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

Abstract

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

1. Introduction

We study a family of 𝑞-difference-differential equations of the form 𝜖𝜕_𝑡𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)+𝑎𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)=_{𝜅=𝜅0,𝜅1∈𝒩}𝑏_𝜅(𝜖,𝑧)𝜕^𝜅0_𝑡𝜕^𝜅1_𝑧𝑋𝜖,𝑞^𝑚𝜅,1𝑡,𝑞^𝑚𝜅,2𝑧 (1) under appropriate initial conditions 𝜕^𝑗_𝑧𝑋(𝜖,𝑡,0)=𝜙_𝑗(𝜖,𝑡),0≤𝑗≤𝑆−1. (2)

Here, 𝑆 is an integer with 𝑆≥1 and 𝑎∈ℂ^⋆∶=ℂ⧵{0}. 𝒩 stands for a finite subset of ℕ², where ℕ∶={0,1,2,…} is the set of nonnegative integers. For every (𝜅₀,𝜅₁)∈𝒩, 𝑏_𝜅(𝜖,𝑧) turns out to be a polynomial in the variable 𝑧 with holomorphic and bounded coefficients in a neighborhood of the origin in the parameter and 𝑚_𝜅,1,𝑚_𝜅,2∈ℕ.

From now on, 𝑞 stands for a fixed real number with 0<𝑞<1.

We construct actual holomorphic solutions 𝑋(𝜖,𝑡,𝑧) for the previous Cauchy problem in ℰ×𝒯×ℂ, where ℰ is a bounded open sector in the complex plane with vertex at the origin and 𝒯 is an unbounded well-chosen open set. The procedure is based on the use of the map 𝑡↦𝑡/𝜖 which was firstly considered by Canalis-Durand et al. in [1] to transform a singularly perturbed equation into an auxiliary regularly perturbed equation, easier to handle. This celebrated technique has also been used in the study of singularly perturbed partial differential equations (see [2, 3], e.g.), 𝑞-difference-differential equations (like in [4] or [5]), and more recently to the study of difference-differential equations (see [6]).

Indeed, the present work is motivated by a previous work [6], where the second author studies a singularly perturbed difference-differential equation with small delay. This work can be seen as a continuation of that one. The dynamics appearing in that previous work involve a small shift in variable 𝑡 with respect to 𝜖, meaning that they are of the form (𝜖,𝑡,𝑧)↦(𝜖,𝑡+𝜅₂𝜖,𝑧), whereas the actual work deals with a shrinking behaviour in both 𝑡 and 𝑧 variables.

In [6], a Gevrey 1+ phenomenon, with estimates associated to the sequence ((𝑛/log𝑛)^𝑛)_𝑛≥0, is observed for the series solution of the problem. This sequence naturally appears when working with difference equations (see [7, 8], e.g.). Now, a 𝑞-Gevrey-like behaviour, related to the sequence of estimates (𝑞^−𝑛2)_𝑛≥0, appears. This behaviour comes up in the context of 𝑞-difference equations (see [9, 10]). One can observe that 1+ sequence is asymptotically upper bounded by Gevrey sequence (𝑛!)_𝑛≥0, and this one is upper bounded by 𝑞-Gevrey sequence (𝑞^−𝑛2)_𝑛≥0.

The main aim of this work is to construct actual holomorphic solutions 𝑋(𝜖,𝑡,𝑧) of (1)+(2) and obtain sufficient conditions for the existence and unicity of a formal power series in the parameter 𝜖, 𝑋(𝜖,𝑡,𝑧)=∑_𝛽≥0𝑋_𝛽(𝑡,𝑧)(𝜖^𝛽/𝛽!), owing its coefficients in an adequate functional space and such that 𝑋 is represented by 𝑋 in a sense to precise (see Theorem 27). This representation is measured in terms of 𝑞-Gevrey bounds due to the appearance of 𝑞-difference operators on the right-hand side in (1).

The Cauchy problem (1)+(2) we consider in this paper comes also within the framework of the asymptotic analysis of linear differential and partial differential equations with multiplicative delays.

In the context of differential equations most of the statements in the literature are dedicated to linear problems of the form 𝑥^′(𝑡)=𝐹𝑡,𝑥𝜆₁𝑡,…,𝑥𝜆_𝑛𝑡,𝑥^′𝜆₁𝑡,…,𝑥^′𝜆_𝑛𝑡, (3) where 𝐹 are vector valued polynomial functions in 𝑡 and linear in its other arguments, where 0<𝜆_𝑗<1, for 1≤𝑗≤𝑛 are real numbers and concern the study of asymptotic behaviour of some of their solutions 𝑥(𝑡) as 𝑡 tends to infinity for given initial data 𝑥(0). When 𝐹 is real or matrix valued and with constant coefficients, we quote [11–14]. For polynomial 𝐹 in 𝑡, we notice [15, 16]. For studies in a complex variable 𝑡, we refer to [17, 18]. For more general delay functional equations, we indicate [19].

In the framework of linear partial differential equations, we mention a series of papers devoted to general results on the existence and unicity of holomorphic solutions to generalized Cauchy-Kowalevski type problems with shrinkings of the form 𝜕^𝑚_𝑡𝑢(𝑡,𝑥)=𝑓𝑡,𝑥,𝑢(𝑡,𝑥),𝜕^𝑙_𝑥𝑢(𝑡,𝑥),𝜕^𝑝_𝑥𝑢(𝛼(𝑡)),𝜕𝑡,𝑥_𝑞𝑥𝑢𝑡,𝛽(𝑡,𝑥)𝑥_{(𝑙,𝑝,𝑞)∈𝐼} (4) for some integer 𝑚≥1, a finite set 𝐼, and where 𝑓 is analytic or of Gevrey type function and such that the functions 𝛼(𝑡) and 𝛽(𝑡,𝑥) satisfy the shrinking constraints |𝛼(𝑡)|<1 and |𝛽(𝑡,𝑥)|<1 for given initial data (𝜕^𝑗_𝑡𝑢)(0,𝑥), 0≤𝑗≤𝑚−1 that belong to some functional space. We refer to [20–22]. For partial differential problems with contractions dealing with less regular solution spaces like Sobolev spaces, we quote [23], for instance.

Let us briefly reproduce the strategy followed. We consider a finite family of sectors with vertex at the origin (ℰ_𝑖)_{0≤𝑖≤𝜈} which provides a good covering at 0 in the variable 𝜖 (see Definition 19). Let 𝑖∈{0,1,…,𝜈−1}. One can consider an auxiliary Cauchy problem as follows: 𝜕^𝑆_𝑧𝑊(𝜖,𝜏,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)(𝑎−𝜏)𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0×𝜕^𝜅1_𝑧𝑊𝜖,𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧 (5) with initial conditions (𝜕^𝑗_𝑧𝑊)(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏), 0≤𝑗≤𝑆−1. We assume 𝑊_𝑗 is a holomorphic function in (𝐷(0,𝑟₀)⧵{0})×𝐷(0,𝑅₀) for some 𝑟₀,𝑅₀>0, for every 0≤𝑗≤𝑆−1, which is upper bounded in terms of 𝑞-Gevrey bounds (see (61)). Moreover, we assume each 𝑊_𝑗 can be extended to ℰ_𝑖×𝒮, where 𝒮 is a sector with vertex at the origin and verifying 𝑞-Gevrey bounds in ℰ_𝑖×𝑆₀ with 𝑆₀∶={𝑧∈𝒮∶|𝑧|≥𝑅₀} (see (31)). Under these hypotheses, one can construct a formal solution to the auxiliary Cauchy problem, 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), where 𝑊_𝛽(𝜖,𝜏) turns out to be a holomorphic function in (𝐷(0,𝑟₀)⧵{0})×(𝐷_𝛽⧵{0}). Here, 𝐷_𝛽 is a disc centered at the origin with radius decreasing to 0 whenever 𝛽 tends to infinity and reproducing 𝑞-Gevrey bounds given by the initial conditions (see (62)). Moreover, each 𝑊_𝛽(𝜖,𝜏) can be extended to ℰ_𝑖×𝑆_𝛽 under 𝑞-Gevrey bounds (see (32)), where 𝑆_𝛽∶={𝑧∈𝒮∶|𝑧|>𝑅_𝛽}, with (𝑅_𝛽)_𝛽≥0 being a sequence of positive numbers that decrease to 0. We assume 𝑆_𝛽∩𝐷_𝛽≠∅ for every 𝛽≥0. The decrease rate of both 𝑅_𝛽 and the radius of 𝐷_𝛽 has to be chosen adequately, in accordance to the elements of a 𝑞-Gevrey sequence such as (𝑞^𝛼𝛽2)_𝛽≥0 for some 𝛼>0.

The main difficulty in this work is the occurrence of propagation of singularities in the coefficients of the auxiliary problem which leads to a small divisor phenomenon. The singular points form a sequence of complex numbers tending to 0. As a result, one can only obtain a formal solution for the auxiliary problem. In [24], a small divisor phenomenon comes from the Fuchsian operator studied in the main Cauchy problem. There, 𝑞∈ℂ is chosen to have |𝑞|>1, whilst in the present work 𝑞∈ℝ with 0<𝑞<1. A suchlike phenomenon also appears in [2], where the asymptotics in the parameter suffers the effect of a small divisor, and it is solved studying a Dirichlet-like series.

General Dirichlet series of the form _𝑛≥0𝑎_𝑛𝑒^{−𝜆𝑛𝑧} (6) have been throughly studied in the case when (𝜆_𝑛)_𝑛≥0 is an increasing sequence of real numbers to ∞ (see [25–27]) or a sequence of complex numbers with |𝜆_𝑛|→∞ (see [28]). This theory has also been developed when working with almost periodic functions, introduced by Bohr (see [29–31]), which are the uniform limits in ℝ of exponential polynomials ∑^𝑛_𝑘=1𝑎_𝑘𝑒^{𝑖𝑠𝑘𝑥}, where the values 𝑠_𝑘 belong to the so-called spectrum Λ⊆ℝ. However, we are more interested in the behaviour of the sum when 𝑥 tends to ∞ in the positive imaginary axis. Our technique rests on the Euler-Mac-Laurin formula, Watson's lemma, and the equivalence between null 𝑞-Gevrey asymptotics. The characterization of 𝑞—exponentially at functions is also considered on the way.

In [2], we solve the problem by means of a Dirichlet series with a spectrum being of the form (1/(𝑘+1)^𝛼)_𝑘≥0. Now, the spectrum which helps us to achieve our purpose is of geometric nature (see Lemma 22).

The growth properties of 𝑊_𝛽 for 𝛽≥0 allow us to apply a Laplace like transform on each of them with respect to the variable 𝜏 in order to provide a holomorphic solution 𝑋_𝑖(𝜖,𝜏,𝑧) of the main problem, defined in ℰ_𝑖×𝒯×ℂ, for some appropriate unbounded open set 𝒯. In addition to this, one has null 𝑞-Gevrey asymptotic bounds for the difference of 𝑋_𝑖 and 𝑋_𝑖+1 when the domain of the variable 𝑧 is restricted to a bounded set, meaning that for every 𝜌>0, there exist 𝐿₁,𝐿₂>0 such that sup_{𝑡∈𝜏𝑧∈𝐷(0,𝜌)}𝑋_𝑖+1(𝜖,𝑡,𝑧)−𝑋_𝑖(𝜖,𝑡,𝑧)≤𝐿₁𝑒^{−𝐿2(1/(−log(𝑞))2)log2|𝜖|}, (7) for every 𝜖∈ℰ_𝑖∩ℰ_𝑖+1.

Finally, a novel version regarding 𝑞-Gevrey asymptotics of Malgrange-Sibuya theorem (Theorem 25) leads us to the main result in the present work (Theorem 27), where we guarantee the existence of a formal power series in 𝜖 as follows: 𝑋(𝜖,𝑡,𝑧)=_𝛽≥0𝑋_𝛽(𝑡,𝑧)𝜖𝛽!^𝛽∈ℍ_𝒯,𝜌[[𝜖]], (8) with coefficients in the Banach space of bounded holomorphic functions defined in 𝒯×𝐷(0,𝜌), which is common for every 0≤𝑖≤𝜈−1 and such that 𝑋_𝑖 admits 𝑋 as its 𝑞-Gevrey asymptotic expansion of some positive type in the variable 𝜖 (see (135)).

It is worth pointing out that a 𝑞-Gevrey version of Malgrange-Sibuya theorem was already obtained in [5], when dealing with 𝑞∈ℂ, |𝑞|>1. The type in the asymptotic expansion suffers some increasement in that previous work. This is so due to the need of extension results in ultradifferentiable classes of functions (see [32, 33]) to be applied along the proof. Here, the geometry of the problem changes so that we are able to maintain the type 𝑞-Gevrey. The proof rests on the classical Malgrange-Sibuya theorem (see [34]).

The paper is organized as follows.

In Sections 2 and 3, we introduce Banach spaces of formal power series in order to solve auxiliary Cauchy problems with the help of fixed point results involving complete metric spaces. In Section 2, this result is achieved when dealing with formal power series with holomorphic coefficients in a product of a finite sector with vertex at the origin times an infinite sector, while in Section 3 the result is obtained when dealing with a product of two punctured discs at 0.

In Section 4, we first recall the definition and main properties of a Laplace-like transform and 𝑞-Gevrey asymptotic expansions (Section 4.1). Next, we construct analytic solutions for the main problem and determine flat 𝑞-Gevrey bounds for the difference of two solutions when the intersection of the domains in the perturbation parameter is not empty (Section 4.2). In the proof, a Dirichlet type series is studied. The section is concluded proving the existence of a formal power series in the perturbation parameter which represents every solution in some sense which is specified (Section 4.3).

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

𝑀, 𝐴₁, 𝐶, and 𝛿₁>0 are fixed positive real numbers throughout the present work. Let 𝑞∈ℝ with 0<𝑞<1 and let (𝑅_𝛽)_𝛽≥0 be a sequence of positive real numbers.

We consider an open and bounded sector ℰ with vertex at the origin and we fix an open and unbounded sector 𝒮 with vertex at the origin having positive distance to a fixed complex number 𝑎∈ℂ^⋆, it is to say, there exists 𝑀₁>0 such that |𝜏−𝑎|>𝑀₁ for every 𝜏∈𝒮. We write 𝑆_𝛽 for the subset of 𝒮 defined by 𝑆_𝛽∶=𝑧∈𝒮∶|𝑧|>𝑅_𝛽. (9)

The incoming definition of Banach spaces of functions and formal power series turns out to be an adaptation of the corresponding one in [5]. Here, the symmetry of these norms at 0 and the point of infinity in the 𝜏 variable has to be removed, so that a Laplace-like transform of the elements in these Banach spaces makes sense.

Definition 1. Let 𝜖∈ℰ and 𝛽∈ℕ. 𝐸_{𝛽,𝜖,𝑆𝛽} denotes the vector space of functions 𝑣∈𝒪(𝑆_𝛽) such that ‖𝑣(𝜏)‖_{𝛽,𝜖,𝑆𝛽}∶=sup_{𝜏∈𝑆𝛽}|𝑣(𝜏)|𝑒^{𝑀log2(|𝜏|/|𝜖|+𝛿1)}𝜏𝜖^−𝐶𝛽𝑞^{−𝐴1𝛽2} (10) is finite.
Let 𝛿>0. 𝐻(𝜖,𝛿,𝒮) denotes the complex vector space of all formal power series 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)(𝑧^𝛽/𝛽!) with 𝑣_𝛽∈𝒪(𝑆_𝛽) for every 𝛽≥0 and such that ‖𝑣(𝜏,𝑧)‖_{(𝜖,𝛿,𝒮)}∶=_𝛽≥0𝑣_𝛽(𝜏)_{𝛽,𝜖,𝑆𝛽}𝛿^𝛽𝛽!<∞. (11)
It is straightforward to check that the pair (𝐻(𝜖,𝛿,𝒮),‖⋅‖_{(𝜖,𝛿,𝒮)}) is a Banach space.

For our purposes, the elements in the sequence (𝑅_𝛽)_𝛽≥0 are chosen to be related to the ones in a 𝑞-Gevrey sequence. This choice would provide that 𝑆_𝛽 tends to 𝒮 when 𝛽→∞.

Let (𝔼_𝛽)_𝛽≥0 be a family of complex functional Banach spaces. For every 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)(𝜏^𝛽/𝛽!)∈(∪_𝛽≥0𝔼_𝛽)[[𝑧]], we consider the formal integration operator 𝜕_𝑧⁻¹ defined on (∪_𝛽≥0𝔼_𝛽)[[𝑧]] by 𝜕_𝑧⁻¹(𝑣(𝜏,𝑧))∶=_𝛽≥1𝑣_𝛽−1(𝜏)𝑧^𝛽.𝛽! (12)

Lemma 2. Let 𝑠, ℓ₀, ℓ₁, 𝑚₁, and 𝑚₂∈ℕ, 𝛿>0, and 𝜖∈ℰ. We assume that 𝐶ℓ₁+𝑠−ℓ₀−2𝑚₁𝑀−log𝑞≥0. (13)
In addition to this, we consider the elements in (𝑅_𝛽)_𝛽≥0 are such that 𝑅_𝛽≥𝑞^𝑚1𝑅_{𝛽−ℓ1−𝑠}, (14) for every 𝛽≥ℓ₁+𝑠. Moreover, we assume there exist constants 𝑑₁,𝑑₂>0 such that 𝑅_𝛽≥𝑑₁𝑞^𝑑2𝛽, (15) for every 𝛽≥0. In addition to this, we assume 𝑚₂−2𝐴₁ℓ₁+𝑠−𝑚₁𝐶+𝑑₂×ℓ₀−2𝑚₁𝑀log𝑞−𝐶ℓ₁+𝑠>0. (16)
Under the previous assumptions, there exists a positive constant 𝐶₁₁, which does not depend on 𝜖 nor 𝛿 such that 𝑧^𝑠−𝜏𝜖^ℓ01𝑞^{𝑚1(ℓ0+1)}𝜕^−ℓ1_𝑧𝑣𝜏𝑞^−𝑚1,𝑧𝑞^𝑚2_{(𝜖,𝛿,𝒮)}≤𝐶₁₁𝛿^ℓ1+𝑠‖𝑣(𝜏,𝑧)‖_{(𝜖,𝛿,𝒮)}, (17) for every 𝑣∈𝐻(𝜖,𝛿,𝒮).

Proof. Let 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)(𝑧^𝛽/𝛽!)∈𝐻(𝜖,𝛿,𝒮). We have 𝑧^𝑠−𝜏𝜖^ℓ01𝑞^{𝑚1(ℓ0+1)}𝜕^−ℓ1_𝑧𝑣𝜏𝑞^−𝑚1,𝑧𝑞^𝑚2_{(𝜖,𝛿,𝒮)}=_{𝛽≥ℓ1+𝑠}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}×𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0𝑧^𝛽𝛽!_{(𝜖,𝛿,𝒮)}. (18)
From (14), one derives that for every 𝜏∈𝑆_𝛽, 𝑣_{𝛽−ℓ1−𝑠}(𝜏𝑞^−𝑚1) is well defined and the function 𝜏↦𝑣_{𝛽−ℓ1−𝑠}(𝑞^−𝑚1𝜏) is holomorphic in 𝑆_𝛽 for every 𝛽≥ℓ₁+𝑠. The expression in (18) equals _{𝛽≥ℓ1+𝑠}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}×𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0_{𝛽,𝜖,𝑆𝛽}𝛿^𝛽.𝛽! (19)
Let 𝛽≥ℓ₁+𝑠. From the definition of the norm ‖⋅‖_{𝛽,𝜖,𝑆𝛽}, we get 𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0_{𝛽,𝜖,𝑆𝛽}=sup_{𝜏∈𝑆𝛽}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1|𝜏|𝑞^−𝑚1|𝜖|^{−𝐶(𝛽−ℓ1−𝑠)}×𝑒^{−𝑀log2((|𝜏|/|𝜖|)𝑞1−𝑚+𝛿1)}×𝜏𝜖^ℓ0|𝜏|𝑞^−𝑚1|𝜖|^{𝐶(𝛽−ℓ1−𝑠)}×𝑒^{𝑀log2(|𝜏|𝑞1−𝑚/|𝜖|+𝛿1)}×𝜏𝜖^−𝐶𝛽𝑒^{−𝑀log2(|𝜏|/|𝜖|+𝛿1)}×𝛽!!𝑞𝛽−𝑠^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}𝑞^{−𝐴1𝛽2}×𝑞^{𝐴1(𝛽−ℓ1−𝑠)2}𝑞^{−𝐴1(𝛽−ℓ1−𝑠)2}. (20)
Direct calculations allow us to obtain the following estimates 𝑒^{−𝑀log2((|𝜏|/|𝜖|)+𝛿1)+𝑀log2((|𝜏|𝑞1−𝑚/|𝜖|)+𝛿1)}≤𝐶₀₁𝑒^{−𝑀log2(|𝜏|/|𝜖|)+𝑀log2(|𝜏|𝑞1−𝑚/|𝜖|)}≤𝐶₀₂|𝜏||𝜖|^{−2𝑚1𝑀log(𝑞)}, (21) for some positive constants 𝐶₀₁ and 𝐶₀₂ only depending on 𝑞, 𝑚₁, and 𝑀. Moreover, |𝜏|𝑞^{−𝑚1𝐶(𝛽−ℓ1−𝑠)}=𝐶₀₃𝑞^{−𝑚1𝐶𝛽}|𝜏|^{𝐶(𝛽−ℓ1−𝑠)}, (22) for some constant 𝐶₀₃>0 depending on 𝑞, 𝑚₁, ℓ₁, and  𝑠. This last equality and (13) yield 𝜏𝜖^ℓ0|𝜏|𝑞^−𝑚1|𝜖|^{𝐶(𝛽−ℓ1−𝑠)}𝜏𝜖^−𝐶𝛽|𝜏||𝜖|^{−2𝑚1𝑀log(𝑞)}=𝐶₀₄|𝜖||𝜏|^{−ℓ0+𝐶(ℓ1+𝑠)+2𝑚1𝑀log(𝑞)}≤𝐶₀₅|𝜏|^{ℓ0−2𝑚1𝑀log(𝑞)−𝐶(ℓ1+𝑠)}𝑞^{−𝑚1𝐶𝛽}, (23) for some positive constants 𝐶₀₄ and 𝐶₀₅ depending on 𝑞, 𝑚, ℓ₀, ℓ₁, 𝑠, 𝐶, and ℰ. From the hypothesis (15) on 𝑅_𝛽, the last expression is upper bounded by 𝐶₀₅𝑞^{(−𝑚1𝐶+𝑑2(ℓ0−2𝑚1𝑀log(𝑞)−𝐶(ℓ1+𝑠)))𝛽}, (24) for some positive constant 𝐶₀₅ only depending on 𝑞, 𝑚₁, ℓ₀, ℓ₁, 𝑠, ℰ, 𝐶, and 𝑑₁. Now, from (16) one gets that 𝛽!/(𝛽−𝑠)!𝑞^𝑝1(𝛽) is upper bounded by a constant 𝐶₀₆>0 which does not depend on 𝛽, where 𝑝₁𝛽=𝑚₂𝛽−𝑠−𝑚₁ℓ₀+1−𝐴₁𝛽²+𝐴₁𝛽−ℓ₁−𝑠²−𝑚₁𝐶𝛽+𝑑₂ℓ₀−2𝑚₁𝑀log𝑞−𝐶ℓ₁+𝑠. (25)
Taking into account all these computations, one achieves that (20) can be upper bounded by 𝐶₀₅sup_{𝜏∈𝑆𝛽}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1|𝜏|𝑞^−𝑚1|𝜖|^{−𝐶(𝛽−ℓ1−𝑠)}×𝑒^{−𝑀log2(|𝜏|𝑞1−𝑚/|𝜖|+𝛿1)}𝑞^{−𝐴1(𝛽−ℓ1−𝑠)}. (26)
The lemma follows bearing in mind (14) and the definition of the norms in 𝐸_{𝛽−(ℓ1+𝑠),𝜖,𝑆1𝛽−(ℓ+𝑠)} and of 𝐻(𝜖,𝛿,𝒮).

Remark 3. The hypotheses made in (14), (15), and (16) are verified if one departs from 𝑅_𝛽=𝑑₁𝑞^𝑑2𝛽 for small enough positive 𝑑₂ and any 𝑑₁>0 provided (13) is satisfied and 𝑚₂−2𝐴₁(ℓ₁+𝑠)−𝑚₁𝐶>0.

Lemma 4. Let 𝐹(𝜖,𝜏) be a holomorphic and bounded function defined on ℰ×𝒮. Then, there exists a constant 𝐶₁₂=𝐶₁₂(𝐹,ℰ,𝒮)>0 such that 𝐹(𝜖,𝜏)𝑣_𝜖(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}≤𝐶₁₂𝑣_𝜖(𝜏,𝑧)_{(𝜖,𝛿,𝒮)} (27) for every 𝜖∈ℰ, every 𝛿>0 and all 𝑣_𝜖∈𝐻(𝜖,𝛿,𝒮).

Proof. Direct calculations on the definition of the norms in the space 𝐻(𝜖,𝛿,𝒮) allow us to conclude when taking 𝐶₁₂∶=max{|𝐹(𝜖,𝜏)|∶𝜖∈ℰ,𝜏∈𝒮}.

Let 𝑆≥1 and let 𝒩 be a finite subset of ℕ². We also fix 𝑎∈ℂ⧵ℝ₊, where ℝ₊ stands for the set {𝑧∈ℂ∶Re(𝑧)≥0,Im(𝑧)=0}.

For every 𝜅=(𝜅₀,𝜅₁)∈𝒩, let 𝑚_𝜅,1, 𝑚_𝜅,2 be nonnegative integers and 𝑏_𝜅(𝜖,𝑧)∈𝒪(𝐷(0,𝑟₀))[𝑧], where 𝑟₀>0 is such that ℰ⊆𝐷(0,𝑟₀). We write 𝑏_𝜅(𝜖,𝑧)=∑_{𝑠∈𝐼𝜅}𝑏_𝜅,𝑠(𝜖)𝑧^𝑠, where 𝐼_𝜅 is a finite subset of ℕ for every 𝜅∈𝒩. We assume that 1≤𝜅₁<𝑆 for every 𝜅=(𝜅₀,𝜅₁)∈𝒩.

We consider the functional equation 𝜕^𝑆_𝑧𝑊(𝜖,𝜏,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)(𝑎−𝜏)𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0×𝜕^𝜅1_𝑧𝑊𝜖,𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧 (28) with initial conditions 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏),0≤𝑗≤𝑆−1, (29) where the function (𝜖,𝜏)↦𝑊_𝑗(𝜖,𝜏) is an element in 𝒪(ℰ×𝒮) for every 0≤𝑗≤𝑆−1.

We make the following assumptions.

Assumption A. For every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and every 𝑠∈𝐼_𝜅, we assume 𝐶𝑆−𝜅₁+𝑠−𝜅₀−2𝑚_𝜅,1𝑀𝑞𝐶−log≥0,𝑆−𝜅₁+𝑠−𝜅₀−2𝑚_𝜅,1𝑀−log𝑞𝑑₂<𝑚_𝜅,2−2𝐴₁𝑆−𝜅₁+𝑠−𝑚_𝜅,1𝐶. (30)

Assumption B. 𝑅_𝛽≥𝑞^𝑚𝜅,1𝑅_{𝛽−𝜅1−𝑠} and there exist 𝑑₁, 𝑑₂>0 with 𝑅_𝛽≥𝑑₁𝑞^𝑑2𝛽, for every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and every 𝑠∈𝐼_𝜅.

Theorem 5. Let Assumption A and Assumption B be fulfilled. We assume that the initial conditions in (29) verify there exist Δ>0 and 0<𝑀<𝑀 such that for every 0≤𝑗≤𝑆−1𝑊_𝑗(𝜖,𝜏)≤Δ𝑒𝑀_log²_{(|𝜏|/|𝜖|+𝛿1)}|𝜖|^𝐾0, (31) for every 𝜏∈𝑆₀, 𝜖∈ℰ, where 𝐾₀=max{𝜅₀∶(𝜅₀,𝜅₁)∈𝒩}. Then, there exists 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), formal solution of (28)+(29), where 𝑊_𝛽∈𝒪(ℰ×𝑆_𝛽).
Then, there exist positive constants 𝐶₁₃ and 𝐶₁₄ (only depending on 𝑞, 𝑑₁, 𝑑₂, 𝐶, 𝑆, 𝛿₁, and  𝐴₁) and 𝛿>0 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₁₃𝐶𝛽!¹⁴𝛿^𝛽𝑒^{𝑀log2(|𝜏|/|𝜖|+𝛿1)}×𝜏𝜖^𝐶𝛽𝑞^𝐴1𝛽2, (32) for every 𝛽≥0, all 𝜖∈ℰ and every 𝜏∈𝑆_𝛽.

Proof. Let 𝜖∈ℰ. We put 𝔼∶={𝒪(𝑆_𝛽)∶𝛽≥0} and define the map 𝒜_𝜖 from 𝔼[[𝑧]] into itself by 𝒜_𝜖𝑊(𝜏,𝑧)∶=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)(𝑎−𝜏)𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0×𝜕^𝜅1_𝑧^−𝑆𝑊𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧+𝜕^𝜅1_𝑧𝑤_𝜖𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧, (33) where 𝑤_𝜖(𝜏,𝑧)=∑^𝑆−1_𝑗=0𝑊_𝑗(𝜖,𝜏)(𝑧^𝑗/𝑗!). For an appropriate choice of 𝛿,Δ>0, the map 𝒜_𝜖 turns out to be a Lipschitz shrinking map.

Lemma 6. There exist 𝑅,𝛿,Δ>0 (not depending on 𝜖) such that (1)‖𝒜_𝜖(𝑊(𝜏,𝑧))‖_{(𝜖,𝛿,𝒮)}≤𝑅 for every 𝑊(𝜏,𝑧)∈𝐵(0,𝑅). 𝐵(0,𝑅) denotes the closed ball centered at 0 with radius 𝑅 in 𝐻(𝜖,𝛿,𝒮). (2)𝒜_𝜖𝑊₁(𝜏,𝑧)−𝒜_𝜖𝑊₂(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}≤12𝑊₁(𝜏,𝑧)−𝑊₁(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}, (34)for every 𝑊₁, 𝑊₂∈𝐵(0,𝑅).

Proof. Let 𝑅>0 and 𝛿>0. In order to prove the first enunciate, we take 𝑊(𝜏,𝑧)∈𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝒮). From Lemmas 2 and 4 we deduce that 𝒜_𝜖(𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}≤_{𝜅=𝜅0,𝜅1∈𝒩𝑠∈𝐼𝜅}𝑀_𝜅𝑠𝑀₁𝐶₀₁𝛿^{𝑆−𝜅1+𝑠}𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}+𝑧^𝑠𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0𝜕^𝜅1𝑤_𝜖×𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧_{(𝜖,𝛿,𝒮)}, (35) with 𝑀_𝜅𝑠=sup_𝜖∈ℰ|𝑏_𝜅𝑠(𝜖)|<∞ for every 𝜅∈𝒩 and 𝑠∈𝐼_𝜅.
Let us fix 𝜅=(𝜅₀,𝜅₁)∈𝒩 and 𝑠∈𝐼_𝜅. Taking into account the definition of 𝐻(𝜖,𝛿,𝒮), we derive 𝑧^𝑠𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0𝜕^𝜅1𝑤_𝜖𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧_{(𝜖,𝛿,𝒮)}=^{𝑆−1−𝜅1−𝑠}_𝑗=𝑠𝑊_{𝑗+𝜅1−𝑠}𝜖,𝑞^{−𝑚𝜅,1}𝜏−𝜏𝜖^𝜅0_{𝑗,𝜖,𝑆𝑗}×𝑞^{𝑚𝜅,2(𝑗−𝑠)−𝑚𝜅,1(𝜅0+1)}𝑗!!𝛿𝑗−𝑠𝑗!^𝑗≤𝐶₁₄^{𝑆−1−𝜅1−𝑠}_𝑗=𝑠sup_{𝜏∈𝑆𝑗}𝑊_{𝑗+𝜅1−𝑠}𝜖,𝑞^{−𝑚𝜅,1}𝜏×𝜏𝜖^𝜅0𝜏𝜖^−𝐶𝑗𝑒^{−𝑀log2(|𝜏|/|𝜖|+𝛿1)}𝛿^𝑗, (36) for some 𝐶₁₄>0 which only depends on the parameters defining (28). The terms of the form |𝜖|^𝐶𝑗 in the previous expression can be upper bounded by an adequate constant. Taking into account (31), usual estimates in (36) derive 𝒜_𝜖𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}≤_{𝜅=𝜅0,𝜅1∈𝒩𝑠∈𝐼𝜅}𝑀_𝜅𝑠𝑀₁𝐶₀₁𝛿^{𝑆−𝜅1+𝑠}𝑊(𝜏,𝑧)_{(𝜖,𝛿,𝒮)}+𝐶₁₅, (37) for some 𝐶₁₅ depending on the parameters defining the equation and such that it tends to 0 whenever both Δ and 𝛿 tend to 0. An appropriate choice for these constants allows us to conclude the first part of the proof.
The second part of the lemma follows similar arguments as before. Let 𝑊₁, 𝑊₂∈𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝒮). One has 𝒜_𝜖(𝑊₁)−𝒜_𝜖(𝑊₂)_{(𝜖,𝛿,𝒮)}≤_{𝜅=(𝜅0,𝜅1)∈𝒩𝑠∈𝐼𝜅}𝑀_𝜅𝑠𝑀₁𝐶₀₁𝛿^{𝑆−𝜅1+𝑠}𝑊₁−𝑊_{2(𝜖,𝛿,𝒮)}. (38) The result is achieved with an adequate choice of 𝛿>0.

Let 𝑅, Δ, and let 𝛿 be as in the previous lemma. Bearing in mind Lemma 6 one can apply the shrinking map theorem on complete metric spaces to guarantee the existence of a fixed point for 𝒜_𝜖 in 𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝒮), say 𝑊_𝜖, which verifies ‖𝑊_𝜖(𝜏,𝑧)‖_{(𝜖,𝛿,𝒮)}≤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). (39)

We put 𝑊(𝜖,𝜏,𝑧)∶=𝑊_𝜖(𝜏,𝑧), and 𝑊(𝜖,𝜏,𝑧)∶=𝜕_𝑧^−𝑆𝑊(𝜖,𝜏,𝑧)+𝑤_𝜖(𝜏,𝑧). Then, 𝑊(𝜖,𝜏,𝑧) can be written as a formal power series in 𝑧 as 𝑊(𝜖,𝜏,𝑧)=_𝛽≥0𝑊_𝛽()𝑧𝜖,𝜏^𝛽,𝛽! (40) where 𝑊_𝛽+𝑆(𝜖,𝜏)=𝑊_𝛽,𝜖(𝜏) for every 𝛽≥0.

From the construction of 𝑊(𝜖,𝜏,𝑧), we have 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!) is a formal solution of (28)+(29). Moreover, from the domain of holomorphy of the initial conditions in (29) and the recursion formula satisfied by the coefficients in 𝑊(𝜖,𝜏,𝑧), we get 𝑊_ℎ+𝑆(𝜖,𝜏)ℎ!=_{𝜅=𝜅0,𝜅1∈𝒩ℎ1+ℎ2=ℎ,ℎ1∈𝐼𝜅}𝑏_𝜅,ℎ1(𝜖−𝜏)𝜖^𝜅0×𝑞^{𝑚𝜅,2ℎ2}(𝑎−𝜏)ℎ₂!𝑞^{𝑚𝜅,1(𝜅0+1)}×𝑊_ℎ2+𝜅1𝜖,𝑞^{−𝑚𝜅,1}𝜏. (41) We can conclude the function (𝜖,𝜏)↦𝑊_𝛽∈𝒪(ℰ×𝒮) for every 𝛽≥0.

Finally, the estimates in (32) are obtained for every 𝛽≥0 from the fact that 𝑊_𝜖∈𝐵(0,𝑅)⊆𝐻(𝜖,𝛿,𝒮). The definition of the elements in 𝐻(𝜖,𝛿,𝒮) lead us to 𝑊_{𝛽,𝜖𝛽,𝜖,𝑆𝛽}1≤𝑅𝛽!𝛿^𝛽, (42) so that 𝑊_𝛽(𝜖,𝜏)=𝑊_{𝛽−𝑆,𝜖}(𝜏!1)≤𝑅𝛽−𝑆𝛿^𝛽−𝑆×𝑒^{𝑀log2(|𝜏|/|𝜖|+𝛿1)}𝜏𝜖^{𝐶(𝛽−𝑆)}𝑞^{𝐴1(𝛽−𝑆)2}, (43) for every 𝛽≥𝑆. In addition to this, Assumption B and usual estimates allow us to refine the previous estimates leading to 𝑊_𝛽(𝜖,𝜏)≤𝐶₁₃𝐶𝛽!¹⁴𝛿^𝛽𝑒^{𝑀log2((|𝜏|/|𝜖|)+𝛿1)}𝜏𝜖^𝐶𝛽𝑞^𝐴1𝛽2, (44) for some constants 𝐶₁₃>0 and 𝐶₁₄>0 which only depend on 𝑞, 𝑑₁, 𝑑₂, 𝐶, 𝑆, 𝛿₁, and 𝐴₁. This is valid for every 𝜖∈ℰ and 𝜏∈𝑆_𝛽. The hypothesis (31) in the enunciate allows us to affirm that (32) is also valid for 0≤𝛽≤𝑆−1.

Remark 7. One derives holomorphy of 𝑊_𝛽 in the variable 𝜏 in the whole sector 𝒮 and not only in 𝑆_𝛽 for every 𝛽≥𝑆 whilst the estimates are only given for 𝜏∈𝑆_𝛽. It is also worth saying that 𝑅>0 can be arbitrarily chosen whenever 𝑠>0 for every 𝑠∈𝐼_𝜅, 𝜅∈𝒩.

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

We provide the solution of a Cauchy problem with analogous equation as the one studied in the previous section, written as a formal power series in 𝑧 with coefficients in an appropriate Banach space of functions in the variable 𝜏 and the perturbation parameter 𝜖. In Section 2, the domain of holomorphy of the coefficients remains invariant from the domain of holomorphy of the initial conditions. This happens because the dilation operator 𝜏↦𝑞⁻¹𝜏 sends points in any infinite sector in the complex plane with vertex at the origin into itself. Now, the domain of holomorphy of the coefficients for the formal solution of the Cauchy problem under study depends on the index considered. More precisely, if the initial conditions present a singularity at some point 𝑎∈ℂ in the variable 𝜏, the coefficients of the formal solution of the Cauchy problem have singularities in 𝜏 that tend to 0, providing a small divisor phenomenon.

For every 𝜌>0, 𝐷̇_𝜌 stands for the set 𝐷(0,𝜌)⧵{0}. We preserve the value of the positive constants 𝑀,𝐴₁,𝐶, and 𝛿₁ from the previous section. Let 𝑟₀>0 with ℰ⊆𝐷(0,𝑟₀) and let (𝑅_𝛽)_𝛽≥0 be a sequence of positive real numbers.

Definition 8. Let 𝛽∈ℕ. For 𝑟₀>0 and 𝜖∈𝐷(0,𝑟₀)⧵{0}; 𝐸²_{̇𝐷𝛽,𝜖,}𝑅_𝛽 stands for the vector space of functions 𝐷̇𝑅𝑣∈𝒪(_𝛽) such that |𝑣(𝜏)|_{̇𝐷𝛽,𝜖,}𝑅_𝛽∶=sup_{̇𝐷𝜏∈}𝑅_𝛽𝑣(𝜏|𝜖||)|^𝐶𝛽𝑒^{𝑀log2(|𝜏|+𝛿1)}𝑞^{−𝐴1𝛽2} (45) is finite. Let 𝛿>0. We write 𝐻₂(𝜖,𝛿) for the vector space of all formal power 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)𝑧^𝛽/𝛽! such that 𝑣_𝛽∈𝐸²_{̇𝐷𝛽,𝜖,}𝑅_𝛽 with |𝑣(𝜏,𝑧)|_(𝜖,𝛿)∶=_𝛽≥0𝑣_𝛽(𝜏)_{̇𝐷𝛽,𝜖,}𝑅_𝛽𝛿^𝛽𝛽!<∞. (46) The pair (𝐻₂(𝜖,𝛿),|⋅|_(𝜖,𝛿)) is a Banach space.

Lemma 9. Let 𝑠, ℓ₀, ℓ₁, 𝑚₁, and 𝑚₂∈ℕ, 𝛿>0, and 𝜖∈𝐷(0,𝑟₀)⧵{0}. We assume that 𝐶ℓ₁+𝑠−ℓ₀≥0,𝑚₂−2𝐴₁ℓ₁+𝑠>0. (47)
Moreover, we assume that the elements of the sequence (𝑅_𝛽)_𝛽≥0 are such that 𝑅_𝛽≤𝑞^𝑚1𝑅_{𝛽−ℓ1−𝑠}, (48) for every 𝛽≥ℓ₁+𝑠.
Under the previous assumptions, there exists a positive constant 𝐶₂₁ which depends on 𝐶, 𝑞, 𝑚₁, 𝑚₂, 𝑠, ℓ₀, ℓ₁, 𝑀, 𝐴₁, 𝛿₁, and 𝑟₀ (not depending on 𝜖 nor 𝛿) such that 𝑧^𝑠−𝜏𝜖^ℓ01𝑞^{𝑚1(ℓ0+1)}𝜕^−ℓ1_𝑧𝑣𝜏𝑞^−𝑚1,𝑧𝑞^𝑚2_(𝜖,𝛿)≤𝐶₂₁𝛿^ℓ1+𝑠|𝑣(𝜏,𝑧)|_(𝜖,𝛿), (49) for every 𝑣∈𝐻₂(𝜖,𝛿).

Proof. Let 𝑣(𝜏,𝑧)=∑_𝛽≥0𝑣_𝛽(𝜏)(𝑧^𝛽/𝛽!) be an element of 𝐻₂(𝜖,𝛿). We have 𝑧^𝑠−𝜏𝜖^ℓ01𝑞^{𝑚1(ℓ0+1)}𝜕^−ℓ1_𝑧𝑣𝜏𝑞^−𝑚1,𝑧𝑞^𝑚2_(𝜖,𝛿)=_{𝛽≥ℓ1+𝑠}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}×𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0𝑧^𝛽𝛽!_(𝜖,𝛿). (50)
From (48), one derives that for every 𝐷̇𝑅𝜏∈_{1𝛽−ℓ−𝑠}, 𝑣_{𝛽−ℓ1−𝑠}(𝜏𝑞^−𝑚1) is well defined. In addition to this, the function 𝜏↦𝑣_{𝛽−ℓ1−𝑠}(𝜏𝑞^−𝑚1) is holomorphic in 𝐷̇𝑅_𝛽 for every 𝛽≥ℓ₁+𝑠. The expression in (50) equals _{𝛽≥ℓ1+𝑠}𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}×𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0_{̇𝐷𝛽,𝜖,}𝑅_𝛽𝛿^𝛽.𝛽! (51)
Let 𝛽≥ℓ₁+𝑠. From the definition of the norm |⋅|_{̇𝐷𝛽,𝜖,}𝑅_𝛽, we get 𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1𝑞^{𝑚2(𝛽−𝑠)−𝑚1(ℓ0+1)}×𝛽!−𝜏(𝛽−𝑠)!𝜖^ℓ0_{̇𝐷𝛽,𝜖,}𝑅_𝛽=sup_{̇𝐷𝜏∈}𝑅_𝛽𝑣_{𝛽−ℓ1−𝑠}𝜏𝑞^−𝑚1×|𝜖|^{𝐶(𝛽−ℓ1−𝑠)}𝑒^{𝑀log2(|𝜏|/𝑞𝑚1+𝛿1)}×𝜏𝜖^ℓ0|𝜖|^{𝐶(ℓ1+𝑠)}×𝑒^{𝑀(log2(|𝜏|/𝑞𝑚1+𝛿1)−log2(|𝜏|+𝛿1))}×𝑞^𝑝1(𝛽)𝛽!!𝑞𝛽−𝑠^{−𝐴1(𝛽−ℓ1−𝑠)2}, (52) with 𝑝₁(𝛽)=𝑚₂(𝛽−𝑠)−𝑚₁(ℓ₀+1)−𝐴₁𝛽²+𝐴₁(𝛽−ℓ₁−𝑠)².
The result follows provided that one is able to estimate the expression 𝑞^𝑝1(𝛽)𝛽!!|𝜏|𝛽−𝑠^ℓ0|𝜖|^{𝐶(ℓ1+𝑠)−ℓ0}×𝑒^{𝑀(log2((|𝜏|/𝑞𝑚1)+𝛿1)−log2(|𝜏|+𝛿1))}. (53)
From the first of the hypotheses made in (47), |𝜖|^{𝐶(ℓ1+𝑠)−ℓ0} is upper bounded by a constant. Also, taking into account (48), there exists 𝑅>0 such that |𝜏|≤𝑅 for every 𝜏∈∪_𝛽≥0𝐷̇𝑅_𝛽, so that |𝜏|^ℓ0exp𝑀log²|𝜏|𝑞^−𝑚1+𝛿₁−log²|𝜏|+𝛿₁≤𝐷₂₁𝑞,𝑚₁,𝛿₁,𝑀,ℓ₀, (54) for some positive constant 𝐷₂₁. The result immediately follows from (47) that guarantees that 𝛽!/(𝛽−𝑠)!𝑞^𝑝1(𝛽) is bounded from above.

Let 𝑅>0 be as in the proof of the previous lemma, that is, 𝑅≥𝑅_𝛽 for every 𝛽≥0.

Lemma 10. Let 𝐹(𝜖,𝜏) be a holomorphic and bounded function defined on 𝐷(0,𝑟₀)×𝐷(0,𝑅).
Then, there exists a constant 𝐶₂₂=𝐶₂₂(𝐹)>0 such that 𝐹(𝜖,𝜏)𝑣_𝜖(𝜏,𝑧)_(𝜖,𝛿)≤𝐶₂₂𝑣_𝜖(𝜏,𝑧)_(𝜖,𝛿), (55) for every 𝜖∈𝐷(0,𝑟₀)⧵{0}, every 𝛿>0 and all 𝑣_𝜖∈𝐻₂(𝜖,𝛿).

Proof. Direct calculations on the definition of the norms in the space 𝐻₂(𝜖,𝛿) allow us to conclude when taking 𝐶₂₂∶=max{|𝐹(𝜖,𝜏)|∶𝜖∈𝐷(0,𝑟₀),𝜏∈𝐷(0,𝑅)}.

Let 𝑆≥1 and let 𝒩 be a finite subset of ℕ². We also fix 𝑎∈ℂ⧵ℝ₊ such that |𝑎|≥𝑅, with 𝑅 as before.

Let 𝑚_𝜅,1, 𝑚_𝜅,2, and let 𝑏_𝜅 be as in Section 2, for every 𝜅=(𝜅₀,𝜅₁).

We consider the functional equation 𝜕^𝑆_𝑧𝑊(𝜖,𝜏,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)(𝑎−𝜏)𝑞^{𝑚𝜅,1(𝜅0+1)}−𝜏𝜖^𝜅0×𝜕^𝜅1_𝑧𝑊𝜖,𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧 (56) with initial conditions 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏),0≤𝑗≤𝑆−1, (57) where the function (𝜖,𝜏)↦𝑊_𝑗(𝜖,𝜏) is an element in 𝒪((𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×₀) for every 0≤𝑗≤𝑆−1.

We make the following assumptions.

Assumption A^′. For every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and every 𝑠∈𝐼_𝜅, we assume 𝐶𝑆−𝜅₁+𝑠−𝜅₀≥0,𝑚_𝜅,2−2𝐴₁𝑆−𝜅₁+𝑠>0. (58)

Remark 11. Observe that Assumption A implies Assumption A′.

Assumption B^′. We assume 𝑅_𝛽≤𝑞^𝑚𝜅,1𝑅_{𝛽−𝜅1−𝑠} for every 𝜅=(𝜅₀,𝜅₁)∈𝒩, every 𝑠∈𝐼_𝜅, and every 𝛽≥𝜅₁+𝑠.

We first state a result which provides a concrete value for the elements in (𝑅_𝛽)_𝛽≥0 under Assumption B′. The choice is made in two respects: first, to clarify how the singularities suffer propagation in the formal solution of (56)+(57), with respect to the variable 𝜏, and second, to provide acceptable domains of holomorphy for such coefficients when regarding this phenomenon of propagation of singularities. Any other appropriate choice for the elements in (𝑅_𝛽)_𝛽≥0 regarding these issues would also be fairish for our purpose.

Lemma 12. Let 𝑑₁,𝑑₂>0, and 𝜖∈𝐷(0,𝑟₀)⧵{0}.
We put 𝑅_𝛽∶=𝑅₀ for 𝛽=0,1,…,𝑆−1, and 𝑅_𝛽=𝑑₁𝑞𝑑_2𝛽 for every 𝛽≥𝑆. Let us assume that (56)+(57) has a formal solution in 𝑧, 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!). Then, there exists 𝑑₂₀ such that for every 𝑑₂≥𝑑₂₀, the function 𝜏↦𝑊_𝛽(𝜖,𝜏) belongs to 𝐷̇𝑅𝒪(_𝛽) for every 𝛽≥𝑆 and all 𝜖∈𝐷(0,𝑟₀)⧵{0}.

Proof. Let 𝑊(𝜖,𝜏,𝑧) be a formal power series in 𝑧 of the form ∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!). One can plug the formal power series into (56) to obtain the recursion formula in (41) for the coefficients (𝑊_𝛽)_𝛽≥𝑆. From this recurrence, one derives that the domain of holomorphy for 𝑊_ℎ+𝑆 in the variable 𝜏 depends on the domain of holomorphy on 𝜏 of 𝑊_ℎ2+𝜅1 and also on 𝑞^{−𝑚𝜅,1} for every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and every 0≤ℎ₂≤ℎ such that ℎ−ℎ₂∈𝐼_𝜅.
The initial conditions 𝑊₀,…,𝑊_𝑆−1 are holomorphic functions in 𝐷̇𝑅₀.

Lemma 13. For every 𝑁≥1 the coefficients 𝑊_{𝑁𝑆−(𝑁−1)𝜅10},…,𝑊_{(𝑁+1)𝑆−𝑁𝜅10} turn out to be holomorphic functions in 𝐷̇_{𝑞𝑁𝑚𝜅,1}𝑅₀, for 𝜅₁₀∶=max{𝜅₁∶(𝜅₀,𝜅₁)∈𝒩} and 𝑚_𝜅,1∶=max{𝑚_𝜅,1∶𝜅∈𝒩}.

Proof. We prove it by recurrence on 𝑁 and regarding the recursion formula (41).
Let 𝑁=1. One has ℎ₂+𝜅₁≤𝑆−1 for any ℎ₂,𝜅₁ as in (41) if and only if ℎ₂≤𝑆−1−𝜅₁ for every (𝜅₀,𝜅₁)∈𝒩, it is to say, if and only if ℎ₂≤𝑆−1−𝜅₁₀. In this case, 𝑊_ℎ+𝑆 only depends on the initial conditions (𝑊_𝑗)_{0≤𝑗≤𝑆−1}. Moreover, ℎ+𝑆∈𝑆,𝑆+1,…,2𝑆−𝜅₁₀−1, (59) and the dilation on the variable 𝜏 allows us to obtain that 𝑊_𝑆,…,𝑊_{2𝑆−𝜅10−1} are holomorphic functions in 𝐷̇_{𝑞𝑚𝜅,1}𝑅₀.
The proof can be followed recursively for every 𝑁≥2 by considering analogous blocks of indices as before.

Regarding Lemma 13, the proof of Lemma 12 is concluded if one can check that for every 𝑁≥1, 𝑅_𝛽≤𝑅₀𝑞^{𝑁𝑚𝜅,1} whenever 𝛽∈𝑁𝑆−(𝑁−1)𝜅₁₀,…,(𝑁+1)𝑆−𝑁𝜅₁₀=𝑁−1𝑆−𝜅₁₀+𝜅₁₀,…,𝑁𝑆−𝜅₁₀+𝑆−1. (60)

Let 𝑁≥1 and 𝛽=𝑁(𝑆−𝜅₁₀)+𝐿, with 𝜅₁₀≤𝐿≤𝑆−1. Let 𝑑₁≤𝑅₀. We have 𝑅_𝛽=𝑑₁𝑞𝑑_{2[𝑁(𝑆−𝜅10)+𝐿]}≤𝑅₀𝑞^{𝑁𝑚𝜅,1} if and only if 𝑁𝑚_𝜅,1≤𝑑₂[𝑁(𝑆−𝜅₁₀)+𝐿]. The result follows for any 𝑑₂≥𝑚_𝜅,1/(𝑆−𝜅₁₀).

Lemma 14. Let 𝑅_𝛽 be defined as in Lemma 12. Then, (𝑅_𝛽)_𝛽≥0 satisfies Assumption B′.

Proof. From the definition of 𝑅_𝛽, the lemma follows when taking 𝑑₂≥𝑚_𝜅,1/(𝜅₁+𝑠) for every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and every 𝑠∈𝐼_𝜅.

Assumption 𝐵^′′. We assume 𝑅_𝛽=𝑅₀ for 0≤𝛽≤𝑆−1 and 𝑅_𝛽∶=𝑑₁𝑞𝑑_2𝛽 for any 𝑑₂≥𝑑₂₀, with 𝑑₂₀>0 as in Lemma 12.

As it has been pointed out before, the Assumption B′ is substituted in the present work by Assumption B′′ with the cost of losing some generality but giving concrete values for 𝑅_𝛽, for every 𝛽≥0. The incoming theorem is valid when considering any other choice of the elements in (𝑅_𝛽)_𝛽≥0 satisfying Assumption B′.

Theorem 15. Let Assumption A′ and Assumption B′′ be fulfilled. We also make the next assumption on the initial conditions (57); there exist Δ>0 and 0<𝑀<𝑀 such that 𝑊_𝑗(𝜖,𝜏)≤Δ𝑒𝑀_log²_{(|𝜏|+𝛿1)}|𝜖|^𝐾0, (61) for every 𝜏∈𝐷(0,𝑅₀), 𝜖∈𝐷(0,𝑟₀)⧵{0} and 0≤𝑗≤𝑆−1, where 𝐾₀=max{𝜅₀∶(𝜅₀,𝜅₁)∈𝒩}. Then, there exists a formal power series 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), with 𝑊_𝛽(𝜖,𝜏)∈𝒪((𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝛽), which provides a formal solution of (56)+(57). Moreover, there exist positive constants 𝐶₂₃ and 𝐶₂₄ (only depending on 𝑟₀, 𝑅₀, 𝑞, 𝐶, 𝑆, 𝐴₁, 𝛿₁, and 𝑀) and 0<𝛿<1 such that 𝑊_𝛽(𝜖,𝜏)≤𝐶₂₃𝐶𝛽!²⁴𝛿^𝛽|𝜖|^−𝐶𝛽×𝑒^{𝑀log2(|𝜏|+𝛿1)}𝑞^𝐴1𝛽2, (62) for every 𝜖∈𝐷(0,𝑟₀)⧵{0}, 𝐷̇𝑅𝜏∈_𝛽 and for every 𝛽∈ℕ.

Proof. The proof follows analogous steps as the one of Theorem 5, so we do not enter into details as to not to repeat arguments.
Let 𝜖∈𝐷(0,𝑟₀)⧵{0} and 0<𝛿<1. The set 𝔼 is taken to be 𝐷̇𝑅{𝒪(_𝛽)∶𝛽≥0}. We consider the map 𝒜_𝜖 from 𝔼[[𝑧]] into itself defined in the same way as in (33).
From Lemma 12 and Assumption B′′, the unique formal solution of (56)+(57) determined by the recursion formula (41), 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), is such that 𝑊_𝛽(𝜖,𝜏)∈𝒪(𝐷̇(0,𝑅_𝛽)) for every 𝛽≥0.
Regarding the initial conditions of the Cauchy problem, one can reduce 𝑑₁, if necessary, so that 𝑅_𝑗+𝑠𝑞^{−𝑚𝜅,1}≤𝑅₀ and so the map 𝜏↦𝑊_𝑗+𝜅1(𝜖,𝑞^{−𝑚𝜅,1}𝜏) is well defined, for every 𝜅=(𝜅₀,𝜅₁)∈𝒩, every 𝑠∈𝐼_𝜅, and 𝑗=0,1,…,𝑆−1−𝜅₁. Moreover, from (61), the expression 𝑧^𝑠𝑞^𝑚𝜅,1𝜅₀−𝜏+1𝜖^𝜅0𝜕^𝜅1_𝑧𝑤_𝜖𝑞^{−𝑚𝜅,1}𝜏,𝑞^𝑚𝜅,2𝑧_(𝜖,𝛿) (63) can be estimated in an analogous manner as in the corresponding step of the proof of Theorem 5, for every 𝜅=(𝜅₀,𝜅₁)∈𝒩 and all 𝑠∈𝐼_𝜅.

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

4.1. Laplace Transform and 𝑞-Gevrey Asymptotic Expansion

In this subsection, we recall some identities for the Laplace transform and state some definitions and first results on 𝑞-Gevrey asymptotic expansions. The next lemma can be found in [6].

Lemma 16. Let 𝑚∈ℕ and let 𝑤₁(𝜏) be a holomorphic function in an unbounded sector 𝑈 such that there exist 𝐶,𝐾>0 with |𝑤(𝜏)|≤𝐶exp(𝐾|𝜏|), (64) for every 𝜏∈𝑈. Let 𝒟 be an unbounded sector with vertex at 0 which verifies that (𝑡)∈−𝜋𝑑+arg2,𝜋2,cos𝑑+arg(𝑡)≥𝛿₂, (65) for some 𝑑∈ℝ and 𝛿₂>0. Then, 𝑡⟼_𝐿𝑑𝑤(𝜏)𝑒^−𝑡𝜏𝑑𝜏 (66) is a holomorphic and bounded function defined for 𝑡∈𝒟∩{|𝑡|>𝐾/𝛿₂}. Moreover, the following identities hold: _𝐿𝑑𝜏^𝑚𝑒^−𝑡𝜏𝑑𝜏=𝑚!𝑡^𝑚+1,𝜕_𝑡𝐿𝑑𝑤(𝜏)𝑒^−𝑡𝜏𝑑𝜏=_𝐿𝑑(−𝜏)𝑤(𝜏)𝑒^−𝑡𝜏𝑑𝜏, (67) where 𝐿_𝑑=ℝ₊𝑒^𝑖𝑑⊆𝑈∪{0}, for all 𝑡∈𝒟∩{|𝑡|>𝐾/𝛿₂}.

In the sequel, we work with functions which satisfy more restrictive bounds that the ones in (64). Indeed, we deal with bounds of the form 𝐶exp(𝐾log²|𝜏|), for some 𝐶,𝐾>0. This alters the asymptotic behaviour of the Laplace transform and cause the appearance of 𝑞-Gevrey asymptotic expansions associated with estimates related to the sequence (𝑞^−𝑛2)_𝑛≥0.

For any open sector 𝑆={𝑧∈ℂ∶𝑎<arg(𝑧)<𝑏,|𝑧|<𝜌} in the complex plane with vertex at 0 with 𝜌 finite or infinite and 0≤𝑎<𝑏≤2𝜋, we say the finite sector 𝑆 with vertex at the origin is a proper subsector of 𝑆, and we denote it as 𝑆≺𝑆, if 𝑆={𝜏∈ℂ^⋆∶𝑎₁<arg(𝜏)<𝑏₁,|𝑧|<𝜌} for some 0≤𝑎<𝑎₁<𝑏₁<𝑏≤2𝜋 and some 𝜌∈ℝ, 0<𝜌<𝜌.

ℍ stands for a complex Banach space.

We preserve the Definition of 𝑞-Gevrey asymptotic expansion established in [5], in order to be coherent with the definitions in that work.

Definition 17. Let 𝑆 be a sector in ℂ^⋆ with vertex at the origin, and 𝐴>0. We say a holomorphic function 𝑓∶𝑆→ℍ admits the formal power series 𝑓=∑_𝑛≥0𝑓_𝑛𝜖^𝑛∈ℍ[[𝜖]] as its 𝑞-Gevrey asymptotic expansion of type 𝐴 in 𝑆 if for every 𝑆≺𝑆 there exist 𝐶₁,𝐻>0 such that 𝑓(𝜖)−^𝑁_𝑛=0𝑓_𝑛𝜖^𝑛_ℍ≤𝐶₁𝐻^𝑁𝑞^{−𝐴(𝑁2/2)}|𝜖|^𝑁+1(𝑁+1)!,𝑁≥0, (68) for every 𝜖∈𝑆.

The next proposition, detailed in [5] in the more general geometry of 𝑞-spirals, characterises null 𝑞-Gevrey asymptotic expansion.

Proposition 18. Let 𝐴>0 and 𝑓∶𝑆→ℍ a holomorphic function in a sector 𝑆 with vertex at the origin. The following holds.(i)If 𝑓 admits the power series with null coefficients, which is denoted by 0, as its 𝑞-Gevrey asymptotic expansion of type 𝐴, then for every 𝑆≺𝑆 there exists 𝐶₁>0 with 𝑓(𝜖)_ℍ≤𝐶₁𝑒^−(1/𝑎_{)(1/2(−log(𝑞)))log}²_|𝜖|, (69)for every 𝜖∈𝑆 and every 𝑎>𝐴. (ii)If for every 𝑆≺𝑆 there exists 𝐶₁>0 with 𝑓(𝜖)_ℍ≤𝐶₁𝑒^{−(1/𝐴)(1/2(−log(𝑞)))log2|𝜖|}, (70)for every 𝜖∈𝑆, then 𝑓 admits 0 as its 𝑞-Gevrey asymptotic expansion of type 𝑎 in 𝑆, for every 𝑎>𝐴.

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

We recall the definition of a good covering.

Definition 19. Let {ℰ_𝑖}_{0≤𝑖≤𝜈−1} be a finite family of open sectors with vertex at the origin and finite radius 𝜖₀. We assume that ℰ_𝑖∩ℰ_𝑖+1≠∅ for 0≤𝑖≤𝜈−1 (we put ℰ_𝜈∶=ℰ₀) and also that 𝐷(0,𝜈₀)⧵{0}⊆∪^𝜈−1_𝑖=0ℰ_𝑖 for some 𝜈₀>0. Then, the family {ℰ_𝑖}_{1≤𝑖≤𝜈−1} is known as a good covering in ℂ^⋆.

Definition 20. Let {ℰ_𝑖}_{0≤𝑖≤𝜈−1} be a good covering in ℂ^⋆. We consider a family {{𝑆_𝑖}_{0≤𝑖≤𝜈−1},𝒯} such that the following holds. (1)There exist 𝑑_𝑖∈[0,2𝜋), 0<𝜃_𝑖<𝜋/2 such that 𝑆_𝑖=𝑆_𝑖𝑑_𝑖,𝜃_𝑖∶=𝑡∈ℂ^⋆∶arg(𝑡)−𝑑_𝑖<𝜃_𝑖2, (71) for every 0≤𝑖≤𝜈−1. (2)𝒯 is an unbounded subset of an open sector with vertex at the origin. We assume |𝑡|≥𝑟_𝒯 for every 𝑡∈𝒯. (3)For every 0≤𝑖≤𝜈−1 and 𝜏∈𝑆_𝑖, there exists 𝛿₃>0 such that |𝑎−𝜏|>𝛿₃. (4)For every 0≤𝑖≤𝜈−1, 𝑡∈𝒯, and 𝜖∈ℰ_𝑖, one has 𝑡/𝜖∈𝑆_𝑖. Under the previous settings, we say the family {{𝑆_𝑖}_{0≤𝑖≤𝜈−1},𝒯} is associated to the good covering {ℰ_𝑖}_{0≤𝑖≤𝜈−1}.

Let us consider a good covering in ℂ^⋆, {ℰ_𝑖}_{0≤𝑖≤𝜈−1}.

Let 𝑆≥1 and 𝑎∈ℂ⧵ℝ₊. We consider a finite subset of ℕ², 𝒩. For every 𝜅=(𝜅₀,𝜅₁)∈𝒩, let 𝑚_𝜅,1,𝑚_𝜅,2∈ℕ and let 𝑏_𝜅(𝜖,𝑧) be a holomorphic and bounded function on 𝐷(0,𝑟₀)×ℂ, for some 𝑟₀>0. For each 0≤𝑖≤𝜈−1, we consider the following main Cauchy problem in the present work: 𝜖𝜕_𝑡𝜕^𝑆_𝑧𝑋_𝑖(𝜖,𝑡,𝑧)+𝑎𝜕^𝑆_𝑧𝑋_𝑖(𝜖,𝑡,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)𝜕^𝜅0_𝑡𝜕^𝜅1_𝑧𝑋_𝑖×𝜖,𝑞^𝑚𝜅,1𝑡,𝑞^𝑚𝜅,2𝑧 (72) with initial conditions 𝜕^𝑗_𝑧𝑋_𝑖(𝜖,𝑡,0)=𝜙_𝑖,𝑗(𝜖,𝑡)0≤𝑗≤𝑆−1, (73) where the functions 𝜙_𝑖,𝑗(𝜖,𝑡) are constructed as follows. Let {{𝑆_𝑖}_{0≤𝑖≤𝜈−1},𝒯} be a family of open sets associated with the good covering {ℰ_𝑖}_{0≤𝑖≤𝜈−1}.

From now on, we assume the values of (𝑅_𝛽)_𝛽≥0 and (𝑅_𝛽)_𝛽≥0 are those in the preceeding sections. If necessary, one can adjust the values of 𝑑₁, 𝑑₂, 𝑑₁, and 𝑑₂ so that 𝑅_𝛽<𝑅_𝛽 for every 𝛽≥0, so that 𝐷̇𝑅_𝛽∩𝑆_𝑖,𝛽≠∅ for every 𝛽≥0 and every 0≤𝑖≤𝜈−1. Here, we have put 𝑆_𝑖,𝛽∶=𝜏∈ℂ^⋆∶𝜏∈𝑆_𝑖,|𝜏|>𝑅_𝛽,𝛽≥0,0≤𝑖≤𝜈−1. (74)

For every 0≤𝑗≤𝜈−1, we assume that (𝜖,𝜏)↦𝑊_𝑗(𝜖,𝜏) is a bounded and holomorphic function on (𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝑗 verifying 𝑊_𝑗(𝜖,𝜏)≤Δ𝑒𝑀_log²_{((|𝜏|/|𝜖|)+𝛿1)}|𝜖|^𝐾0, (75) for every (𝜖,𝜏)∈(𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝑗. Here 𝑀, 𝐾₀, Δ, and 𝛿₁ are the constants provided in Theorem 5. Assume that 𝑊_𝑗(𝜖,𝜏) can be extended to an analytic function (𝜖,𝜏)↦𝑊_{ℰ𝑖,𝑆𝑗,𝑗}(𝜖,𝜏) defined on ℰ_𝑖×𝑆_𝑖,𝑗 and 𝑊_ℰ𝑖,𝑆_{𝑖,𝑗,𝑗}(𝜖,𝜏)≤Δ𝑒𝑀^{log2(|𝜏|+𝛿}₁⁾|𝜖|^𝐾0, (76) for every (𝜖,𝜏)∈ℰ_𝑖×𝑆_𝑖,𝑗.

Take 𝛾_𝑖 such that ℝ₊𝑒^{𝛾𝑖√−1}𝑅⊆𝐷_𝑗∪𝑆_𝑖,𝑗. We put 𝜙_𝑖,𝑗(𝑡,𝜖)∶=_𝐿𝛾𝑖𝑊_{ℰ𝑖,𝑆𝑗,𝑗}(𝜖,𝜏)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏, (77) for every (𝜖,𝑡)∈ℰ_𝑖×𝒯. One can check that 𝜙_𝑖,𝑗 is well defined and holomorphic in 𝒯×ℰ_𝑖. Indeed, there exists 𝛿₂>0 such that cos(𝛾_𝑗−arg(𝑡/𝜖))>𝛿₂ for every (𝑡,𝜖)∈ℰ_𝑖×𝒯. Moreover, from the growth properties of 𝑊_ℰ𝑖,𝑆_{𝑖,𝑗,𝑗}(𝜖,𝜏), one deduces _𝐿𝛾𝑖𝑊_{ℰ𝑖,𝑆𝑗,𝑗}(𝜖,𝜏)𝑒^{−𝑡𝜏/𝜖}≤𝑑𝜏_∞0𝑊_{ℰ𝑖,𝑆𝑗,𝑗}𝜖,𝑠𝑒^{√−1𝛾𝑖}𝑒^{−|𝑡|𝛿2𝑠/|𝜖|}|𝜖|𝑑𝑠≤Δ^𝐾0∞₀𝑒𝑀_log²_{(𝑠/|𝜖|+𝛿1)}𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|}𝑑𝑠, (78) which is convergent for every (𝜖,𝑡)∈ℰ_𝑖×𝒯.

Theorem 21. Let Assumptions A, B, and B′′ be fulfilled. For every 0≤𝑖≤𝜈−1, we consider the problem (72)+(73) with initial conditions constructed as above. Then, the problem (72)+(73) admits a solution 𝑋_𝑖(𝜖,𝑡,𝑧) which is holomorphic and bounded in ℰ_𝑖×𝒯×ℂ.
Moreover, for every 0≤𝑖≤𝜈−1 and for every Δ>1 there exists 𝐸₁>0 (not depending on 𝜖) such that sup_{𝑡∈𝒯𝑧∈𝐷(0,𝜌)}𝑋_𝑖+1(𝜖,𝑡,𝑧)−𝑋_𝑖(𝜖,𝑡,𝑧)≤𝐸₁𝑒^{−((𝐴1/(2𝑑22Δ))(1/(−log(𝑞))))log2|𝜖|}, (79) for every 𝜖∈ℰ_𝑖∩ℰ_𝑖+1 (where, by convention, 𝑋_𝜈∶=𝑋₀).

Proof. Let 0≤𝑖≤𝜈−1 and 𝜖∈ℰ_𝑖. We consider the Cauchy problem (28) with initial conditions given by 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_ℰ𝑖,𝑆_{𝑖,𝑗,𝑗}()𝜖,𝜏,0≤𝑗≤𝑆−1. (80) Theorem 5 shows that the problem (31)+(80) has a formal solution 𝑊(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑊_𝛽(𝜖,𝜏)(𝑧^𝛽/𝛽!), with 𝑊_𝛽∈𝒪(ℰ_𝑖×𝑆_𝑖,𝛽) for every 𝛽≥0. Moreover, for every 𝛽≥0 one has 𝑊_𝛽(𝜖,𝜏)≤𝐶₁₃𝐶𝛽!¹⁴𝛿^𝛽×𝑒^{𝑀log2(|𝜏|/|𝜖|+𝛿1)}𝜏𝜖^𝐶𝛽𝑞^𝐴1𝛽2, (81) for every (𝜖,𝜏)∈ℰ_𝑖×𝑆_𝑖,𝛽, where 𝐶₁₃,𝐶₁₄, and 𝛿 are positive constants provided in the proof of Theorem 5. In a parallel direction, one can consider the same Cauchy problem with initial conditions given by 𝜕^𝑗_𝑧𝑊(𝜖,𝜏,0)=𝑊_𝑗(𝜖,𝜏),0≤𝑗≤𝑆−1, (82) where 𝑊_𝑗∈𝒪((𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝑗) are as previously shown.
From Theorem 15, one concludes that the formal power series 𝑊(𝜖,𝜏,𝑧) is such that 𝑊_𝛽 can be extended to a holomorphic function defined in (𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝛽, for every 𝛽≥0. We preserve notation for these extensions. Moreover, for every 𝛽≥0 one has 𝑊_𝛽(𝜖,𝜏)≤𝐶₂₃𝐶𝛽!²⁴𝛿^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2(|𝜏|+𝛿1)}𝑞^𝐴1𝛽2, (83) for every (𝜖,𝜏)∈(𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝛽, and some positive constants 𝐶₂₃ and 𝐶₂₄ determined in the proof of Theorem 15.
We put 𝑋_𝑖(𝜖,𝜏,𝑧)=∑_𝛽≥0𝑋_𝑖,𝛽(𝑡,𝜖)(𝑧^𝛽/𝛽!), where 𝑋_𝑖,𝛽(𝜖,𝑡)∶=_𝐿𝛾𝑖𝑊_𝛽(𝜏,𝜖)𝑒^{−𝜏𝑡/𝜖}𝑑𝜏. (84)
We fist check that 𝑋_𝑖 is, at least formally, a solution of (72)+(73). From (67), one can check by inserting the formal power series 𝑋_𝑖 in (72), that it turns out to be a formal solution in the variable 𝑧 of (72)+(73) if and only if 𝑊(𝜖,𝜏,𝑧) is a formal solution of (28)+(29) and (56)+(57).
Bearing in mind that 𝑊_𝛽 verifies (82) and (83), one derives 𝑋_𝑖,𝛽 is well defined in ℰ_𝑖×𝒯, for every 𝛽≥0. We now state a proof for the fact that (𝜖,𝜏,𝑧)↦𝑋_𝑖(𝜖,𝜏,𝑧) is indeed a holomorphic solution of (72)+(73) in ℰ_𝑖×𝒯×ℂ. Let 𝜖∈ℰ_𝑖, 𝑡∈𝒯, and 𝛽∈ℕ. One has _𝐿𝛾𝑖𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏≤_{𝐿𝛾𝑖,1}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}+𝑑𝜏_{𝐿𝛾𝑖,2}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}+𝑑𝜏_{𝐿𝛾𝑖,3}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏, (85) where 𝐿_𝛾𝑖,1∶=𝐿_𝛾𝑖𝑅∩𝐷_𝛽, 𝐿_𝛾𝑖,2∶=𝐿_𝛾𝑖∩𝑆_𝑖,𝛽, and 𝐿_𝛾𝑖,3∶=𝐿_𝛾𝑖∩𝐷̇𝑅_𝛽∩𝑆_𝑖,𝛽. We only give details on the first and second integrals appearing on the right-hand side of the previous inequality. The first integral on the right-hand side of (85) can be upper bounded by means of (83) and the choice of direction 𝛾_𝑖.
Consider _{𝐿𝛾𝑖,1}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝜏/𝜖𝑡}≤𝑅𝑑𝜏_𝛽0𝑊_𝛽𝑠𝑒^{√−1𝛾𝑖},𝜖𝑒^{−(𝑠|𝑡|/|𝜖|)cos(𝛾𝑖−arg(𝑡/𝜖))}𝑑𝑠≤𝐶₂₃𝐶𝛽!²⁴𝛿^𝛽𝑞^𝐴1𝛽2×_𝛽𝑅₀|𝜖|^−𝐶𝛽𝑒^{𝑀log2(𝑠/|𝜖|+𝛿1)}𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|}𝑑𝑠. (86)
One has |𝜖|^−𝐶𝛽𝑒^{𝑀log2(𝑠/|𝜖|+𝛿1)}𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|}≤𝐶₂₃|𝜖|^−𝐶𝛽𝑒^{−𝑠𝛿2𝑟𝒯/2|𝜖|}, (87) for some 𝐶₂₃>0. Now, the function 𝑥↦𝑥^−𝐶𝛽𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|} attains its maximum at 𝑥=𝐶𝛽2/𝑠𝛿₂𝑟_𝒯. One can reduce 𝑟₀, if necessary, to conclude that this function is increasing for 𝑥∈[0,𝜖₀]. The expression in (86) is upper bounded by 𝐶₂₃𝐶𝛽!²⁴𝛿𝑟^𝐶₀^𝛽𝑞^{𝐴1𝛽2∞}₀𝑒^{−𝑠𝛿2𝑟𝒯/2𝑟0}𝑑𝑠. (88)
This yields _{𝐿𝛾𝑖,1}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏≤𝐶₃₁𝛽!𝐶^𝛽₃₂𝑞^𝐴1𝛽2, (89) for some constants 𝐶₃₁,𝐶₃₂>0 only depending on 𝛿, 𝑟₀, 𝑅₀, 𝑞, 𝑆, 𝐴₁, 𝐶, 𝛿₁, 𝑀, 𝑟_𝒯, and 𝛿₂. We now consider the second integral appearing on the right-hand side of (85). From (81) and similar estimates as before we get _{𝐿𝛾𝑖,2}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏≤𝐶₁₃𝐶𝛽!¹⁴𝛿^𝛽𝑞^𝐴1𝛽2×^∞_𝑅𝛽𝑒^{𝑀log2((𝑠/|𝜖|)+𝛿1)}𝑠|𝜖|^𝐶𝛽𝑒^{−(𝑠𝛿2𝑟𝒯)/|𝜖|}𝑑𝑠. (90)
The function 𝑥↦𝑔₁(𝑥)=𝑒^{𝑀log2(𝑥+𝛿1)}𝑥^𝐶𝛽, 𝑥≥0 is such that 𝑔₁(𝑥)≤𝑔₂(𝑥) for all 𝑥≥0, where 𝑔₂(𝑥)=𝐶₁₃𝑒^{𝑀log2(𝑥)}𝑥^𝐶𝛽, for some positive constant 𝐶₁₃, not depending on 𝛽. 𝑔₂ attains its maximum value at 𝑥₀=exp(−𝐶𝛽/2𝑀) so that 𝑔(𝑥)≤𝑔(𝑥₀)=exp(−𝐶²𝛽²/4𝑀), for every 𝑥>0. This implies 𝑒^{𝑀log2(𝑠/|𝜖|+𝛿1)}𝑠|𝜖|^𝐶𝛽𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|}≤𝑒^{−(𝐶2/4𝑀)𝛽2}𝑒^{−𝑠𝛿2𝑟𝒯/|𝜖|}. (91)
From (90) we derive _{𝐿𝛾𝑖,2}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏≤𝐶₁₃^∞₀𝑒^{−𝑠𝛿2𝑟𝒯/2𝑟0}𝑑𝑠𝛽!𝑞^𝐴1𝛽2𝑒^{−(𝐶2/4𝑀)𝛽2}𝑒^{−𝑅2𝑟𝒯𝛽𝛿/2𝑟0}=𝐶₁₄𝛽!𝑞^{(𝐴1−𝐶2/4𝑀log(𝑞))𝛽2}, (92) for some 𝐶₁₃>0.
From (89) and (92), we lead to the existence of positive constants 𝐶₄₁,𝐶₄₂, not depending on 𝛽, such that _𝛽≥0𝑋_𝑖,𝛽()𝑧𝑡,𝜖^𝛽𝛽!≤𝐶_41𝛽≥0𝐶^𝛽₄₂𝑞^𝐴1𝛽2|𝑧|^𝛽, (93) for every 𝑧∈ℂ. This allows us to conclude the first part of the proof.
Let 0≤𝑖≤𝜈−1 and 𝜌>0. For every (𝜖,𝑡,𝑧)∈(ℰ_𝑖∩ℰ_𝑖+1)×𝒯×𝐷(0,𝜌) we have 𝑋_𝑖+1(𝜖,𝑡,𝑧)−𝑋_𝑖(𝜖,𝑡,𝑧)≤_𝛽≥0𝑋_𝑖+1,𝛽(𝜖,𝑡)−𝑋_𝑖,𝛽(𝜌𝜖,𝑡)^𝛽.𝛽! (94)
We can write 𝑋_𝑖+1,𝛽(𝜖,𝑡)−𝑋_𝑖,𝛽(𝜖,𝑡)=_{𝐿𝛾𝑖+1,2}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}−𝑑𝜏_{𝐿𝛾𝑖,2}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}+𝑑𝜏_{𝐿𝛾𝑖+1,4−𝐿𝛾𝑖,4}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏, (95) where 𝐿_𝛾𝑖+1,4−𝐿_𝛾𝑖,4 stands for the path consisting of two parts: the first one going from 𝑅_𝛽𝑒^{√−1𝛾𝑖+1} to 0 along the segment [0,𝑅_𝛽𝑒^{√−1𝛾𝑖+1}] and the path going from 0 to 𝑅_𝛽𝑒^{√−1𝛾𝑖} following direction 𝛾_𝑖.
This integral has already been estimated in (92), for the first part of the proof, so we omit the details. We also omit the details on the integral concerning the path 𝐿_𝛾𝑖+1,2 which is analogous.
In order to estimate the integral along the path 𝐿_𝛾𝑖+1,4−𝐿_𝛾𝑖,4, one can observe that the function involved in the integrand does not depend on the index 𝑖 considered, for this function is well defined for (𝜖,𝜏)∈(𝐷(0,𝑟₀𝐷̇𝑅)⧵{0})×_𝛽. One can apply Cauchy Theorem to derive _𝐿𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏=0, (96) where 𝐿=𝐿_𝛾𝑖+1,4−𝐿_𝛾𝑖,4−𝐿₁ is the closed path with 𝑠∈[𝛾_𝑖,𝛾_𝑖+1]→𝐿₁(𝑠)=𝑅_𝛽𝑒^√−1𝑠. Moreover, |∫_{𝐿𝛾𝑖+1,4−𝐿𝛾𝑖,4}𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏| equals _𝐿1𝑊_𝛽(𝜏,𝜖)𝑒^{−𝑡𝜏/𝜖}𝑑𝜏≤𝐶₂₃𝐶𝛽!²⁴𝛿^𝛽|𝜖|^−𝐶𝛽𝑒^{𝑀log2(𝑅𝛽+𝛿1)}𝑞^𝐴1𝛽2𝑅_𝛽×^𝛾𝑖+1_𝛾𝑖𝑒^{−𝑅𝛽|𝑡|cos(𝜃−arg(𝑡/𝜖))/|𝜖|}≤𝐶𝑑𝜃₂₃𝐶𝛽!²⁴𝛿^𝛽|𝜖|^−𝐶𝛽𝑞^𝐴1𝛽2𝑅_𝛽×^𝛾𝑖+1_𝛾𝑖𝑒^{−𝑅𝛽𝑟𝒯𝛿2/2|𝜖|}≤𝐶̆𝑑𝑠₂₃𝐶𝛽!²⁴𝛿^𝛽𝑞^𝐴1𝛽2|𝜖|^−𝐶𝛽×𝑒^{−𝑅𝛽𝑟2𝒯𝛿/2|𝜖|}𝑒^{−𝑅𝛽𝑟2𝒯𝛿/2|𝜖|}, (97) for some 𝐶₂₃,𝐶̆₂₃>0. It only rests to take into account that the function 𝑥∈(0,𝑟₀)↦𝑥^−𝐶𝛽𝑒^{−𝑅𝛽𝑟𝒯𝛿2/2𝑥} is monotonely increasing in (0,𝑟₀), so that |𝜖|^−𝐶𝛽𝑒^{−𝑅𝛽𝑟𝒯𝛿2/2|𝜖|} can be included in the constants 𝐶̆₂₃ and 𝐶₂₄.
From (92) and (97) one gets the existence of positive constants 𝐶₆and𝐶₇ such that 𝑋_𝑖+1,𝛽(𝜖,𝑡)−𝑋_𝑖,𝛽(𝜖,𝑡)≤𝐶₆𝛽!𝐶^𝛽₇𝑞^𝐴1𝛽2𝑒^{−(𝑑1𝛿2𝑟𝒯/2)(𝑞𝑑2𝛽/|𝜖|)}, (98) for every (𝜖,𝑡)∈(ℰ_𝑖∩ℰ_𝑖+1)×𝒯. Taking this last estimate into the expression of 𝑋_𝑖+1−𝑋_𝑖 one can conclude that 𝑋_𝑖+1(𝜖,𝑡,𝑧)−𝑋_𝑖(𝜖,𝑡,𝑧)≤𝐶_6𝛽≥0𝐶₇𝜌^𝛽𝑞^𝐴1𝛽2𝑒^{−(𝑑1𝛿2𝑟𝒯/2)(𝑞𝑑2𝛽/|𝜖|)}, (99) for every (𝜖,𝑡,𝑧)∈(ℰ_𝑖∩ℰ_𝑖+1)×𝒯×𝐷(0,𝜌). The proof of the second statement in the theorem leans on the incoming lemma whose proof is left until the end of the current section. It provides information on the estimates for a Dirichlet type series. A similar argument concerning a Dirichlet series of different nature can be found in [2], Lemma 16, when dealing with Gevrey asymptotic expansions.

Lemma 22. Let 𝐴₁, 𝐷₁, 𝐷₂, and let   𝑑₂ be positive constants, with 𝐷₂>1. Then, for every Δ>1 there exist 𝐸₁>0 and 𝛿>0 such that _𝛽≥0𝐷^𝛽₁𝑞^𝐴1𝛽2𝑒^{−𝐷2(𝑞𝑑2𝛽/𝜖)}≤𝐸₁𝑒^{−(𝐴1/𝑑22Δ)(1/(−log(𝑞))2)log2𝜖}, (100) for every 𝜖∈(0,𝛿].

The proof of Lemma 22 heavily rests on the 𝑞-Gevrey version of some preliminary results which are classical in Gevrey case (see [2] and the references therein). Their proofs do not differ from the classical ones, so we omit them.

Lemma 23. Let 𝑏>0 and 𝑓∶[0,𝑏]→ℂ a continuous function having the formal expansion ∑_𝑛≥0𝑎_𝑛𝑡^𝑛∈ℂ[[𝑡]] as its q-asymptotic expansion of type 𝐴₁>0 at 0, meaning there exist 𝐶,𝐻>0 such that 𝑓(𝑡)−^𝑁−1_𝑛=0𝑎_𝑛𝑡^𝑛≤𝐶𝐻^𝑁𝑞^{−𝐴1𝑁2/2}|𝑡|^𝑁,𝑁! (101) for every 𝑁≥1 and 𝑡∈[0,𝛿], for some 0<𝛿<𝑏.
Then, the function 𝐼(𝑥)=^𝑏₀𝑓(𝑠)𝑒^−𝑠/𝑥𝑑𝑠 (102) admits the formal power series ∑_𝑛≥0𝑎_𝑛𝑛!𝜖^𝑛+1∈ℂ[[𝜖]] as its 𝑞-Gevrey asymptotic expansion of type 𝐴₁ at 0. It is to say, there exist 𝐶,𝐻>0 such that 𝐼(𝑥)−^𝑁−1_𝑛=0𝑎_𝑛𝑛!𝑥^𝑛+1≤𝐶𝐻^𝑁+1𝑞^{−𝐴1(𝑁+1)2/2}|𝑥|^𝑁+1()!,𝑁+1 (103) for every 𝑁≥0 and 𝑥∈[0,𝛿^′] for some 0<𝛿^′<𝑏.

One can adapt the proof of Proposition  4 in [5] in our framework.

Lemma 24. Let 𝐴₁,𝛿>0, and let 𝜓∶[0,𝛿]→ℂ be a continuous function. The following holds. (1)If there exist 𝐶,𝐻>0 such that |𝜓(𝑥)|≤𝐶𝐻^𝑛𝑞^{−𝐴1𝑛2/2}|𝑥|^𝑛/𝑛!, for every 𝑛∈ℕ, 𝑛≥0 and 𝑥∈[0,𝛿], then for every 𝐴₁>𝐴₁ there exists 𝐶>0 such that 𝜓(𝑥)≤𝐶𝑒^−((1/𝐴_{1)(1/(−log(𝑞))2))log}²_|𝑥|, (104) for every 𝑥∈(0,𝛿]. (2)If there exists 𝐶>0 such that |𝜓(𝑥)|≤𝐶𝑒^{−((1/𝐴1)(1/(−log(𝑞))2))log2|𝑥|}, for every 𝑛∈ℕ, and 𝑥∈[0,𝛿], then for every 𝐴₁>𝐴₁ there exists 𝐶,𝐻>0 such that 𝜓(𝑥)≤𝐶𝐻^𝑛𝑞⁻𝐴_1𝑛²_/2|𝑥|^𝑛,𝑛! (105) for every 𝑛∈ℕ and for every 𝑥∈(0,𝛿].

Proof of Lemma 22. Let 𝑓∶[0,+∞)→ℝ be a 𝒞¹ function. For every 𝑛∈ℕ, one can apply the Euler-Mac-Laurin formula ^𝑛_𝜅=0𝑓(𝜅)=12𝑓(0)+𝑓(𝑛)+^𝑛₀𝑓(𝑡)+𝑑𝑡_𝑛0𝐵₁(𝑡−⌊𝑡⌋)𝑓^′(𝑡)𝑑𝑡, (106) where 𝐵₁(𝑠)=𝑠−1/2 is the Bernoulli polynomial and ⌊⋅⌋ stands for the floor function to 𝑓(𝑠)=𝐷^𝑠₁𝑞^𝐴1𝑠2𝑒^{−𝐷2(𝑞𝑑2𝑠/𝜖)}. One has ^𝑛_𝜅=0𝐷^𝜅₁𝑞^𝐴1𝜅2𝑒^{−𝐷2(𝑞𝑑2𝜅/𝜖)}=12𝑒^{−𝐷2/𝜖}+𝐷^𝑛₁𝑞^𝐴1𝑛2𝑒^{−𝐷2(𝑞𝑑2𝑛/𝜖)}+^𝑛₀𝐷^𝑡₁𝑞^𝐴1𝑡2𝑒^{−𝐷2(𝑞𝑑2𝑡/𝜖)}+𝑑𝑡_𝑛0𝐵₁(𝑡−⌊𝑡⌋)𝐷^𝑡₁𝑞^𝐴1𝑡2𝑒^{−𝐷2(𝑞𝑑2𝑡/𝜖)}×log𝐷₁+log𝑞𝐴₁2𝑡−𝐷₂log𝑞𝑑₂𝜖𝑑𝑡. (107)
Taking the limit when 𝑛 tends to infinity in the previous expression we arrive at the following equality for a convergent series: ^∞_𝜅=0𝐷^𝜅₁𝑞^𝐴1𝜅2𝑒^{−𝐷2(𝑞𝑑2𝜅/𝜖)}=12𝑒^{−𝐷2/𝜖}+^∞₀𝐷^𝑡₁𝑞^𝐴1𝑡2𝑒^{−𝐷2(𝑞𝑑2𝑡/𝜖)}+𝑑𝑡_∞0𝐵₁(𝑡−⌊𝑡⌋)𝐷^𝑡₁𝑞^𝐴1𝑡2𝑒^{−𝐷2(𝑞𝑑2𝑡/𝜖)}×log𝐷₁+log𝑞𝐴₁2𝑡−𝐷₂log𝑞𝑑₂𝜖𝑑𝑡. (108)
Let 𝐼₁∶=∫^∞₀𝑓(𝑡)𝑑𝑡 and 𝐼₂∶=∫^∞₀𝐵₁(𝑡−⌊𝑡⌋)(log(𝐷₁)+log(𝑞)𝐴₁2𝑡−𝐷₂(log(𝑞)𝑑₂/𝜖)𝑞^{𝐷2𝑡/𝜖})𝑓(𝑡)𝑑𝑡.
From the fact that 𝐵₁(𝑡−⌊𝑡⌋)≤1/2 for every 𝑡≥0 and the change of variable 𝐷₂𝑞^𝑑2𝑡=𝑢, one gets 𝐼₁=^𝐷2₀𝑓₁(𝑢)𝑒^−𝑢/𝜖𝑓𝑑𝑢,where₁(𝑢)∶=𝐷^{log(𝑢/𝐷2)/log(𝑞)𝑑2}₁𝑞^{𝐴1log2(𝑢/𝐷2)/log2(𝑞)𝑑22}×1𝑑₂𝑞𝑢,𝐼−log₂≤12^𝐷2₀𝑓₂(𝑢)+𝑓₃(𝑢)+𝑓₄(𝑢)𝑒^−𝑢/𝜖𝑑𝑢, (109) with 𝑓₂(𝑢)∶=log(𝐷₁)𝑓₁(𝑢), 𝑓₃(𝑢)∶=(2𝐴₁/𝑑₂)log(𝑢/𝐷₂)𝑓₁(𝑢), and 𝑓₄∶=((−log(𝑞))𝑑₂/𝜖)𝑒^{𝐷2log(𝑢/𝐷2)/𝑑2𝜖}𝑓₁(𝑢), for every 𝑢∈(0,𝐷₂]. Bearing in mind that 𝑓₃(𝑢)<0 for 𝑢∈(0,𝐷₂] and from usual estimates we derive 𝐼₂≤𝐶₁(1+1/𝜖)∫^𝐷2₀𝑓₁(𝑢)𝑒^−𝑢/𝜖𝑑𝑢 for some 𝐶₁>0. The proof is complete if one can estimate 𝑒^{−𝐷2/𝜖}, 𝐼₁, and 1/𝜖𝐼₁ appropriately. The first expression is clearly upper bounded according to (100).
From usual estimates we arrive at 𝐼₁≤𝐶₃^𝐷2₀𝑒^{−(𝐴1/(−log(𝑞))𝑑22)log2(𝑢/𝐷2)}𝑒^−𝑢/𝜖𝑑𝑢=𝐶₃^𝐷3₀𝑓(𝑢)𝑒^−𝑢/𝜖, (110) for some 𝐶₃>0. From Lemma 24, the function 𝑔 defined by 𝑢∈[0,1]↦𝑓(𝐷₂𝑢) (extended by continuity to 𝑢=0) is such that for every Δ>1 there exist 𝐶,𝐻>0 with 𝑔(𝑢)≤𝐶𝐻^𝑛𝑞^−((𝑑22Δ_{2)/𝐴1)𝑛}²|𝑢|^𝑛,𝑛! (111) for every 𝑢∈[0,𝐷₂] and for every 𝑛≥0. From Lemma 23, the functions 𝐼₁(𝜖) and (1/𝜖)𝐼₁(𝜖) admit the series with null coefficients as 𝑞-asymptotic expansion of type (𝑑²₂Δ)/𝐴₁. Again, from Lemma 24, one can conclude that for every Δ>Δ there exists 𝐶₄>0 such that both 𝐼₁ and (1/𝜖)𝐼₁ are upper bounded by 𝐶₄𝑒^{−((𝐴1/𝑑22Δ)(1/(−log(𝑞))2))log2(𝜖)}, for every 𝜖∈(0,𝜖₁], for some 𝜖₁>0.

4.3. Existence of Formal Series Solutions in the Complex Parameter

In this last subsection we obtain a 𝑞-Gevrey version of a Malgrange-Sibuya type theorem. A result in this direction has already been obtained by the authors in [5] when dealing with 𝑞∈ℂ, |𝑞|>1. In that work, the geometry of the problem differs from the one in the present work. Indeed, the result is settled in terms of discrete 𝑞-spirals tending to the origin and with 𝑞∈ℂ.

Given 𝑞∈ℂ with 0<|𝑞|<1 and a nonempty open subset 𝑈⊂ℂ^⋆, the discrete 𝑞-spiral associated with 𝑈 and 𝑞 consists of the products of an element in 𝑈 and 𝑞^𝑚, for some 𝑚∈ℕ. For our purpose, 𝑞 is a real number and 𝑈 is chosen in such a way that the discrete 𝑞-spiral turns out to be a sector with vertex at the origin.

The proof of the 𝑞-Gevrey version of Malgrange-Sibuya theorem in [5] is based on the use of extension results on ultradifferential spaces of weighted functions which preserve the information of 𝑞-Gevrey bounds but causes the 𝑞-Gevrey type involved in the 𝑞-Gevrey asymptotic to suffer an increasement. Here, one can follow similar steps as for the classical proof Malgrange-Sibuya theorem based on Cauchy-Heine transform, so that the 𝑞-Gevrey type is preserved. In [6], an analogous demonstration for the Gevrey version of the result can be found. We have decided to include the whole proof of the result in order to facilitate comprehension and clarity of the result.

Theorem 25 (𝑞-MS). Let (𝔼,‖⋅‖_𝔼) be a Banach space over ℂ and let {ℰ_𝑖}_{0≤𝑖≤𝜈−1} be a good covering in ℂ^∗. For all 0≤𝑖≤𝜈−1, let 𝐺_𝑖 be a holomorphic function from ℰ_𝑖 into the Banach space (𝔼,‖⋅‖_𝔼) and let the cocycle Δ_𝑖(𝜖)=𝐺_𝑖+1(𝜖)−𝐺_𝑖(𝜖) be a holomorphic function from the sector 𝑍_𝑖=ℰ_𝑖+1∩ℰ_𝑖 into 𝔼 (with the convention that ℰ_𝜈=ℰ₀ and 𝐺_𝜈=𝐺₀). We make the following assumptions.(1)The functions 𝐺_𝑖(𝜖) are bounded as 𝜖∈ℰ_𝑖 tends to the origin in ℂ, for all 0≤𝑖≤𝜈−1.(2)Δ_𝑖 has a 𝑞-exponential decreasing of some type 𝐿>0, for every 0≤𝑖≤𝜈−1, meaning there exists 𝐶_𝑖>0 such that Δ_𝑖(𝜖)_𝔼≤𝐶_𝑖𝑒^{−((1/𝐿)(1/(−log(𝑞))2))log2|𝜖|}, (112)

Then, there exists a formal power series 𝐺(𝜖)∈𝔼[[𝜖]] such that 𝐺_𝑖(𝜖) admits 𝐺(𝜖) as its 𝑞-Gevrey asymptotic expansion of type 𝐿 on ℰ_𝑖, for every 0≤𝑖≤𝜈−1.

Proof. We first state an auxiliary result.

Lemma 26. For all 0≤𝑖≤𝜈−1, there exist bounded holomorphic functions Ψ_𝑖∶ℰ_𝑖→ℂ such that Δ_𝑖(𝜖)=Ψ_𝑖+1(𝜖)−Ψ_𝑖(𝜖), (113) for all 𝜖∈𝑍_𝑖, where by convention Ψ_𝜈(𝜖)=Ψ₀(𝜖). Moreover, there exist 𝜑_𝑚∈𝔼, 𝑚≥0, such that for each 0≤𝑙≤𝜈−1 any 𝐿>𝐿 and every 𝒲≺ℰ_𝑙, there exist 𝐾_𝑙,𝑀_𝑙>0 with Ψ_𝑙(𝜖)−^𝑀−1_𝑚=0𝜑_𝑚𝜖^𝑚_𝔼≤𝐾_𝑙𝑀_𝑙^𝑀𝑞⁻𝐿_((𝑀−1)²_/2)|𝜖|^𝑀𝑀! (114) for all 𝜖∈𝒲 and all 𝑀≥2.

Proof. We follow analogous arguments as in Lemma XI-2-6 from [34] with appropriate modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform.
For all 0≤𝑙≤𝜈−1, we choose a segment 𝒞_𝑙=𝑡𝑒^{√−1𝜃𝑙},𝑡∈[0,𝑟]⊂ℰ_𝑙∩ℰ_𝑙+1. (115)
These 𝜈 segments divide the open punctured disc 𝐷(0,𝑟)⧵{0} into 𝜈 open sectors ℰ₀,…,ℰ_𝜈−1, where ℰ_𝑙=ℂ𝜖∈^∗𝜃_𝑙−1<arg(𝜖)<𝜃_𝑙,|𝜖|<𝑟,0≤𝑙≤𝜈−1, (116) where by convention 𝜃₋₁=𝜃_𝜈−1. Let Ψ_𝑙(𝜖)=−12𝜋√−1^𝜈−1_ℎ=0^𝐶_ℎΔ_ℎ(𝜉)𝜉−𝜖𝑑𝜉 (117) for all 𝜖∈ℰ_𝑙, for 0≤𝑙≤𝜈−1, be defined as a sum of Cauchy-Heine transforms of the functions Δ_ℎ(𝜖). By deformation of the paths 𝐶_𝑙−1 and 𝐶_𝑙 without moving their endpoints and letting the other paths 𝐶_ℎ, ℎ≠𝑙−1,and𝑙 untouched (with the convention that 𝐶₋₁=𝐶_𝜈−1), one can continue analytically the function Ψ_𝑙 onto ℰ_𝑙. Therefore, Ψ_𝑙 defines a holomorphic function on ℰ_𝑙, for all 0≤𝑙≤𝜈−1.
Now, take 𝜖∈ℰ_𝑙∩ℰ_𝑙+1. In order to compute Ψ_𝑙+1(𝜖)−Ψ_𝑙(𝜖), we write Ψ_𝑙(𝜖)=−12𝜋√−1𝐶_𝑙Δ_𝑙(𝜉)𝜉−𝜖𝑑𝜉+−12𝜋√×−1_{𝜈−1ℎ=0,ℎ≠𝑙𝐶ℎ}Δ_ℎ(𝜉)Ψ𝜉−𝜖𝑑𝜉,_𝑙+1(𝜖)=−12𝜋√−1^̆𝐶_𝑙Δ_𝑙(𝜉)𝜉−𝜖𝑑𝜉+−12𝜋√×−1_{𝜈−1ℎ=0,ℎ≠𝑙𝐶ℎ}Δ_ℎ(𝜉)𝜉−𝜖𝑑𝜉, (118) where the paths 𝐶_𝑙 and 𝐶̆_𝑙 are obtained by deforming the same path 𝐶_𝑙 without moving its endpoints in such a way that(a)𝐶_𝑙⊂ℰ_𝑙∩ℰ_𝑙+1 and 𝐶̆_𝑙⊂ℰ_𝑙∩ℰ_𝑙+1,(b)Γ_𝑙,𝑙+1∶=−𝐶̆_𝑙+𝐶_𝑙 is a simple closed curve with positive orientation whose interior contains 𝜖.
Therefore, due to the residue formula, we can write Ψ_𝑙+1(𝜖)−Ψ_𝑙(𝜖)=12𝜋√−1^Γ_𝑙,𝑙+1Δ_𝑙(𝜉)𝜉−𝜖𝑑𝜉=Δ_𝑙(𝜖), (119) for all 𝜖∈ℰ_𝑙∩ℰ_𝑙+1 and for all 0≤𝑙≤𝜈−1 (with the convention that Ψ_𝜈=Ψ₀).
In a second step, we derive asymptotic properties of Ψ_𝑙. We fix an 0≤𝑙≤𝜈−1 and a proper closed sector 𝒲 contained in ℰ_𝑙. Let 𝐶_𝑙 (resp., 𝐶_𝑙−1) be a path obtained by deforming 𝐶_𝑙 (resp., 𝐶_𝑙−1) without moving the endpoints so that 𝒲 is contained in the interior of the simple closed curve 𝐶_𝑙−1+𝛾_𝑙−𝐶_𝑙 (which is itself contained in ℰ_𝑙), where 𝛾_𝑙 is a circular arc joining the two points 𝑟𝑒^{√−1𝜃𝑙−1} and 𝑟𝑒^{√−1𝜃𝑙}. We get the representation Ψ_𝑙(𝜖)=−12𝜋√−1𝐶_𝑙Δ_𝑙(𝜉)𝜉−𝜖𝑑𝜉+−12𝜋√×𝐶−1_𝑙−1Δ_𝑙−1(𝜉)+𝜉−𝜖𝑑𝜉−12𝜋√−1^𝜈−1_{ℎ=0,ℎ≠𝑙,𝑙−1}^𝐶_ℎΔ_ℎ(𝜉)𝜉−𝜖𝑑𝜉, (120) for all 𝜖∈𝒲. One assumes that the path 𝐶_𝑙 is given as the union of a segment 𝐿_𝑙={𝑡𝑒^{√−1𝑤𝑙}∶𝑡∈[0,𝑟₁]}, where 𝑟₁<𝑟 and 𝑤_𝑙>𝜃_𝑙 and a curve Γ_𝑙={𝜇_𝑙(𝜏)∶𝜏∈[0,1]} such that 𝜇_𝑙(0)=𝑟₁𝑒^{√−1𝑤𝑙}, 𝜇_𝑙(1)=𝑟𝑒^{√−1𝜃𝑙}, and 𝑟₁≤|𝜇_𝑙(𝜏)|<𝑟 for all 𝜏∈[0,1). We also assume that there exists a positive number 𝜎<1 with |𝜖|≤𝜎𝑟₁ for all 𝜖∈𝒲. By construction of the path Γ_𝑙, we get that the function 𝜖↦(1/2𝜋)√−1∫_Γ𝑙(Δ_𝑙(𝜉)/(𝜉−𝜖))𝑑𝜉 defines an analytic function on the open disc 𝐷(0,𝑟₁).
It remains to give estimates for the integral (1/2𝜋√−1)∫_𝐿𝑙(Δ_𝑙(𝜉)/(𝜉−𝜖))𝑑𝜉. Let 𝑀≥0 be an integer. From the usual geometric series expansion, one can write 12𝜋√−1^𝐿_𝑙Δ_𝑙(𝜉)𝜉−𝜖𝑑𝜉=^𝑀_𝑚=0𝛼_𝑙,𝑚𝜖^𝑚+𝜖^𝑀+1𝐸_𝑙,𝑀+1(𝜖), (121) where 𝛼_𝑙,𝑚=12𝜋√−1^𝐿_𝑙Δ_𝑙(𝜉)𝜉^𝑚+1𝐸𝑑𝜉,_𝑙,𝑀+1(𝜖)=12𝜋√−1^𝐿_𝑙Δ_𝑙(𝜉)𝜉^𝑀+1(𝜉−𝜖)𝑑𝜉, (122) for all 𝜖∈𝒲.
Gathering (112) and (122), we get 𝛼_{𝑙,𝑚𝔼}≤𝐾_𝑙2𝜋^𝑟1₀𝑒^{−((1/𝐿)(1/(−log(𝑞))2))log2𝜏}𝜏^𝑚+1𝑑𝜏. (123) The changes of variable log(𝜏)=𝑠 first, and 𝑠=2𝐿(−log(𝑞))𝑡 afterwards, transform the right-hand side of (123) into 𝐾_𝑙2𝜋^log(𝑟1)_−∞𝑒^{−((1/𝐿)(1/(−log(𝑞))2))𝑠2}𝑒^𝑠𝑚=𝐾𝑑𝑠_𝑙2𝐿−log𝑞×2𝜋_{log(𝑟1)/2𝐿(−log(𝑞))−∞}exp−𝑡²−𝑚𝑞𝑡≤𝐾2𝐿−log𝑑𝑡_𝑙2𝐿−log𝑞×2𝜋_∞−∞exp−𝑡²−𝑚2𝐿−log𝑞𝑡𝑑𝑡. (124)
The application of 𝑒^𝑎2/4√𝜋=^∞_−∞𝑒^{−𝑥2−𝑎𝑥}𝑑𝑥, (125) for every 𝑎∈ℝ, which can be found in [35] (Chapter 10, page 498), leads us to 𝛼_{𝑙,𝑚𝔼}≤2𝐿−log𝑞𝐾_𝑙2√𝜋𝑞^{−𝐿(𝑚2/2)}. (126)
Moreover, as previously described, one can choose a positive number 𝜂>0 (depending on 𝒲) such that |𝜉−𝜖|≥|𝜉|sin(𝜂) for all 𝜉∈𝐿_𝑙 and all 𝜖∈𝒲. Again by (112) and (122) and following analogous calculations as before, we obtain 𝐸_𝑙,𝑀+1(𝜖)_𝔼≤𝐾_𝑙2𝜋sin𝜂^𝑟1₀𝑒^{−((1/𝐿)(1/2(−log(𝑞))))log2𝜏}𝜏^𝑀+2≤𝑑𝜏2𝐿−log𝑞𝐾_𝑙2sin(𝜈)√𝜋𝑞^{−𝐿((𝑀+1)2/2)}, (127) for all 𝜖∈𝒲. Using comparable arguments, one can give analogous estimates when estimating the other integrals −12𝜋√−1𝐶_𝑙−1Δ_𝑙−1(𝜉)𝜉−𝜖𝑑𝜉,−12𝜋√−1^𝐶_ℎΔ_ℎ(𝜉)𝜉−𝜖𝑑𝜉, (128) for all ℎ≠𝑙,𝑙−1.
As a consequence, for any 0≤𝑙≤𝜈−1, there exist 𝜑_𝑙,𝑚∈𝔼, for all 𝑚≥0 and a constant 𝐾_𝑙>0 such that Ψ_𝑙(𝜖)−^𝑀−1_𝑚=0𝜑_𝑙,𝑚𝜖^𝑚_𝔼≤𝐾_𝑙𝑞^{−𝐿(𝑀2/2)}|𝜖|^𝑀, (129) for all 𝑀≥2 and all 𝜖∈𝒲.
Taking into account Proposition 18, we deduce that for every ℰ_𝑖,𝑖+1≺ℰ_𝑙∩ℰ_𝑙+1 and for every 𝐿>𝐿, the function Ψ_𝑙+1(𝜖)−Ψ_𝑙(𝜖) has the formal series 0 as 𝑞-Gevrey asymptotic expansion of type 𝐿 in ℰ_𝑖,𝑖+1. From the unicity of the asymptotic expansions on sectors, we deduce that all the formal series ∑_𝑚≥0𝜑_𝑙,𝑚𝜖^𝑚, 0≤𝑙≤𝜈−1, are equal to some formal series denoted 𝐺(𝜖)=∑_𝑚≥0𝜑_𝑚𝜖^𝑚∈𝔼[[𝜖]].

We consider now the bounded holomorphic functions 𝑎_𝑖(𝜖)=𝐺_𝑖(𝜖)−Ψ_𝑖(𝜖), (130) for all 0≤𝑖≤𝜈−1 and all 𝜖∈ℰ_𝑖. By definition, for any 𝑖∈{0,…,𝜈−1}, we have that 𝑎_𝑖+1(𝜖)−𝑎_𝑖(𝜖)=𝐺_𝑖+1(𝜖)−𝐺_𝑖(𝜖)−Δ_𝑖(𝜖)=0, (131) for all 𝜖∈𝑍_𝑖. Therefore, each 𝑎_𝑖(𝜖) is the restriction on ℰ_𝑖 of a holomorphic function 𝑎(𝜖) on 𝐷(0,𝑟)⧵{0}. Since 𝑎(𝜖) is bounded on 𝐷(0,𝑟)⧵{0}, the origin turns out to be a removable singularity for 𝑎(𝜖) which, as a consequence, defines a convergent power series on 𝐷(0,𝑟).

Finally, one can write 𝐺_𝑖(𝜖)=𝑎(𝜖)+Ψ_𝑖(𝜖), (132) for all 𝜖∈ℰ_𝑖, all 0≤𝑖≤𝜈−1. Moreover, 𝑎(𝜖) is a convergent power series, and for every 𝐿>𝐿, Ψ_𝑖(𝜖) has the series 𝐺(𝜖)=∑_𝑚≥0𝜑_𝑚𝜖^𝑚 as 𝑞-Gevrey asymptotic expansion of type 𝐿 on ℰ_𝑖, for all 0≤𝑖≤𝜈−1.

We are under conditions to enunciate the main result in the present work.

Theorem 27. Let 𝜌>0. Under the same hypotheses as in Theorem 21, we denote ℍ_𝒯,𝜌 the Banach space of holomorphic and bounded functions in 𝒯×𝐷(0,𝜌) with the supremum norm. Then, there exists a formal power series 𝑋(𝜖,𝑡,𝑧)=_𝑘≥0𝑋_𝑘(𝑡,𝑧)𝜖𝑘!^𝑘∈ℍ_𝒯,𝜌[[𝜖]], (133) which is formal solution of 𝜖𝜕_𝑡𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)+𝑎𝜕^𝑆_𝑧𝑋(𝜖,𝑡,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩}𝑏_𝜅(𝜖,𝑧)𝜕^𝜅0_𝑡𝜕^𝜅1_𝑧𝑋×𝜖,𝑞^𝑚𝜅,1𝑡,𝑞^𝑚𝜅,2𝑧. (134) Moreover, for every 0≤𝑖≤𝜈−1 and every 𝐿₂>𝑑²₂/𝐴₁, the function 𝑋_𝑖(𝜖,𝑡,𝑧) constructed in Theorem 21 admits 𝑋(𝜖,𝑡,𝑧) as its 𝑞-Gevrey asymptotic expansion of type 𝐿₂ in ℰ_𝑖, meaning that for every 0≤𝑖≤𝜈−1 and ℰ_𝑖≺ℰ_𝑖, there exist 𝐿₀,𝐿₁>0 such that sup_{𝑡∈𝒯𝑧∈𝐷(0,𝜌)}𝑋(𝜖,𝑡,𝑧)−^𝑁_𝑘=0𝑋_𝑘(𝑡,𝑧)𝜖𝑘!^𝑘≤𝐿₀𝐿^𝑁₁𝑞^{−𝐿2(𝑁2/2)}|𝜖|^𝑁+1()!,𝑁+1 (135) for every 𝑁≥0 and all 𝜖∈ℰ_𝑖.

Proof. Let us consider the family (𝑋_𝑖(𝜖,𝑡,𝑧))_{0≤𝑖≤𝜈−1} constructed in Theorem 21. For every 0≤𝑖≤𝜈−1, we define the function 𝜖∈ℰ_𝑖↦𝐺_𝑖(𝜖)∶=𝑋_𝑖(𝜖,𝑡,𝑧), which belongs to the space 𝔼_𝒯,𝜌. From (79), we derive the cocycle Δ_𝑖∶=𝐺_𝑖+1(𝜖)−𝐺_𝑖(𝜖) verifies (112) in Theorem 25, with 1/𝐿∶=𝐴₁/𝑑²₂Δ for some fixed 0<Δ<1. Theorem 25 guarantees the existence of a formal power series 𝑋(𝜖)∈ℍ_𝒯,𝜌[[𝜖]], such that for every 𝐿₂>𝐿, 𝐺_𝑖(𝜖) admits 𝐺(𝜖) as its 𝑞-Gevrey asymptotic expansion of type 𝐿₂ on ℰ_𝑖. This is valid for every 0≤𝑖≤𝜈−1. This concludes the second part of the result.
It only rests to verify that 𝑋 is a formal solution of (72)+(73).
If we write 𝑋(𝜖,𝑡,𝑧)∶=∑_𝑘≥0(𝑋_𝑘(𝑡,𝑧)/𝑘!)𝜖^𝑘, we have lim_{𝜖→0𝜖∈}ℰ_𝑖sup_{(𝑡,𝑧)∈𝒯×𝐷(0,𝜌)}𝜕^ℓ_𝜖𝑋_𝑖(𝑡,𝑧,𝜖)−𝑋_ℓ(𝑡,𝑧)=0, (136) for every 0≤𝑖≤𝜈−1 and all ℓ≥0.
Let 0≤𝑖≤𝜈−1. By construction, 𝑋_𝑖 satisfies (72)+(73). We differentiate in the equality (72) ℓ≥1 times with respect to 𝜖. By Leibniz’s rule, we deduce that 𝜕^ℓ_𝜖𝑋_𝑖(𝜖,𝑡,𝑧) satisfies 𝜖𝜕^ℓ_𝜖𝜕_𝑡𝜕^𝑆_𝑧𝑋_𝑖(𝜖,𝑡,𝑧)+ℓ𝜕_𝜖^ℓ−1𝜕_𝑡𝜕^𝑆_𝑧𝑋_𝑖(𝜖,𝑡,𝑧)+𝑎𝜕^𝑆_𝑧𝑋_𝑖(𝜖,𝑡,𝑧)=_{𝜅=(𝜅0,𝜅1)∈𝒩ℓ0+ℓ1=ℓ}ℓ!ℓ₀!ℓ₁!𝜕^ℓ1_𝜖𝑏_𝜅()×𝜕𝜖,𝑧_ℓ1𝜖𝜕^𝜅0_𝑡𝜕^𝜅1_𝑧𝑋_𝑖𝜖,𝑞^𝑚𝜅,1𝑡,𝑞^𝑚𝜅,2𝑧, (137) for every (𝜖,𝑡,𝑧)∈ℰ_𝑖×𝒯×𝐷(0,𝜌). Let 𝜖→0 in the previous expression. From (136) we obtain 𝜕_𝑡𝜕^𝑆_𝑧𝑋_ℓ−1(𝑡,𝑧)(ℓ−1)!+𝑎𝜕^𝑆_𝑧𝑋_ℓ(𝑡,𝑧)ℓ!=_{𝜅=(𝜅0,𝜅1)∈𝒩ℓ0+ℓ1=ℓ}𝜕^ℓ0_𝜖𝑏_𝜅(0,𝑧)ℓ₀!×𝜕^𝜅0_𝑡𝜕^𝜅1_𝑧𝑋_ℓ1𝑞^𝑚𝜅,1𝑡,𝑞^𝑚𝜅,2𝑧ℓ₁!. (138)𝑏_𝜅(𝑧,𝜖) is holomorphic wih respect to 𝜖 for every 𝜅∈𝒩. This entails 𝑏_𝜅(𝜖,𝑧)=∑_ℎ≥0(𝜕^ℎ_𝜖𝑏_𝜅(0,𝑧)/ℎ!)𝜖^ℎ, for every (𝜖,𝑧) in a neighborhood of the origin in ℂ². From this and (138), we deduce 𝑋(𝜖,𝑡,𝑧)=∑_𝑘≥0(𝑋_𝑘(𝑡,𝑧)/𝑘!)𝜖^𝑘∈ℍ_𝒯,𝜌[[𝜖]] is a formal solution of (72)+(73).

Conflict of Interests

The authors certify that there is no conflict of interests between the authors and any mentioned identity in the submitted work.

Acknowledgments

Alberto Lastra is partially supported by the project MTM2012-31439 of Ministerio de Ciencia e Innovacion (Spain) and Caja de Burgos Obra Social. Stéphane Malek is partially supported by the french ANR-10-JCJC 0105 project and the PHC Polonium 2013 Project no. 28217SG.