑞^{𝑚(𝑚+1)/2}𝑥^𝑚Θ(𝑥),𝑥∈ℂ,𝑥≠0 (4.2) for every 𝑚∈ℤ. The following lower bounds for the Jacobi Theta function will be useful in the sequel.

Lemma 4.1. Let 𝛿>0. There exists 𝐶>0 (not depending on 𝛿) such that |Θ(𝑥)|≥𝐶𝛿𝑒^{(log2|𝑥|)/2log|𝑞|}|𝑥|^1/2, (4.3) for every 𝑥∈ℂ^⋆ such that |1+𝑥𝑞^𝑘|>𝛿 for all 𝑘∈ℤ.

Proof. Let 𝛿>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+𝑥𝑞^𝑘|>𝛿 for all 𝑘∈ℤ. Now, Θ_|𝑞|(|𝑥|)=_𝑛∈ℤ𝑞^{−𝑛(𝑛−1)/2}|𝑥|^𝑛≥max_𝑛∈ℤ𝑞^{−𝑛(𝑛−1)/2}|𝑥|^𝑛. (4.4) Let us fix |𝑥|. The function 𝑓(𝑡)−1=exp2𝑡(𝑡−1)log𝑞+𝑡log|𝑥| (4.5) 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 max_𝑛∈ℤ𝑞^{−𝑛(𝑛−1)/2}|𝑥|^𝑛≥𝑓𝑡₀=𝑓𝑡₀𝑞^{−(⌊𝑡0⌋−𝑡0)2/2}≥𝑓𝑡₀𝑞^−1/2, (4.6) one can conclude the result. Here ⌊⋅⌋ stands for the entire part.

Corollary 4.2. Let 𝛿>0. For any 𝜉∈(0,1) there exists 𝐶_𝜉=𝐶_𝜉(𝛿)>0 such that |Θ(𝑥)|≥𝐶_𝜉𝑒^{𝜉log2|𝑥|/2log|𝑞|}, (4.7) for every 𝑥∈ℂ^⋆ such that |1+𝑥𝑞^𝑘|>𝛿, for all 𝑘∈ℤ.

From now on, (ℍ,‖⋅‖_ℍ) stands for a complex Banach space.

For any 𝜆∈ℂ and 𝛿>0ℛ_{𝜆,𝑞,𝛿}∶=𝑧∈ℂ^⋆∶𝜆1+𝑧𝑞^𝑘>𝛿,∀𝑘∈ℝ. (4.8)

The following definition corresponds to a 𝑞-analog of the Laplace transform and can be found in [7] when working with sectors in the complex plane.

Proposition 4.3. Let 𝛿>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 𝐷̇_𝜌0 with values in ℍ such that let can be extended to a function 𝐹 defined in 𝐷̇_𝜌0∪𝑉𝑞^ℝ+ and ‖𝐹(𝑥)‖_ℍ≤𝐶₁𝑒^{𝑀log2|𝑥|},𝑥∈𝐷̇_𝜌0∪𝑉𝑞^ℝ+, (4.9) for positive constants 𝐶₁>0 and 0<𝑀<1/2log|𝑞|.
Let 𝜋_𝑞=log(𝑞)∏_𝑛≥0(1−𝑞^−𝑛−1)⁻¹, and put ℒ^𝜆_𝑞;1𝐹(𝑧)=1𝜋_𝑞0^∞𝜆𝐹(𝜉)Θ(𝜉/𝑧)𝑑𝜉𝜉, (4.10) where the path [0,∞𝜆] is given by 𝑡∈(−∞,∞)↦𝑞^𝑡𝜆. Then, ℒ^𝜆_𝑞;1𝐹 defines a holomorphic function in ℛ_{𝜆,𝑞,𝛿} and it is known as the 𝑞-Laplace transform of 𝑓 following direction [𝜆].

Proof. Let 𝐾⊆ℛ_{𝜆,𝑞,𝛿} be a compact set and 𝑧∈𝐾. From the parametrization of the path [0,∞𝜆] we have ₀^∞𝜆𝐹(𝜉)Θ(𝜉/𝑧)𝑑𝜉𝜉=log𝑞^∞_−∞𝐹𝑞^𝑡𝜆Θ𝑞^𝑡𝜆/𝑧𝑑𝑡. (4.11) Let 0<𝜉₁<1 such that 0<𝑀<𝜉₁/2log|𝑞|, and let 𝑡∈ℝ. We have that 𝑤=𝑞^𝑡𝜆/𝑧 satisfies |1+𝑞^𝑘𝑤|>𝛿 for every 𝑘∈ℤ. Corollary 4.2 and (4.9) yield ^∞_−∞𝐹𝑞^𝑡𝜆Θ𝑞^𝑡𝜆/𝑧_ℍ𝑑𝑡≤^∞_−∞𝐶₁𝑒^{𝑀log2|𝑞𝑡𝜆|}𝐶_𝜉1𝑒^{(𝜉1/2log|𝑞|)log2|𝑞𝑡𝜆/𝑧|}𝑑𝑡≤𝐿₁^∞_−∞𝑞^𝑡𝜆^{𝜉1log|𝑧|/log|𝑞|}𝑒^{(𝑀−𝜉1/2log|𝑞|)log2|𝑞𝑡𝜆|}𝑑𝑡, (4.12) 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 𝐿₁^{−log|𝜆|/log|𝑞|}_−∞𝑞^𝑡𝜆^{𝜉1log𝐴/log|𝑞|}𝑒^{(𝑀−𝜉1/2log|𝑞|)log2|𝑞𝑡𝜆|}𝑑𝑡+𝐿₁^∞_{|𝜆|𝑞−log/log}𝑞^𝑡𝜆^{𝜉1log𝐵/log|𝑞|}𝑒^{(𝑀−𝜉1/2log|𝑞|)log2|𝑞𝑡𝜆|}𝑑𝑡. (4.13) The result follows from this last expression.

Remark 4.4. If we let 𝑀=1/2log|𝑞|, then ℒ^𝜆_𝑞;1𝐹 will only remain holomorphic in ℛ_{𝜆,𝑞,𝛿}∩𝐷(0,𝑟₁) for certain 𝑟₁>0.
In the next proposition, we recall a commutation formula for the 𝑞-Laplace transform and the multiplication by a polynomial.

Proposition 4.5. Let 𝑉 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 𝑉𝑞^ℝ+∪𝐷̇_𝜌0 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 𝜙(𝑥)_ℍ<𝐶₁𝑒^{𝑀log2|𝑥|},𝑥∈𝐷̇_𝜌0∪𝑉𝑞^ℝ+. (4.14) Then, the function 𝑚𝜙(𝜏)=𝜏𝜙(𝜏) is holomorphic on 𝑉𝑞^ℝ+∪𝐷̇_𝜌0 and satisfies estimates in the shape above. Let 𝜆∈𝑉∩𝐷(0,𝜌₀) and 𝛿>0. One has the following equality: ℒ^𝜆_𝑞;1𝑚𝜙(𝑡)=𝑡ℒ^𝜆_𝑞;1𝜙𝑞𝑡 (4.15) for every 𝑡∈ℛ_{𝜆,𝑞,𝛿}.

Proof. It is direct to prove that 𝑚𝜙 is a holomorphic function in 𝑉𝑞^ℝ+∪𝐷̇_𝜌0 and also that 𝑚𝜙 verifies bounds as in (4.14). From (4.2) we have Θ(𝑥)=𝑥Θ(𝑥/𝑞), 𝑥∈ℂ^⋆, so ℒ^𝜆_𝑞;1(𝑡)=1𝑚𝜙𝜋_𝑞0^∞𝜆𝑚𝜙(𝜉)Θ(𝜉/𝑡)𝑑𝜉𝜉=1𝜋_𝑞0^∞𝜆𝜙(𝜉)Θ()=1𝜉/𝑡𝑑𝜉𝜋_𝑞0^∞𝜆𝜙(𝜉)(𝜉/𝑡)Θ𝜉/𝑞𝑡𝑑𝜉=𝑡ℒ^𝜆_𝑞;1𝜙𝑞𝑡, (4.16) for every 𝑡∈ℛ_{𝜆,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. Let I=(I₁,I₂) be a pair of open intervals in ℝ each one of length smaller than 1/4, and let U_I be the corresponding open bounded set in ℂ^⋆ defined by 𝑈_𝐼=𝑒^{2𝜋𝑢𝑖}𝑞^𝑣∈ℂ^⋆∶𝑢∈𝐼₁,𝑣∈𝐼₂. (4.17) Let ℐ be a finite family of tuple 𝐼 as above verifying(1)⋃_𝐼∈ℐ(𝑈_𝐼𝑞^−ℕ)=𝜈⧵{0}, where 𝜈 is a neighborhood of 0 in ℂ, (2)the open sets 𝑈_𝐼𝑞^−ℕ, 𝐼∈ℐ are four-by-four disjoint. Then, we say that (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ is a good covering.

Definition 4.7. Let (U_Iq^−ℕ)_I∈ℐ be a good covering. Let 𝛿>0. We consider a family of open bounded sets {(V_I)_I∈ℐ,𝒯} in ℂ^⋆ such that(1)there exists 1<𝜌₀ with 𝑉_𝐼∩𝐷(0,𝜌₀)≠∅, for all 𝐼∈ℐ,(2)for every 𝐼∈ℐ and 𝜏∈𝑉_𝐼𝑞^ℝ,|𝜏+1|>𝛿,(3)for every 𝐼∈ℐ,𝑡∈𝒯,𝜖_𝑢∈𝑈_𝐼, and 𝜆_𝑣∈𝑉_𝐼∩𝐷(0,𝜌₀), we have 𝜆1+^𝑣𝜖_𝑢𝑡𝑞^𝑟>𝛿, (4.18) for every 𝑟∈ℝ,(4)|𝑡|≤1 for every 𝑡∈𝒯. We say that the family {(𝑉_𝐼)_𝐼∈ℐ,𝒯} is associated to the good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ.
Let 𝑆≥1 be an integer. For every 0≤𝑘≤𝑆−1, let 𝑚_0,𝑘,𝑚_1,𝑘 be positive integers and 𝑏_𝑘(𝜖,𝑧)=∑_{𝑠∈𝐼𝑘}𝑏_𝑘𝑠(𝜖)𝑧^𝑠 a polynomial in 𝑧, where 𝐼_𝑘 is a subset of ℕ and 𝑏_𝑘𝑠(𝜖) are bounded holomorphic functions on some disc 𝐷(0,𝑟₀) in ℂ,0<𝑟₀≤1. Let (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ be a good covering such that 𝑈_𝐼𝑞^−ℕ⊆𝐷(0,𝑟₀) for every 𝐼∈ℐ.

Assumption B. We have 1𝑀<𝑞.2log (4.19)

Definition 4.8. Let 𝜌₀>1 such that 𝑉∩𝐷(0,𝜌₀)≠∅. Let Δ,𝑀>0 such that 𝑀<𝑀, and that (𝜖,𝜏)↦𝑊(𝜖,𝜏) be a bounded holomorphic function on (𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0 verifying|𝑊(𝜖,𝜏)|≤Δ𝑒𝑀_log²_|𝜏/𝜖|, (4.20) for every (𝜖,𝜏)∈(𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0. Assume moreover that 𝑊(𝜖,𝜏) can be extended to an analytic function (𝜖,𝜏)↦𝑊_𝑈𝑉(𝜖,𝜏) on 𝑈𝑞^−ℕ×(𝑉𝑞^ℝ+∪𝐷̇_𝜌0) and 𝑊_𝑈𝑉(𝜖,𝜏)≤Δ𝑒𝑀_log²_|𝜏/𝜖|, (4.21) for every (𝜖,𝜏)∈𝑈𝑞^−ℕ×(𝑉𝑞^ℝ+∪𝐷̇_𝜌0). One says that the set {𝑊,𝑊_𝑈𝑉,𝜌₀} is admissible.
Let ℐ be a finite family of indices. For every 𝐼∈ℐ, we consider the following singularly perturbed Cauchy problem: 𝜖𝑡𝜕^𝑆_𝑧𝑋_𝐼𝜖,𝑞𝑡,𝑧+𝜕^𝑆_𝑧𝑋_𝐼(𝜖,𝑡,𝑧)=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)𝑡𝜎_𝑞^𝑚0,𝑘𝜕^𝑘_𝑧𝑋_𝐼𝜖,𝑡,𝑧𝑞^{−𝑚1,𝑘} (4.22) with 𝑏_𝑘 as in (2.13), and with initial conditions 𝜕^𝑗_𝑧𝑋_𝐼(𝜖,𝑡,0)=𝜙_𝐼,𝑗(𝜖,𝑡),0≤𝑗≤𝑆−1, (4.23) where the functions 𝜙_𝐼,𝑗(𝜖,𝑡) are constructed as follows. Let {(𝑉_𝐼)_𝐼∈ℐ,𝒯} be a family of open sets associated to the good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ. For every 0≤𝑗≤𝑆−1 and 𝐼∈ℐ, let {𝑊_𝑗,𝑊_{𝑈𝐼,𝑉𝐼,𝑗},𝜌₀} be an admissible set. Let 𝜆_𝐼 be a complex number in 𝑉_𝐼∩𝐷(0,𝜌₀). We can assume that 𝑟₀<1<|𝜆_𝐼|. If not, we diminish 𝑟₀ as desired. We put 𝜙_𝐼,𝑗(𝜖,𝑡)∶=ℒ^𝜆𝐼_𝑞;1𝜏⟼𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(𝜖,𝜏)(𝜖,𝜖𝑡). (4.24)

Lemma 4.9. The function (𝜖,𝑡)↦𝜙_𝐼,𝑗(𝜖,𝑡), constructed as above, turns out to be holomorphic and bounded on 𝑈_𝐼𝑞^−ℕ×𝒯 for every 𝐼∈ℐ and all 0≤𝑗≤𝑆−1.

Proof. Let 𝐼∈ℐ and 0≤𝑗≤𝑆−1. From (4.21), one has 𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(𝜖,𝜏)≤Δ𝑒𝑀_log²_|𝜏/𝜖|=Δ𝑒𝑀_log²_|𝜖||𝜏|⁻²𝑀_log|𝜖|𝑒𝑀_log²_|𝜏|, (4.25) for every (𝜖,𝜏)∈𝑈_𝐼𝑞^−ℕ×(𝑉_𝐼𝑞^ℝ+∪𝐷̇_𝜌0). Let 𝜖∈𝑈_𝐼𝑞^−ℕ and 𝑀<𝑀₂<1/2log|𝑞|. Then, (4.25) can be upper bounded by Δexp(𝑀₂log²|𝜏|), for some Δ=Δ(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 𝜙_𝐼,𝑗 on 𝑈_𝐼𝑞^−ℕ×𝒯.
We now prove the boundness of 𝜙_𝐼,𝑗 in its domain of definition. One has 𝜙_𝐼,𝑗(𝜖,𝑡)=ℒ^𝜆𝐼_𝑞;1𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(𝜖,𝜖𝑡)≤ℒ^𝜆𝐼_𝑞;1,+𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(𝜖,𝜖𝑡)+ℒ^𝜆𝐼_𝑞;1,−𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(𝜖,𝜖𝑡), (4.26) for every (𝜖,𝑡)∈𝑈_𝐼𝑞^−ℕ×𝒯, where ℒ^𝜆𝐼_𝑞;1,+𝑊_{𝑈𝐼,𝑉𝐼,𝑗}()=𝑞𝜖,𝜖𝑡log𝜋_𝑞^∞₀𝑊_{𝑈𝐼,𝑉𝐼,𝑗}𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼ℒ/𝜖𝑡𝑑𝑠,_{𝜆𝐼𝑞;1}𝑊_{𝑈𝐼,𝑉𝐼,𝑗,−}()=𝑞𝜖,𝜖𝑡log𝜋_𝑞⁰_−∞𝑊_{𝑈𝐼,𝑉𝐼,𝑗}𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼/𝜖𝑡𝑑𝑠. (4.27) We only give bounds for the first integral. The estimates for the second one can be deduced following a similar procedure.
Let 0<𝜉<1 such that 𝑀<𝜉/2log|𝑞|. From Corollary 4.2 and (4.21) we deduce ℒ^𝜆𝐼_𝑞;1,+𝑊_{𝑈𝐼,𝑉𝐼,𝑗}(≤𝑞𝜖,𝜖𝑡)log𝜋_𝑞^∞0𝑊_{𝑈𝐼,𝑉𝐼,𝑗}𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼≤𝑞Δ/𝜖𝑡𝑑𝑠log𝜋_𝑞𝐶_𝜉^∞0𝑒𝑀_log²_|𝑞^𝑠_{𝜆𝐼/𝜖|}𝑒^{𝜉log2|𝑞𝑠𝜆𝐼/𝜖𝑡|/2log|𝑞|}≤𝑞Δ𝑑𝑠log𝜋_𝑞𝐶_𝜉𝑒⁽𝑀_{−𝜉/2log|𝑞|)log}²_{|𝜆𝐼/𝜖|}𝑒^{−𝜉log2|𝑡|/2log|𝑞|}𝑒^{𝜉log|𝜆𝐼/𝜖|log|𝑡|/log|𝑞|}×^∞₀𝑒²⁽𝑀_{−(𝜉/2log|𝑞|))log}²_|𝑞|𝑠²𝑒⁽𝑀_{−𝜉/2log|𝑞|)log|𝑞|log|𝜆𝐼/𝜖|𝑠}𝑒^{𝜉log|𝑡|𝑠}𝑑𝑠≤𝐶_𝑗, (4.28) for some 𝐶_𝑗>0 which does not depend on 𝜖 nor 𝑡.

The following assumption is related to technical reasons appearing in the proof of Lemma 4.9 and Theorem 4.11.

Assumption C. There exist 𝑎₁,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+𝐶²/4𝜉log|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 of 𝑞 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 𝜖𝑡𝜕²_𝑧𝑋_𝐼𝜖,𝑞𝑡,𝑧+𝜕²_𝑧𝑋_𝐼(𝜖,𝑡,𝑧)=𝑏₀₀(𝜖)+𝑏₀₁(𝜖)𝑧𝑡²𝑋_𝐼𝜖,𝑞²𝑡,𝑧𝑞⁻³⁰+𝑏₁₀(𝜖)𝑡𝜕_𝑧𝑋_𝐼𝜖,𝑞𝑡,𝑧𝑞⁻¹⁰, (4.29) with 𝑏₀₀,𝑏₀₁,𝑏₁₀ being holomorphic functions near the origin.

Theorem 4.11. Let Assumption A be fulfilled by the integers 𝑚_0,𝑘,𝑚_1,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 𝑋_𝐼(𝜖,𝑡,𝑧) which is holomorphic and bounded in 𝑈_𝐼𝑞^−ℕ×𝒯×ℂ.
Moreover, for every 𝜌>0, if 𝐼,𝐼^′∈ℐ are such that 𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ≠∅, then there exists a positive constant 𝐶₁=𝐶₁(𝜌)>0 such that 𝑋_𝐼(𝜖,𝑡,𝑧)−𝑋_𝐼′(𝜖,𝑡,𝑧)≤𝐶₁𝑒^{−(1/𝐴)log2|𝜖|},(𝜖,𝑡,𝑧)∈𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ×𝒯×𝐷0,𝜌, (4.30) with 1/𝐴=(1−𝜉)(𝜉/2log|𝑞|−𝑀) with 𝜉,𝜉 chosen as in Assumption C.

Proof. Let 𝛿>0 and 𝐼∈ℐ. We consider the Cauchy problem (3.8) with initial conditions (𝜕^𝑗_𝑧𝑊)(𝜖,𝜏,0)=𝑊_𝑗(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≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽)∈𝒪((𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0)[[𝑧]] and positive constants 𝐶₃>0 and 0<𝛿₁<1 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₃𝑞𝛽!^2𝐴1𝑆𝛿₁^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2},𝛽≥0, (4.31) for (𝜖,𝜏)∈(𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0.
Moreover, from Theorem 2.4 we get that the coefficients 𝑊_𝛽(𝜖,𝜏) can be extended to holomorphic functions defined in 𝑈_𝐼𝑞^−ℕ×𝑉_𝐼𝑞^ℝ+ and also the existence of positive constants 𝐶₂ and 0<𝛿₂<1 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₂𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝜏𝜖^𝐶𝛽𝑒^{𝑀log2|𝜏/𝜖|}𝑞^{−𝐴1𝛽2},𝛽≥0, (4.32) for (𝜖,𝜏)∈𝑈_𝐼𝑞^−ℕ×𝑉_𝐼𝑞^ℝ+.
We choose 𝜆_𝐼∈𝑉_𝐼∩𝐷(0,𝜌₀). In the following estimates we will make use of the fact that |𝜖|≤|𝜆_𝐼| for every 𝜖∈𝐷(0,𝑟₀⧵{0}). Proposition 4.3 allows us to calculate the 𝑞-Laplace transform of 𝑊_𝛽 with respect to 𝜏 for every 𝛽≥0,ℒ^𝜆𝐼_𝑞;1(𝑊_𝛽)(𝜖,𝜏). It defines a holomorphic function in 𝑈_𝐼𝑞^−ℕ×ℛ_{𝜆𝐼,𝑞,𝛿}. From the fact that {(𝑉_𝐼)_𝐼∈ℐ,𝒯} is chosen to be a family associated to the good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ we derive that the function (𝜖,𝑡)⟼ℒ^𝜆𝐼_𝑞;1𝑊_𝛽(𝜖,𝜖𝑡) (4.33) is a holomorphic and bounded function defined in 𝑈_𝐼𝑞^−ℕ×𝒯. We can define, at least formally, 𝑋_𝐼(𝜖,𝑡,𝑧)∶=_𝛽≥0ℒ^𝜆𝐼_𝑞;1𝑊_𝛽()𝑧𝜖,𝜖𝑡^𝛽,𝛽! (4.34) in 𝒪(𝑈_𝐼𝑞^−ℕ×𝒯)[[𝑧]]. If 𝑋_𝐼(𝜖,𝑡,𝑧) were a holomorphic function in 𝑈_𝐼𝑞^−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 𝑈_𝐼𝑞^−ℕ×𝒯×ℂ. Let (𝜖,𝑡)∈𝑈_𝐼𝑞^−ℕ×𝒯 and 𝛽≥0. We have ℒ^𝜆𝐼_𝑞;1𝑊_𝛽(𝜖,𝜖𝑡)≤ℒ^𝜆𝐼_𝑞;1,+𝑊_𝛽(𝜖,𝜖𝑡)+ℒ^𝜆𝐼_𝑞;1,−𝑊_𝛽(𝜖,𝜖𝑡), (4.35) where ℒ^𝜆𝐼_𝑞;1,+𝑊_𝛽()=𝑞𝜖,𝜖𝑡log𝜋_𝑞^∞₀𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼ℒ/𝜖𝑡𝑑𝑠,_{𝜆𝐼𝑞;1,−}𝑊_𝛽()=𝑞𝜖,𝜖𝑡log𝜋_𝑞⁰_−∞𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼/𝜖𝑡𝑑𝑠. (4.36) We now establish bounds for both integrals: ℒ^𝜆𝐼_𝑞;1,+𝑊_𝛽(𝜖,𝜖𝑡)≤log𝑞𝜋_𝑞^∞0𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼/𝜖𝑡𝑑𝑠. (4.37)
Let 0<𝜉<1 as in Assumption C. From (4.32) and (4.7), the previous integral is bounded by log𝑞𝜋_𝑞^∞0𝐶₂𝛽!𝑞^2𝐴1𝑆/𝛿₂^𝛽𝑞^𝑠𝜆_𝐼/𝜖^𝐶𝛽𝑒^{𝑀log2|𝑞𝑠𝜆𝐼/𝜖|}𝑞^{−𝐴1𝛽2}𝐶_𝜉exp𝜉log²𝑞^𝑠𝜆_𝐼𝑞≤/𝜖𝑡/2log𝑑𝑠log𝑞𝜋_𝑞𝐶₂𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝜆_𝐼𝜖^𝐶𝛽𝑞^{−𝐴1𝛽2∞}₀𝑞^𝐶𝑠𝛽𝑒^{𝑀log2|𝑞𝑠𝜆𝐼/𝜖|}exp𝜉log²𝑞^𝑠𝜆_𝐼/𝜖𝑡/2log𝑞𝑑𝑠. (4.38)
Let 𝑎₁,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 𝒜^∞₀𝑞^−𝐵𝑠2𝑒^{(𝑀−𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖|}𝑒^{((2𝑀log|𝑞|−𝜉)log|𝜆𝐼/𝜖|+𝜉log|𝑡|)𝑠}𝑑𝑠, (4.39) where 0<𝐵=𝜉/2−𝑀log|𝑞|−𝐶𝑎₁/2𝑎₂ and 𝒜=log𝑞𝜋_𝑞𝐶₂𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝜆_𝐼𝜖^𝐶𝛽𝑞^{−𝐴1𝛽2+𝐶𝑎2𝛽2/2𝑎1}𝑒^{−𝜉log2|𝑡|/2log|𝑞|}𝑒^{𝜉log|𝜆𝐼/𝜖|log|𝑡|/log|𝑞|}. (4.40)
The previous integral is uniformly bounded for 𝜖∈𝐷(0,𝑟₀)⧵{0} and 𝑡∈𝒯 from the hypotheses made on these sets. The expression in (4.39) can be bounded by log𝑞𝜋_𝑞𝐶^′₂𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝜆_𝐼𝜖^𝐶𝛽𝑒^{(𝑀−𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖|}𝑞^{−𝐴1𝛽2+𝐶𝑎2𝛽2/2𝑎1}×𝑒^{−𝜉log2|𝑡|/2log|𝑞|}𝑒^{𝜉log|𝜆𝐼/𝜖|log|𝑡|/log|𝑞|}, (4.41) for an appropriate constant 𝐶^′₂>0.
The function 𝑠↦𝑠^𝛾𝛽𝑒^{−𝛼log2(𝑠)} takes its maximum at 𝑠=𝑒^{𝛾𝛽/(2𝛼)} so each element in the image set is bounded by 𝑒^{(𝛾𝛽)2/(4𝛼)}. Taking this to the expression above we get ℒ^𝜆𝐼_𝑞;1,+𝑊_𝛽(𝜖,𝜖𝑡)≤log𝑞𝜋_𝑞𝐶₂^′′𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝑞^{−𝐴1𝛽2+𝐶𝑎2𝛽2/2𝑎1+𝐶2𝛽2/4log|𝑞|(𝜉/(2log|𝑞|)−𝑀)}, (4.42) for certain 𝐶₂^′′>0.
Assumption (C.3) applied to the last term in the previous expression allows us to deduce that the sum _𝛽≥0ℒ^𝜆𝐼_𝑞;1,+𝑊_𝛽(|𝑧|𝜖,𝜖𝑡)^𝛽𝛽! (4.43) converges in the variable 𝑧 uniformly in the compact sets of ℂ.
We now study ℒ^𝜆𝐼_𝑞;1,−𝑊_𝛽(𝜖,𝜖𝑡). We have ℒ^𝜆𝐼_𝑞;1,−𝑊_𝛽(𝜖,𝜖𝑡)≤log𝑞𝜋_𝑞^0−∞𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼/𝜖𝑡𝑑𝑠. (4.44) From (3.11) and (4.7) the previous integral is bounded by log𝑞𝜋_𝑞^0−∞𝐶₃𝛽!𝑞^2𝐴1𝑆/𝛿₁^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2|𝑞𝑠𝜆𝐼/𝜖|}𝑞^{−𝐴1𝛽2}𝐶_𝜉𝑒^{𝜉log2|𝑞𝑠𝜆𝐼/𝜖𝑡|/2log|𝑞|}𝑑𝑠. (4.45)
Similar calculations to those in the first part of the proof resting on Assumption C can be made so that the series _𝛽≥0ℒ^𝜆𝐼_𝑞;1,−𝑊_𝛽()𝑧𝜖,𝜖𝑡^𝛽𝛽! (4.46) is uniformly convergent with respect to the variable 𝑧 in the compact sets of ℂ, for (𝜖,𝑡)∈𝑈_𝐼𝑞^−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.
Let 𝐼,𝐼′∈ℐ such that 𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ≠∅ and 𝜌>0. For every (𝜖,𝑡,𝑧)∈(𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ)×𝒯×𝐷(0,𝜌) we have 𝑋_𝐼(𝜖,𝑡,𝑧)−𝑋_𝐼′(𝜖,𝑡,𝑧)≤_𝛽≥0ℒ^𝜆𝐼_𝑞;1𝑊_𝛽(𝜖,𝜖𝑡)−ℒ^𝜆𝐼′_𝑞;1𝑊_𝛽(𝜌𝜖,𝜖𝑡)^𝛽.𝛽! (4.47)
We can write ℒ^𝜆𝐼_𝑞;1𝑊_𝛽(𝜖,𝜖𝑡)−ℒ^𝜆𝐼′_𝑞;1𝑊_𝛽()=1𝜖,𝜖𝑡𝜋_𝑞𝛾1𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉−_𝛾2𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉+_{𝛾3−𝛾4}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉, (4.48) where the path 𝛾₁ is given by 𝑠∈(0,∞)↦𝑞^𝑠𝜆_𝐼, 𝛾₂ by 𝑠∈(0,∞)↦𝑞^𝑠𝜆_𝐼′, 𝛾₃ by 𝑠∈(−∞,0)↦𝑞^𝑠𝜆_𝐼, and 𝛾₄ by 𝑠∈(−∞,0)↦𝑞^𝑠𝜆_𝐼′.
Without loss of generality, we can assume that |𝜆_𝐼|=|𝜆_𝐼′|.
For the first integral we deduce ₁𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉≤log𝑞^∞₀𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼/𝜖𝑡𝑑𝑠. (4.49) Similar estimates as in the first part of the proof lead us to bound the right part of the previous inequality by 𝐶₂^′′′𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₂^𝛽𝜆_𝐼𝜖^𝐶𝛽𝑞^{−𝐴1𝛽2+(𝐶𝑎2/2𝑎1)𝛽2}𝑒^{(𝑀−𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖|}, (4.50) for certain 𝐶₂^′′′>0. For any 𝜉∈(0,1) we have 𝜆_𝐼𝜖^𝐶𝛽𝑒^{𝜉(𝑀−𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖|}≤𝑒^{𝐶2𝛽2/4𝜉(𝜉/2log|𝑞|−𝑀)},𝛽≥0. (4.51) This yields _𝛾1𝑊_𝛽𝜖,𝑞^𝑠𝜆_𝐼Θ𝑞^𝑠𝜆_𝐼≤𝐶/𝜖𝑡𝑑𝑠_2′′′𝐶_𝜉𝛽!|𝑞|^2𝐴1𝑆𝛿₂^𝛽𝑞^{(−𝐴1+(𝐶𝑎2/2𝑎1)+𝐶2/4𝜉log|𝑞|((𝜉/2log|𝑞|)−𝑀))𝛽2}𝑒^{(1−𝜉)(𝑀−(ss))log2|𝜆𝐼/𝜖|}. (4.52) We choose 𝜉 as in Assumption C.
The integral corresponding to the path 𝛾₂ can be bounded following identical steps.
We now give estimates concerning 𝛾₃−𝛾₄. It is worth saying that the function in the integrand is well defined for (𝜖,𝜏)∈(𝐷(0,𝑟₀)⧵{0})×𝐷̇_𝜌0 and does not depend on the index 𝐼∈ℐ. This fact and the Cauchy theorem allow us to write for any 𝑛∈ℕ_Γ𝑛𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=0, (4.53) where Γ_𝑛=𝛾_𝑛,1+𝛾₅−𝛾_𝑛,2−𝛾_𝑛,3 is the closed path defined in the following way: 𝑠∈[−𝑛,0]↦𝛾_𝑛,1(𝑠)=𝜆_𝐼𝑞^𝑠,𝛾₅ is the arc of circumference from 𝜆_𝐼 to 𝜆_𝐼′, 𝑠∈[−𝑛,0]↦𝛾_𝑛,2(𝑠)=𝜆_𝐼′𝑞^𝑠, and 𝛾_𝑛,3 is the arc of circumference from 𝜆_𝐼𝑞^−𝑛 to 𝜆_𝐼′𝑞^−𝑛. Taking 𝑛→∞ we derive 0=lim_{𝑛→∞Γ𝑛}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=lim_{𝑛→∞𝛾𝑛,1+𝛾5−𝛾𝑛,2}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉−lim_{𝑛→∞𝛾𝑛,3}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉. (4.54) Usual estimates lead us to prove that lim_{𝑛→∞𝛾𝑛,3}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=0. (4.55) Moreover, lim_{𝑛→∞𝛾𝑛,1+𝛾5−𝛾𝑛,2}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=_{𝛾3+𝛾5−𝛾4}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉. (4.56) From (4.54), (4.55), and (4.56) we obtain _{𝛾3−𝛾4}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=_−𝛾5𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉=^𝜃𝐼_𝜃𝐼′𝑊_𝛽𝜖,𝜆_𝐼𝑒^𝑖𝜃Θ𝜆_𝐼𝑒^𝑖𝜃/𝜖𝑡𝑑𝜃, (4.57) where 𝜃_𝐼=arg(𝜆_𝐼),𝜃_𝐼′=arg(𝜆_𝐼′). 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 length𝛾₅𝐶₃𝐶_𝜉𝑞𝛽!^2𝐴1𝑆𝛿₁^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2|𝜆𝐼/𝜖|}𝑒^{(𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖𝑡|}𝑞^{−𝐴1𝛽2}≤𝐶^′₃𝑞𝛽!^2𝐴1𝑆𝛿₁^𝛽|𝜖|^−𝐶𝛽𝑒^{(𝑀−𝜉/2log|𝑞|)log2|𝜆𝐼/𝜖|}𝑞^{−𝐴1𝛽2}≤𝐶^′₃𝑞𝛽!^2𝐴1𝑆𝛿₁^𝛽|𝜖|^−𝐶𝛽𝑒^{𝜉(𝑀−𝜉/2log|𝑞|)log2|𝜖|}𝑞^{−𝐴1𝛽2}𝑒^{(1−𝜉)(𝑀−𝜉/2log|𝑞|)log2|𝜖|} (4.58) for adequate positive constants 𝐶₃,𝐶^′₃. From the standard estimates we achieve _{𝛾3−𝛾4}𝑊_𝛽(𝜖,𝜉)Θ(𝜉/𝜖𝑡)𝑑𝜉𝜉≤𝐶^′₃𝑞𝛽!^2𝐴1𝑆𝛿₁^𝛽𝑞^{−𝐴1𝛽2}𝑒^{𝐶2/4𝜉(𝜉/2log|𝑞|−𝑀)𝛽2}𝑒^{(1−𝜉)(𝑀−𝜉/2log|𝑞|)log2|𝜖|}. (4.59)
From (4.47), (4.48), (4.52), (4.59) and Assumption (C.3) we conclude the existence of a positive constant 𝐶^′₁>0 such that 𝑋_𝐼(𝜖,𝑡,𝑧)−𝑋_𝐼′(𝜖,𝑡,𝑧)≤𝐶^′_1𝛽≥0𝛽!|𝑞|^2𝐴1𝑆𝛿₀^𝛽𝑞^{(−𝐴1+(𝐶𝑎2/2𝑎1)+𝐶2/4𝜉log|𝑞|(𝜉/2log|𝑞|−𝑀))𝛽2}×𝑒^{(1−𝜉)(𝑀−𝜉/2log|𝑞|)log2|𝜖|}𝜌^𝛽𝛽!≤𝐶₁𝑒^{(1−𝜉)(𝑀−𝜉/2log|𝑞|)log2|𝜖|}, (4.60) for every (𝜖,𝑡,𝑧)∈(𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ)×𝒯×𝐷(0,𝜌), with 𝛿₀=min{𝛿₁,𝛿₂}.

5. A 𝑞-Gevrey Malgrange-Sibuya-Type Theorem

In this section we obtain a 𝑞-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,𝑋_𝐼 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, although 𝑞-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].

Let ℍ be a complex Banach space.

Definition 5.1. Let U be a bounded open set in ℂ^⋆ and 𝐴>0. One says a holomorphic function 𝑓∶𝑈𝑞^−ℕ→ℍ admits 𝑓=∑_𝑛≥0𝑓_𝑛𝜖^𝑛∈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 𝑓(𝜖)−^𝑁_𝑛=0𝑓_𝑛𝜖^𝑛_ℍ≤𝐶₁𝐻^𝑁𝑞^{𝐴(𝑁2/2)}|𝜖|^𝑁+1(𝑁+1)!,𝑁≥0, (5.1) for every 𝜖∈𝐾𝑞^−ℕ.
The following proposition can be found, under slight modifications in Section  4 of [17].

Proposition 5.2. Let 𝐴>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 𝑓∈ℍ[[𝜖]] as its 𝑞-Gevrey asymptotic expansion of type 𝐴 in 𝑈𝑞^−ℕ. Then, if 𝑓^(𝑘) stands for the 𝑘th formal derivative of 𝑓 for every 𝑘∈ℕ, one has that 𝑓^(𝑘) admits 𝑓^(𝑘) as its 𝑞-Gevrey asymptotic expansion of type 𝐴 in 𝑈𝑞^−ℕ.

Proposition 5.3. Let 𝐴>0, and let 𝑓∶𝑈𝑞^−ℕ→ℍ a holomorphic function in 𝑈𝑞^−ℕ. Then, (i)if 𝑓 admits 0 as its 𝑞-Gevrey expansion of type 𝐴, then for every compact set 𝐾⊆𝑈 there exists 𝐶₁>0 with 𝑓(𝜖)_ℍ≤𝐶₁𝑒^−(1/𝑎_{)(1/2log|𝑞|)log}²_|𝜖|, (5.2) for every 𝜖∈𝐾𝑞^−ℕ and every 𝑎>𝐴;(ii)if for every compact set 𝐾⊆𝑈 there exists 𝐶₁>0 with 𝑓(𝜖)_ℍ≤𝐶₁𝑒^{−(1/𝐴)(1/2log|𝑞|)log2|𝜖|}, (5.3) for every 𝜖∈𝐾𝑞^−ℕ, then 𝑓 admits 0 as its 𝑞-Gevrey asymptotic expansion of type 𝑎 in 𝑈𝑞^−ℕ, for every 𝑎>𝐴.

Proof. Let 𝐶₁,𝐻,𝐴>0 and 𝜖∈ℂ^⋆. The function 𝐺(𝑥)=𝐶₁(𝐻)𝑞𝐴explog𝑥+log2𝑥²+(𝑥+1)log|𝜖| (5.4) 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 (U_Iq^−ℕ)_I∈ℐ be a good covering at 0 (see Definition 4.6) and 𝑔_𝐼,𝐼′∶𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ→ℍ a holomorphic function in 𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ for 𝐼,𝐼^′∈ℐ when the intersection is not empty. The family (𝑔_𝐼,𝐼′)_{(𝐼,𝐼′)∈ℐ2} is a q-Gevrey ℍ-cocycle of type 𝐴>0 attached to a good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ if the following properties are satisfied.(1)𝑔_𝐼,𝐼′ admits 0 as its 𝑞-Gevrey asymptotic expansion of type 𝐴>0 on 𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ for every (𝐼,𝐼^′)∈ℐ.(2)𝑔_𝐼,𝐼′(𝜖)=−𝑔_𝐼′,𝐼(𝜖) for every (𝐼,𝐼^′)∈ℐ, and 𝜖∈𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ.(3)One has 𝑔_{𝐼,𝐼′′}(𝜖)=𝑔_𝐼,𝐼′(𝜖)+𝑔_{𝐼′,𝐼′′}(𝜖) for all 𝜖∈𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ∩𝑈_𝐼′′𝑞^−ℕ, 𝐼,𝐼^′,𝐼^′′∈ℐ.

Let 𝜌>0 and 𝒯⊆ℂ^⋆ be an open and bounded set. ℍ_𝒯,𝜌 stands for the Banach space of holomorphic and bounded functions in 𝒯×𝐷(0,𝜌) with the supremum norm.

Proposition 5.5. Let 𝜌>0. We consider the family (𝑋_𝐼(𝜖,𝑡,𝑧))_𝐼∈ℐ constructed in Theorem 4.11. Then, the set of functions (𝑔_𝐼,𝐼′(𝜖))_{(𝐼,𝐼′)∈ℐ2} defined by 𝑔_𝐼,𝐼′(𝜖)∶=(𝑡,𝑧)∈𝒯×𝐷0,𝜌⟼𝑋_𝐼′(𝜖,𝑡,𝑧)−𝑋_𝐼(𝜖,𝑡,𝑧) (5.5) for 𝐼,𝐼^′∈ℐ is a 𝑞-Gevrey ℍ_𝒯,𝜌-cocycle of type 𝐴 for every 1𝐴>𝐴∶=1−𝜉𝑞𝑞=1𝜉/2log−𝑀2log1−𝜉𝑞,𝜉−2𝑀log (5.6) attached to the good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ.

Proof. The first property in Definition 5.4 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 function 𝑤∶[0,∞)→[0,∞) is a weight function if it satisfies the following:(𝛼) there exists 𝑘≥1 with 𝑤(2𝑡)≤𝑘(𝑤(𝑡)+1) for all 𝑡≥0,(𝛽)∫^∞₀𝑤(𝑡)/(1+𝑡²)𝑑𝑡<∞, (𝛾)lim_𝑡→∞(log𝑡)/𝑤(𝑡)=0, (𝛿)𝜙∶𝑡↦𝑤(𝑒^𝑡) is convex. The Young conjugate associated to 𝜙,𝜙^⋆∶[0,∞)→ℝ is defined by 𝜙^⋆𝑦∶=sup𝑥𝑦−𝜙(𝑥)∶𝑥≥0. (5.7)

Definition 5.7. Let 𝐾 be a nonempty compact set in ℝ². A jet on 𝐾 is a family 𝐹=(𝑓^𝛼)_𝛼∈ℕ2 where 𝑓^𝛼∶𝐾→ℂ is a continuous function on K for each 𝛼∈ℕ².
Let 𝑤 be a weight function. A jet 𝐹=(𝑓^𝛼)_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 𝑓_𝐾,1/𝑚∶=sup_{𝑥∈𝐾,𝛼∈ℕ2}𝑓^𝛼(𝑥−1)exp𝑚𝜙^⋆(𝑚|𝛼|)≤𝑀, (5.8) and for every 𝑙∈ℕ,𝛼∈ℕ² with |𝛼|≤𝑙 and 𝑥,𝑦∈𝐾 one has 𝑅^𝑙_𝑥𝐹_𝛼𝑦≤𝑀𝑥−𝑦^{𝑙+1−|𝛼|}(|𝛼!1𝑙+1−|)exp𝑚𝜙^⋆(𝑚(𝑙+1)), (5.9) where (𝑅^𝑙_𝑥𝐹)_𝛼(𝑦)∶=𝑓^𝛼(𝑦)−∑_{|𝛼+𝛽|≤𝑙}(1/𝛽!)𝑓^𝛼+𝛽(𝑥)(𝑦−𝑥)^𝛽.

ℰ_{𝑤}(𝐾) denotes the linear space of 𝑤-Whitney jets on 𝐾.

Definition 5.8. Let 𝐾⊆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, (𝑓^𝛼)_𝛼∈ℕ2 such that 𝑓^(0,0)=𝑓.
For an open set Ω∈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 in ℰ_{𝑤}(𝐾) can be extended to an element in ℰ_{𝑤}(ℝ²).

Theorem 5.9 (Corollary  3.10, [13]). For a given weight function 𝑤, the following statements are equivalent.(1)For every nonempty closed set 𝐾 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,lim_𝜖→0+lim_𝑡→∞𝜖𝑤(𝑡)𝑤(𝜖𝑡)=0. (5.10)

Let 𝑘₁=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 of 𝑤₀ one has𝜙^⋆_𝑤0𝑦=sup𝑥𝑦−𝜙_𝑤0(𝑥)𝑥∶𝑥≥0=sup𝑥𝑦−²𝑞𝑞𝑦4log∶𝑥≥0=log²,𝑦≥0. (5.11)

The spaces appearing in Definition 5.7 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=(𝑓^𝛼)_𝛼∈ℕ2 on 𝐾 such that there exist 𝑚∈ℕ,𝑀>0 with𝑓^𝛼(𝑥)≤𝑀𝑞^𝑚|𝛼|2,𝑥∈𝐾,𝛼∈ℕ², (5.12)

and such that for every 𝑙∈ℕ and 𝛼∈ℕ² with |𝛼|≤𝑙 one has𝑅^𝑙_𝑥𝐹_𝛼𝑦≤𝑀𝑥−𝑦^{𝑙+1−|𝛼|}(|𝛼!𝑞𝑙+1−|)^{𝑚(𝑙+1)2},𝑥,𝑦∈𝐾. (5.13)

We derive that ℰ_{𝑤0}(𝐾) consists of the Whitney jets on 𝐾 such that there exist 𝐶₁,𝐻>0 with 𝑓^𝛼(𝑥)≤𝐶₁𝐻^|𝛼|𝑞^{𝐴(|𝛼|2/2)},𝑥∈𝐾,𝛼∈ℕ², (5.14) and for every 𝑥,𝑦∈𝐾 and all 𝑙∈ℕ,𝛼∈ℕ² with |𝛼|≤𝑙,𝑅^𝑙_𝑥𝐹_𝛼𝑦≤𝐶₁𝐻^𝑙𝑞^{𝐴(𝑙2/2)}𝑥−𝑦^{𝑙+1−|𝛼|}(|𝛼!.𝑙+1−|) (5.15)

Theorem 5.11. 𝑤₀ is a strong weight function so that Theorem 5.9 holds.

Proof. We have lim_𝜖→0+lim_𝑡→∞𝜖𝑤(𝑡)𝑤(𝜖𝑡)=lim_𝜖→0+lim_𝑡→∞𝜖𝑘₁log²(𝑡)𝑘₁log²(𝜖𝑡)=lim_𝜖→0+𝜖=0. (5.16)

Remark 5.12. A continuous function 𝑓 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 𝑓^𝑘𝑥,𝑦=𝜕^𝑘1_𝑥𝜕^𝑘2_𝑦𝑓𝑥,𝑦, (5.17) 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. Let 𝑈 be an open set in ℂ^⋆ and 𝑓∶𝑈𝑞^−ℕ→ℍ a holomorphic function with 𝑓=∑_ℎ≥0𝑎_ℎ𝜖^ℎ∈ℍ[[𝜖]] being its 𝑞-Gevrey asymptotic expansion of type 𝐴>0 in 𝑈𝑞^−ℕ. Then, for any 𝑛∈ℕ, the family 𝜕^𝑛_𝜖𝑓(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 𝜕^𝑘_𝜖𝑓𝜖_𝑎−^𝑚−𝑘_ℎ=0𝜕_𝜖^𝑘+ℎ𝑓𝜖_𝑏𝜖ℎ!^𝑎−𝜖_𝑏^ℎ_ℍ≤𝐶₁𝐻^𝑚𝑞^{𝐴(𝑚2/2)}𝜖_𝑎−𝜖_𝑏^{𝑚+1−𝑘}()!,𝑚+1−𝑘 (5.18) for every 𝜖_𝑎,𝜖_𝑏∈𝐾𝑞^−ℕ∪{0}. Here, one writes 𝜕^𝑙_𝜖𝑓(0)=𝑙!𝑎_𝑙  for 𝑙∈ℕ.

Proof. We will first state the result when 𝜖_𝑏=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.
Let 𝑚∈ℕ, 𝐾 be a compact set in 𝑈, and consider another compact set 𝐾₁ such that 𝐾⊆𝐾₁⊆𝑈. We define 𝑅_𝑚(𝜖)∶=𝜖^−𝑚−1𝑓(𝜖)−^𝑚_ℎ=0𝜕^ℎ_𝜖𝑓(0)𝜖ℎ!^ℎ,𝜖∈𝐾𝑞^−ℕ, (5.19) where 𝜕^ℎ_𝜖𝑓(0) denotes the limit of 𝜕^ℎ_𝜖𝑓(𝜖) for 𝜖∈𝐾𝑞^−ℕ tending to 0. Then, we have that 𝜕_𝜖𝑓(𝜖)=^𝑚_ℎ=1𝜕^ℎ_𝜖𝑓(0)ℎ!ℎ𝜖^ℎ−1+𝜕_𝜖𝑅_𝑚(𝜖)𝜖^𝑚+1+(𝑚+1)𝑅_𝑚(𝜖)𝜖^𝑚. (5.20) Moreover, from Definition 5.1, there exist 𝐶,𝐻>0 such that ‖𝑅_𝑚(𝜖)‖≤𝐶𝐻^𝑚|𝑞|^{𝐴(𝑚2/2)}/(𝑚+1)! for every 𝜖∈𝐾₁𝑞^−ℕ.
Lemma 5.14 (Lemma  4.4.1 [17]). There exists 𝜌>0 such that 𝐷(𝜖,𝜌|𝜖|)⊆𝐾₁𝑞^−ℕ for every 𝜖∈𝐾𝑞^−ℕ.
The Cauchy’s integral formula and 𝑞-Gevrey expansion of 𝑓 guarantee the existence of a positive constant 𝐶₂>0 such that 𝜕_𝜖𝑅_𝑚(𝜖)_ℍ≤𝐶₂𝐻^𝑚𝑞^{𝐴(𝑚2/2)}()!1𝑚+1𝜌|𝜖|,𝜖∈𝐾𝑞^−ℕ, (5.21) This yields the existence of 𝐶₃>0 such that 𝜖^−𝑚𝜕_𝜖𝑓(𝜖)−^𝑚−1_ℎ=0𝜕_𝜖^ℎ+1𝑓(0)𝜖ℎ!^ℎ_ℍ≤𝜕_𝜖𝑅_𝑚(𝜖)_ℍ|𝜖|+(𝑚+1)𝑅_𝑚(𝜖)_ℍ≤𝐶₂𝐴^𝑚₁𝑞^{𝐴(𝑚2/2)}𝑚!,𝜖∈𝐾𝑞^−ℕ. (5.22) An induction reasoning is sufficient to conclude the proof for every 𝑚≥0.
We now study the case where 𝜖_𝑏≠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 𝜕^𝑘_𝜖𝑓.
If 𝜖_𝑏≠0 we treat two cases:
if |𝜖_𝑎−𝜖_𝑏|≤𝜌|𝜖_𝑏|, then [𝜖_𝑎,𝜖_𝑏] is contained in 𝐾₁𝑞^−ℕ and we conclude from the Cauchy integral formula.
If |𝜖_𝑎−𝜖_𝑏|>𝜌|𝜖_𝑏|, then we bear in mind that the result is obvious when 𝑓 is a polynomial and write 𝑓(𝜖)=𝜖^𝑚+1𝑅_𝑚(𝜖)+𝑝(𝜖) where 𝑝(𝜖)=∑^𝑚_ℎ=0(𝜕^ℎ_𝜖𝑓(0)/ℎ!𝜖^ℎ). So, it is sufficient to prove (5.18) when 𝑓(𝜖)∶=𝜖^𝑚+1𝑅_𝑚(𝜖). The result follows from 𝑞-Gevrey bounds for ‖𝜕^𝑘_𝜖𝑅_𝑚‖_ℍ, 𝑘=0,…,𝑛 and usual estimates.

The following lemma generalizes Lemma  6 in [4].

Lemma 5.15. Let 𝑓∶𝑈𝑞^−ℕ→ℍ be a holomorphic function having 𝑓(𝜖)=∑_ℎ≥0𝑎_ℎ𝜖^ℎ∈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 𝐾^′=𝜖₁,𝜖₂∈ℝ²∶𝜖₁+𝑖𝜖₂∈𝐾𝑞^−ℕ∪{0}. (5.23)

Proof. We consider the set of functions (𝜙^{(𝑘1,𝑘2)})_{(𝑘1,𝑘2)∈ℕ2} defined by 𝜙^{(𝑘1,𝑘2)}∶=𝑖^𝑘2𝜕^𝑘1+𝑘2_𝜖𝑓(𝜖),𝑘₁,𝑘₂∈ℕ²,𝜖₁,𝜖₂∈𝐾^′. (5.24) 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, 1𝑖^𝑘2𝜙^{(𝑘1,𝑘2)}𝑥₁,𝑦₁−^{𝑚−|(𝑘1,𝑘2)|}_{𝑝=0ℎ1+ℎ2=𝑝}𝜙^{(𝑘1+ℎ1,𝑘2+ℎ2)}𝑥₂,𝑦₂𝑖^𝑘2+ℎ2×𝑝!𝑝!ℎ₁!ℎ₂!𝑥₁−𝑥₂^ℎ1𝑖^ℎ2𝑦₁−𝑦₂^ℎ2_ℍ≤𝐶₁𝐻^𝑚𝑞^{𝐴(𝑚2/2)}𝑥₁−𝑥₂,𝑦₁−𝑦₂^{𝑚+1−|(𝑘1,𝑘2}_ℝ^)|₂𝑚+1−𝑘₁,𝑘₂! (5.25) for (𝑥₁,𝑦₁),(𝑥₂,𝑦₂)∈𝐾^′. Expression (5.14) can be directly checked from (5.24) and (5.18) for 𝜖_𝑏=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 in ℰ_{𝑤0}(𝐾), for a given compact set 𝐾.

Theorem 5.16 ([19]. Theorem II.1.3). Let 𝐾₁,𝐾₂ be compact sets in ℝ². The following statements are equivalent. (i)The sequence0⟶ℰ_{𝑤0}𝐾₁∪𝐾⟶₂^𝜋ℰ_{𝑤0}𝐾₁⊕ℰ_{𝑤0}𝐾⟶₂^𝛿ℰ_{𝑤0}𝐾₁∩𝐾₂⟶0 (5.26) is exact. 𝜋(𝑓)=(𝑓∣_𝐾1,𝑓∣_𝐾2) and 𝛿(𝑓,𝑔)=𝑓∣_{𝐾1∩𝐾2}−𝑔∣_{𝐾1∩𝐾2}.(ii)Let 𝑓₁∈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)If 𝐾₁∩𝐾₂≠∅, then there exist 𝐴₃,𝐴₄>0 such that 𝑀𝐴₃dist𝑥,𝐾₁∩𝐾₂≤𝐴₄𝑀dist𝑥,𝐾₂, (5.27) for every 𝑥∈𝐾₁. Here, 𝑀 denotes the function given by 𝑀(0)=0 and 𝑀(𝑡)=inf_𝑛∈ℕ𝑡^𝑛𝑀_𝑛 for 𝑡>0. dist(𝑥,𝐾) stands for the distance from 𝑥 to the set 𝐾.

Corollary 5.17 ([17], Lemma  4.3.6). Given 𝐾₁,𝐾₂ nonempty compact sets in ℂ^⋆, if one puts 𝐾_𝑗∶={(𝜖₁,𝜖₂)∈ℝ²∶𝜖₁+𝑖𝜖₂∈𝐾_𝑗𝑞^−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 Proposition 5.5 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 (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ be a good covering, and let (𝑔_𝐼,𝐼′(𝜖))_{(𝐼,𝐼′)∈ℐ2} be the 𝑞-Gevrey ℍ_𝒯,𝜌-cocycle of type 𝐴 constructed in Proposition 5.5. One chooses a family of compact sets 𝐾_𝐼⊆𝑈_𝐼 for 𝐼∈ℐ, with Int(𝐾_𝐼)≠∅, in such a way that ⋃_𝐼∈ℐ(𝐾_𝐼𝑞^−ℕ) is 𝒰⧵{0}, where 𝒰 is a neighborhood of 0 in ℂ.
Then, for all 𝐼∈ℐ, there exists a 𝑤₀−𝒞^∞ function 𝑓_𝐼(𝜖₁,𝜖₂) in the sense of Whitney on the compact set 𝐴_𝐼={(𝜖₁,𝜖₂)∈ℝ²∶𝜖₁+𝑖𝜖₂∈𝐾_𝐼𝑞^−ℕ∪{0}}, with values in the Banach space ℍ_𝒯,𝜌, such that 𝑔_𝐼,𝐼′𝜖₁+𝑖𝜖₂=𝑓_𝐼′𝜖₁,𝜖₂−𝑓_𝐼𝜖₁,𝜖₂ (5.28) for all 𝐼,𝐼^′∈ℐ such that 𝐴_𝐼∩𝐴_𝐼′≠∅ and, for every (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′.

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.
Let 𝐼,𝐼^′∈ℐ such that 𝐴_𝐼∩𝐴_𝐼′≠∅. From Lemma 5.15, we have that the function (𝜖₁,𝜖₂)↦𝑔_𝐼,𝐼′(𝜖₁+𝑖𝜖₂) is a 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼∩𝐴_𝐼′. In the following we provide the construction of 𝑓_𝐼 for 𝐼∈ℐ verifying (5.28).
Let us fix any 𝐼∈ℐ. We consider any 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼. By definition of the good covering (𝑈_𝐼𝑞^−ℕ)_𝐼∈ℐ the following cases are possible.
Case 1. If there is at least one 𝐼^′∈ℐ,𝐼≠𝐼^′, such that 𝐴_𝐼∩𝐴_𝐼′≠∅ but 𝐴_𝐼∩𝐴_𝐼′∩𝐴_𝐼′′=∅ for every 𝐼^′′∈ℐ with 𝐼^′′≠𝐼^′≠𝐼, then we define 𝑒_𝐼,𝐼′(𝜖₁,𝜖₂)=𝑓_𝐼(𝜖₁,𝜖₂)+𝑔_𝐼,𝐼′(𝜖₁+𝑖𝜖₂) for every (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′. 𝑒_𝐼,𝐼′ is a 𝑤₀−𝒞^∞ function in the sense of Whitney in 𝐴_𝐼∩𝐴_𝐼′. From Theorems 5.9 and 5.11, we can extend 𝑒_𝐼,𝐼′ to a 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼′. This extension is called 𝑓_𝐼′. We have 𝑔_𝐼,𝐼′𝜖₁+𝑖𝜖₂=𝑓_𝐼′𝜖₁,𝜖₂−𝑓_𝐼𝜖₁,𝜖₂,𝜖₁,𝜖₂∈𝐴_𝐼∩𝐴_𝐼′. (5.29)Case 2. There exist two different 𝐼^′,𝐼^′′∈ℐ with 𝐼^′≠𝐼≠𝐼^′′ such that 𝐴_𝐼∩𝐴_𝐼′∩𝐴_𝐼′′≠∅. We first construct a 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼′, 𝑓_𝐼′(𝜖₁,𝜖₂), verifying 𝑔_𝐼,𝐼′𝜖₁+𝑖𝜖₂=𝑓_𝐼′𝜖₁,𝜖₂−𝑓_𝐼𝜖₁,𝜖₂,𝜖₁,𝜖₂∈𝐴_𝐼∩𝐴_𝐼′. (5.30) We define 𝑒_{𝐼,𝐼′′}(𝜖₁,𝜖₂)=𝑓_𝐼(𝜖₁,𝜖₂)+𝑔_{𝐼,𝐼′′}(𝜖₁+𝑖𝜖₂) for every (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′′ and 𝑒_{𝐼′,𝐼′′}(𝜖₁,𝜖₂)=𝑓_𝐼′(𝜖₁,𝜖₂)+𝑔_{𝐼′,𝐼′′}(𝜖₁+𝑖𝜖₂) whenever (𝜖₁,𝜖₂)∈𝐴_𝐼′∩𝐴_𝐼′′. From (5.30), we have 𝑒_{𝐼,𝐼′′}(𝜖₁,𝜖₂)=𝑒_{𝐼′,𝐼′′}(𝜖₁,𝜖₂) for every (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′∩𝐴_𝐼′′. From this, we can define 𝑒_𝐼′′𝜖₁,𝜖₂𝑒∶=^{𝐼,𝐼′′}𝜖₁,𝜖₂if𝜖₁,𝜖₂∈𝐴_𝐼∩𝐴_𝐼′′,𝑒_{𝐼′,𝐼′′}𝜖₁,𝜖₂if𝜖₁,𝜖₂∈𝐴_𝐼′∩𝐴_𝐼′′. (5.31) From Theorem 5.16 and Corollary 5.17 we deduce that 𝑒_𝐼′′(𝜖₁,𝜖₂) can be extended to a 𝑤₀−𝒞^∞ function in the sense of Whitney in 𝐴_𝐼′′,𝑓_𝐼′′(𝜖₁,𝜖₂). It is straightforward to check, from the way 𝑓_𝐼′′ was constructed, that 𝑓_𝐼′′(𝜖₁,𝜖₂)=𝑓_𝐼(𝜖₁,𝜖₂)+𝑔_{𝐼,𝐼′′}(𝜖₁+𝑖𝜖₂) when (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′′ and also 𝑓_𝐼′′(𝜖₁,𝜖₂)=𝑓_𝐼′(𝜖₁,𝜖₂)+𝑔_{𝐼′,𝐼′′}(𝜖₁+𝑖𝜖₂) for (𝜖₁,𝜖₂)∈𝐴_𝐼′∩𝐴_𝐼′′.These two cases solve completely the problem since nonempty intersection of four different compacts in (𝐴_𝐼)_𝐼∈ℐ is not allowed when working with a good covering. The functions in (𝑓_𝐼)_𝐼∈ℐ satisfy (5.28).

6. Existence of Formal Series Solutions and 𝑞-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 to ℍ_𝒯,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. Let 𝑈 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 𝜕_𝜖𝑔𝜖₁,𝜖₂1∶=2𝜕_𝜖1+𝑖𝜕_𝜖2𝑔𝜖₁+𝑖𝜖₂=ℎ𝜖₁,𝜖₂,𝜖₁,𝜖₂∈𝑈. (6.1) Then, 𝑔 also verifies bounds such as those in (5.14) and (5.15) for (𝜖₁,𝜖₂)∈𝑈.

Proof. Let ℎ₁ 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 𝑔𝜖₁,𝜖₂1∶=−𝜋_ℝ²ℎ₁(𝑥)𝜖𝑥−𝜖𝑑𝜉𝑑𝜂,₁,𝜖₂∈𝑈, (6.2) 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 𝜕^𝛼1+𝛼2𝑔𝜕𝜖^𝛼1₁𝜕𝜖^𝛼2₂𝜖₁,𝜖₂1=−𝜋_ℝ²𝜕^𝛼1+𝛼2ℎ₁𝜕𝜖^𝛼1₁𝜕𝜖^𝛼2₂(𝑥)1𝑥−𝜖𝑑𝜉𝑑𝜂, (6.3) 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 functions 𝑋_𝐼 constructed in Theorem 4.11. The procedure is adapted from [4] under 𝑞-Gevrey settings. For every 𝐼∈ℐ, we write 𝑋_𝐼(𝜖)∶𝑈_𝐼𝑞^−ℕ→ℍ_𝒯,𝜌 for the holomorphic function given by 𝑋_𝐼(𝜖)∶=(𝑡,𝑧)↦𝑋_𝐼(𝜖,𝑡,𝑧).

Proposition 6.2. There exist a 𝑤₀−𝒞^∞ function 𝑢(𝜖₁,𝜖₂) and a holomorphic function 𝑎(𝜖₁+𝑖𝜖₂) defined on the neighborhood Int(⋃_𝐼∈ℐ𝐴_𝐼) of 0 such that 𝑋_𝐼𝜖₁+𝑖𝜖₂=𝑓_𝐼𝜖₁,𝜖₂+𝑢𝜖₁,𝜖₂+𝑎𝜖₁+𝑖𝜖₂,𝜖₁,𝜖₂∈Int𝐴_𝐼, (6.4) for every 𝐼∈ℐ.

Proof. From the definition of the cocycle (𝑔_𝐼,𝐼′)_{(𝐼,𝐼′)∈ℐ2} in Proposition 5.5 and from Proposition 5.18 we derive 𝑋_𝐼𝜖₁+𝑖𝜖₂−𝑓_𝐼𝜖₁,𝜖₂=𝑋_𝐼′𝜖₁+𝑖𝜖₂−𝑓_𝐼′𝜖₁,𝜖₂,𝜖₁,𝜖₂∈𝐴_𝐼∩𝐴_𝐼′⧵{(0,0)}, (6.5) whenever (𝐼,𝐼^′)∈ℐ² and 𝐴_𝐼∩𝐴_𝐼′≠∅. The function 𝑋−𝑓 given by 𝑋−𝑓𝜖₁,𝜖₂∶=𝑋_𝐼𝜖₁+𝑖𝜖₂−𝑓_𝐼𝜖₁,𝜖₂,𝜖₁,𝜖₂∈𝐴_𝐼⧵{(0,0)}, (6.6) is well defined on 𝑊⧵{(0,0)}, where 𝑊=⋃_𝐼∈ℐ𝐴_𝐼 is a closed neighborhood of (0,0).
For every 𝐼∈ℐ,𝑋_𝐼 is a holomorphic function on 𝑈_𝐼𝑞^−ℕ so that the Cauchy-Riemann equations hold: 𝜕_𝜖𝑋_𝐼𝜖₁+𝑖𝜖₂=0,𝜖₁,𝜖₂∈𝐴_𝐼⧵{(0,0)}. (6.7) This yields 𝜕_𝜖(𝑋−𝑓)(𝜖₁,𝜖₂)=−𝜕_𝜖𝑓_𝐼(𝜖₁,𝜖₂) for every 𝐼∈ℐ and (𝜖₁,𝜖₂)∈Int(𝐴_𝐼).
We have that −𝜕_𝜖𝑓_𝐼(𝜖₁,𝜖₂) can be extended to a 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼. This yields that 𝑓_𝐼 is 𝑤₀−𝒞^∞ in the sense of Whitney on 𝐴_𝐼. 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 𝐴_𝐼 for every 𝐼∈ℐ and also that 𝜕_𝜖𝑓_𝐼(𝜖₁,𝜖₂)=𝜕_𝜖𝑓_𝐼′(𝜖₁,𝜖₂) for every (𝜖₁,𝜖₂)∈Int(𝐴_𝐼∩𝐴_𝐼′) and every 𝐼,𝐼^′∈ℐ because 𝑔_𝐼,𝐼′(𝜖) is a holomorphic function on 𝑈_𝐼𝑞^−ℕ∩𝑈_𝐼′𝑞^−ℕ. The previous equality is also true for (𝜖₁,𝜖₂)∈𝐴_𝐼∩𝐴_𝐼′ from the fact that 𝑓_𝐼 is 𝑤₀−𝒞^∞ in the sense of Whitney on 𝐴_𝐼.
From Theorem 5.16 and Corollary 5.17 we derive that 𝜕_𝜖(𝑋−𝑓) is a 𝑤₀−𝒞^∞ function in the sense of Whitney on ⋃_𝐼∈ℐ𝐴_𝐼.
Taking into account Lemma 6.1 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 𝜕_𝜖𝑢𝜖₁,𝜖₂=𝜕_𝜖𝑋−𝑓𝜖₁,𝜖₂,𝜖₁,𝜖₂∈Int(𝑊). (6.8) From this last expression we have that 𝑢(𝜖₁,𝜖₂)−(𝑋−𝑓)(𝜖₁,𝜖₂) defines a holomorphic function on Int(𝑊)⧵{(0,0)}.
For every 𝐼∈ℐ, 𝑋_𝐼 is a bounded ℍ_𝒯,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 Theorem 4.11, there exists a formal power series 𝑋(𝜖,𝑡,𝑧)=_𝑘≥0𝑋_𝑘(𝑡,𝑧)𝜖𝑘!^𝑘∈ℍ_𝒯,𝜌[[𝜖]], (6.9) formal solution of 𝜖𝑡𝜕^𝑆_𝑧𝑋𝜖,𝑞𝑡,𝑧+𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)=^𝑆−1_𝑘=0𝑏_𝑘(𝜖,𝑧)𝑡𝜎_𝑞^𝑚0,𝑘𝜕^𝑘_𝑧𝑋𝜖,𝑡,𝑧𝑞^{−𝑚1,𝑘}. (6.10) Moreover, let 𝐼∈ℐ, and let 𝐾_𝐼 be any compact subset of Int(𝐾_𝐼). There exists 𝐵>0 such that the function 𝑋_𝐼(𝜖,𝑡,𝑧) constructed in Theorem 4.11 admits 𝑋(𝜖,𝑡,𝑧) as its 𝑞-Gevrey asymptotic expansion of type 𝐵 in 𝐾_𝐼𝑞^−ℕ.

Proof. Let 𝐼∈ℐ, and let 𝐾_𝐼 be any compact subset of Int(𝐾_𝐼).
From Proposition 6.2 we can extend 𝑋_𝐼(𝜖₁+𝑖𝜖₂) to a 𝑤₀−𝒞^∞ function in the sense of Whitney on 𝐴_𝐼={(𝜖₁,𝜖₂)∈ℝ²∶𝜖₁+𝑖𝜖₂∈𝐾_𝐼𝑞^−ℕ∪{0}}⊆Int(𝐴_𝐼)∪{(0,0)}. Let us fix 𝐼∈ℐ. We consider the family (𝑋^(ℎ1,ℎ2)(𝜖₁,𝜖₂))_{(ℎ1,ℎ2)∈ℕ2} associated to 𝑋_𝐼 by Definition 5.7. We have 𝑋^(ℎ1,ℎ2)_𝐼𝜖₁,𝜖₂=𝜕^ℎ1_𝜖1𝜕^ℎ2_𝜖2𝑋_𝐼𝜖₁+𝑖𝜖₂=𝑖^ℎ2𝜕^ℎ1+ℎ2_𝜖𝑋_𝐼(𝜖),𝜖₁,𝜖₂∈𝐴_𝐼⧵{(0,0)}, (6.11) because 𝑋_𝐼(𝜖) is holomorphic on Int(𝐾_𝐼)𝑞^−ℕ.
We have that 𝑋^(ℎ1,ℎ2)_𝐼(𝜖₁,𝜖₂) is continuous at (0,0) for every (ℎ₁,ℎ₂)∈ℕ² so we can define 𝑘≥0𝑋_𝑘,𝐼𝑋∶=^{(ℎ1,ℎ2)𝐼}(0,0)𝑖^ℎ2∈ℍ_𝒯,𝜌, (6.12) 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 𝑋_𝐼𝜖₁+𝑖𝜖₂−^𝑚_𝑝=0𝑋_𝑝,𝐼𝜖𝑝!¹+𝑖𝜖₂^𝑝_ℍ𝒯,𝜌≤𝐶₁𝐻^𝑚𝑞^{𝐵(𝑚2/2)}𝜖₁+𝑖𝜖₂^𝑚+1()!,𝑚+1 (6.13) for every 𝑚≥0 and 𝜖₁+𝑖𝜖₂∈𝐾_𝐼𝑞^−ℕ. As a matter of fact, this shows that 𝑋_𝐼 admits 𝑋_𝐼(𝜖)=∑_𝑘≥0(𝑋_𝑘/𝑘!)𝜖^𝑘 as its 𝑞-Gevrey expansion of type 𝐵>0 in 𝐾_𝐼𝑞^−ℕ.
The formal power series 𝑋_𝐼 does not depend on 𝐼∈ℐ. Indeed, from Theorem 4.11 we have that 𝑋_𝐼(𝜖)−𝑋_𝐼′(𝜖) admits both 0 and 𝑋_𝐼′−𝑋_𝐼 as 𝑞-asymptotic expansion on 𝐾_𝐼𝑞^−ℕ∩𝐾_𝐼′𝑞^−ℕ whenever this intersection is not empty. We put 𝑋∶=𝑋_𝐼 for any 𝐼∈ℐ. The function 𝑋_𝑘,𝐼=𝑋_𝑘,𝐼(𝑡,𝑧)∈ℍ_𝒯,𝜌 does not depend on 𝐼 for every 𝑘≥0. We write 𝑋_𝑘∶=𝑋_𝑘,𝐼 for 𝑘≥0. 𝑋_𝐼 admits 𝑋=∑_𝑘≥0(𝑋_𝑘/𝑘!)𝜖^𝑘 as its 𝑞-Gevrey asymptotic expansion of type 𝐵>0 in 𝐾_𝐼𝑞^−ℕ for all 𝐼∈ℐ.
In order to achieve the result, it only remains to prove that 𝑋(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 𝜕^𝑙_𝜖𝑋_𝐼(𝜖,𝑡,𝑧) is a solution of 𝜖𝑡𝜕^𝑆_𝑧𝜕^𝑙_𝜖𝑋_𝐼𝜖,𝑞𝑡,𝑧+𝑡𝜕^𝑆_𝑧𝑙𝜕_𝜖^𝑙−1𝑋_𝐼𝜖,𝑞𝑡,𝑧+𝜕^𝑆_𝑧𝜕^𝑙_𝜖𝑋_𝐼()=𝜖,𝑡,𝑧_{𝑆−1𝑘=0𝑙1+𝑙2=𝑙}𝑙!𝑙₁!𝑙₂!𝜕^𝑙1_𝜖𝑏_𝑘(𝜖,𝑧)𝜕^𝑙2_𝜖𝑡𝜎_𝑞^𝑚0,𝑘𝜕^𝑘_𝑧𝑋_𝐼𝜖,𝑡,𝑧𝑞^{−𝑚1,𝑘}. (6.14) for every 𝑙≥1,(𝑡,𝑧)∈𝒯×𝐷(0,𝜌) and 𝜖∈𝐾_𝐼𝑞^−ℕ. Letting 𝜖 tend to 0 in (6.14), we obtain 𝑡𝜕^𝑆_𝑧𝑋_𝑙−1𝑞𝑡,𝑧(𝑙−1)!+𝜕^𝑆_𝑧𝑋_𝑙(𝑡,𝑧)=𝑙!_{𝑆−1𝑘=0𝑙1+𝑙2=𝑙}𝜕^𝑙1_𝜖𝑏_𝑘(𝜖,𝑧)∣_𝜖=0𝑙₁!𝑡𝜎_𝑞^𝑚0,𝑘𝜕^𝑘_𝑧𝑋_𝑙2𝑡,𝑧𝑞^{−𝑚1,𝑘}𝑙₂! (6.15) for every 𝑙≥1,(𝑡,𝑧)∈𝒯×𝐷(0,𝜌). The holomorphy of 𝑏_𝑘(𝜖,𝑧) with respect to 𝜖 at 0 implies 𝑏_𝑘(𝜖,𝑧)=_𝑙≥0𝜕^𝑙_𝜖𝑏_𝑘(𝜖,𝑧)|_𝜖=0𝜖𝑙!^𝑙, (6.16) for 𝜖 near 0 and for every 𝑧∈ℂ. Statements (6.14) and (6.15) conclude that 𝑋(𝜖,𝑡,𝑧)=∑_𝑘≥0𝑋_𝑘(𝑡,𝑧)(𝜖^𝑘/𝑘!) is a formal solution of (6.10).

Acknowledgments

A. Lastra is partially supported by Ministerio de Educación, Programa Nacional de Movilidad de Recursos Humanos, Plan Nacional I-D+i 2008–2011. S. Malek is partially supported by the french ANR-10-JCJC 0105 project.