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^𝜉