Gevrey versus q−Gevrey Asymptotic Expansions for Some Linear q−Difference-Differential Cauchy Problem

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Gevrey versus q−Gevrey Asymptotic Expansions for Some Linear q−Difference-Differential Cauchy Problem

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Submitted:

18 December 2023

Posted:

19 December 2023

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Abstract

The asymptotic behavior of the analytic solutions of a family of singularly perturbed q-difference-differential equations in the complex domain is studied. Different asymptotic expansions with respect to the perturbation parameter and to the time variable are provided: one of Gevrey nature, and another of mixed type Gevrey and q-Gevrey. This asymptotic phenomena is observed due to the modification of the norm established on the space of coefficients of the formal solution. The techniques used are based on the adequate path deformation of the difference of two analytic solutions, and the application of several versions of Ramis-Sibuya theorem.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 35R10; 35C10; 35C15; 35C20

1. Introduction

In the present study, a family of singularly perturbed linear q-difference-differential equations of the form

P (ϵ^{k} t^{k + 1} \partial_{t}) \partial_{z}^{S} u (t, z, ϵ) = P (t, z, ϵ, \partial_{t}, \partial_{z}, σ_{q}) u (t, z, ϵ),

(1)

under initial data

(\partial_{z}^{j} u) (t, 0, ϵ) = φ_{j} (t, ϵ), 0 \leq j \leq S - 1,

(2)

is studied. In the previous problem,

ϵ

acts as a small complex perturbation parameter, and

σ_{q}

stands for the dilation operator on t variable defined by

σ_{q} f (t) = f (q t)

, for some fixed

q > 1

. In (1),

S, k

are positive integers,

P (τ) \in C [τ]

, and the symbol

P (t, z, ϵ, τ_{1}, τ_{2}, τ_{3})

is a polynomial in

(t, z, τ_{1}, τ_{2}, τ_{3})

with holomorphic coefficients on some neighborhood of the origin with respect to the perturbation parameter. The functions

φ_{j} (t, ϵ)

are polynomials with respect to t variable, with holomorphic coefficients on some neighborhood of the origin in

ϵ

. It is worth mentioning the irregular nature of the differential operator in both

P (ϵ^{k} t^{k + 1} \partial_{t})

, and inside

P

in the form of polynomial operators in

ϵ^{k} t^{k + 1} \partial_{t}

. The precise shape of the main problem and the concrete assumptions considered in it are determined in detail in Section 2.1.

There is an increasing interest on the study of the asymptotic behavior of the solutions to

q -

difference-differential equations in the complex domain. This is the case of the recent works by H. Tahara [14], H. Yamazawa [18] and H. Tahara and H. Yamazawa [15]; the authors and J. Sanz [8] and with T. Dreyfus [3]. A different approach via Nevalinna theory is developped in [16,17]. The importance of applications of q-difference equations in the knowledge of wavelets or tsunami and rogue waves is evidenced in recent advances in the field such as [12,13].

In the present work, we construct solutions

u (t, z, ϵ)

of the main problem (1), (2) which are bounded holomorphic functions defined on

T \times D_{R} \times E

, where

T

and

E

stand for finite sectors of the complex plane with vertex at the origin, and

D_{R}

stands for the open disc of radius

R > 0

. The main purpose of the present work is to show that the function

E ∋ ϵ \mapsto u (t, z, ϵ)

shows different asymptotic expansions with respect to the perturbation parameter

ϵ

when modifying the norm considered in the space of coefficients, the space of holomorphic and bounded functions defined on

T \times D_{R}

. The symmetric situation with respect to

T ∋ t \mapsto u (t, z, ϵ)

is also considered.

The appearance of such phenomena is due to the existence of small divisors involved in the main equation, in contrast to [10]. In the seminal work [10], the second author studies the asymptotic solutions with respect to

ϵ

of equations of the form

Q (\partial_{z}) u (q t, z, ϵ) = \sum_{ℓ = 1}^{D} ϵ^{Δ_{ℓ}} t^{d_{ℓ}} c_{ℓ} (t, z, ϵ) R_{ℓ} (\partial_{z}) u (q^{δ_{ℓ}} t, z, ϵ) + f (t, z, ϵ),

(3)

where

q > 1

D \geq 2

δ_{ℓ}, Δ_{ℓ}, d_{ℓ}

are non-negative integers for

1 \leq ℓ \leq D

. The functions

c_{ℓ} (t, z, ϵ)

for

1 \leq ℓ \leq D

and

f (t, z, ϵ)

are bounded holomorphic functions on

D_{r} \times {z \in C : | Im (z) | \leq β^{'}} \times D_{ϵ_{0}}

, for some

r, β^{'}, ϵ_{0} > 0

Q (τ)

and

R_{ℓ} (τ)

for

1 \leq ℓ \leq D

are polynomials with complex coefficients. All these elements are subjected to further hypotheses not mentioned here for the sake of simplicity.

In [10], the technique used to solve asymptotically the problem is to construct the analytic solutions to (3) in the form of a

q -

Laplace transform of order k, for some

k > 0

which depends on the elements of the problem. This causes the absence of two distinguished asymptotic expansions (even when modifying the norm considered for the function spaces involving the variables

(t, z)

) in contrast to the present situation. In that work, no small divisor appear in the main problem under study, all the asymptotic expansions obtained there being of

q -

Gevrey type with respect to

ϵ

The main inspiration of the present study is [1], where the authors deal with the formal solutions to systems of dimension

N \geq 1

of the form

ϵ^{α} x^{p} σ_{q, x} (y) (x, ϵ) = F (x, ϵ, y),

(4)

where

p, α

are non-negative integers,

q \in C

is such that

| q | > 1

α > 0

, and

F (x, ϵ, y)

is analytic on some neighborhood of the origin in

C \times C \times C^{N}

, with

F (0, 0, 0) = 0

and

D F_{y} (0, 0, 0)

being an invertible matrix. Indeed, the unique formal solution of (4)

\hat{y} (x, ϵ) = \sum_{n = 0}^{\infty} y_{n} (ϵ) x^{n} = \sum_{n = 0}^{\infty} u_{n} (x) ϵ^{n}

is such that

(1): for $p > 0$ , all $y_{n}$ converge on some common neighorhood of the origin, whereas $u_{n}$ converges on the disc of radius $r / {| q |}^{⌊n / α⌋}$ for some $r > 0$ and there exist $C = C (q), A = A (q) > 0$ such that for all $n \geq 0$ one has

$sup_{| ϵ | \leq r} | y_{n} (ϵ) | \leq C A^{n} {| q |}^{\frac{n^{2}}{2 p}}, sup_{| ϵ | \leq r / {| q |}^{⌊n / α⌊}} | u_{n} (x) | \leq C A^{n} .$
(2): for $p = 0$ , $y_{n}$ and $u_{n}$ converge on the disc of radius $r / {| q |}^{n / α}$ and $r / {| q |}^{⌊n / α⌋}$ , respectively, for some $r > 0$ , and there exist $C = C (q), A = A (q) > 0$ such that for all $n \geq 0$ one has

$sup_{| ϵ | \leq r / {| q |}^{n / α}} | y_{n} (ϵ) | \leq C A^{n}, sup_{| ϵ | \leq r / {| q |}^{⌊n / α⌊}} | u_{n} (x) | \leq C A^{n} .$

Observe in the previous result that the coefficients of the formal solutions might be defined in shrinking neighborhoods of the origin, determining power series which have null radius of convergence.

The procedure followed to solve (1), (2) analytically is to search for solutions in the form of a Laplace transform of order k (see Section 5) which transforms the main problem into an auxiliary convolution equation, whose analytic solution satisfies appropriate bounds in order to recover an analytic solution to the main equation via Laplace transform (see Proposition 1). Sharp bounds satisfied by the solutions to the auxiliary convolution equation are also available, leading to the construction of a finite family of analytic solutions to (1), (2), say

{(u_{p, 1} (t, z, ϵ))}_{0 \leq p \leq ς_{1} - 1}

, for some integer

ς_{1} \geq 2

, with

u_{p, 1} \in O_{b} (T \times D \times E_{p})

for all

0 \leq p \leq ς_{1} - 1

. Here,

T

stands for a bounded sector in

C

with vertex at the origin, D is a neighborhood of the origin, and

E_{p}

is a bounded sector with vertex at the origin belonging to a good covering in

C^{★}

(see Definition 8). Another finite family of analytic solutions to (1), (2), say

{(u_{p, 2} (t, z, ϵ))}_{0 \leq p \leq ς_{2} - 1}

, for some integer

ς_{2} \geq 2

, is also constructed. For every

0 \leq p \leq ς_{2} - 1

, the solution

u_{p, 2} (t, z, ϵ)

remains analytic on

T_{p} \times D \times E

, where

E

is some bounded sector in

C

with vertex at the origin, and so it is

T_{p}

, which is an element of a good covering in

C^{★}

The main results of the present work determine the asymptotic behavior of the two families of analytic solutions from two radically different topological points of view. It is proved in Theorem 3 the existence of a formal power series in the perturbation parameter, with coefficients in some Banach space of functions which asymptotically approaches each of the analytic solutions in

{(u_{p, 1} (t, z, ϵ))}_{0 \leq p \leq ς_{1} - 1}

. The asymptotic approximation is measured by means of a

L_{1} - q -

relative-sup-norm. Such norm is defined on a larger set of formal power series in one of their variables with coefficients being holomorphic functions on some shrinking neighborhood of the origin (see Definition 3). Under this measurement, the asymptotic behavior is of Gevrey nature (see (29)). On the other hand, when incorporating the classical

L_{1}

-sup norm in the asymptotic approximation, then mixed Gevrey and

q -

Gevrey asymptotic expansions emerge, as it is proved in Theorem 4. We recall that previous results in the field have also observed such multiscaled asymptotics, such as [7]. Theorems 3 and 4 are put forward in a symmetric manner regarding time variable, leading to Gevrey and q-Gevrey asymptotic relations for the analytic solutions

{(u_{p, 2} (t, z, ϵ))}_{0 \leq p \leq ς_{2} - 1}

, in Theorems 5 and 6. The technique used in the preceeding results leans on the application of the classical version of the so-called Ramis-Sibuya theorem (Theorem 7 (RS) in Section 6) and a

q -

analog of Ramis-Sibuya theorem (Theorem 8 (

q -

RS) in Section 6).

In brief, the work states different asymptotic expansions with respect to

ϵ

and t regarding different sets of analytic solutions to the main problem under study (1), (2): one of Gevrey order

1 / k

and another of mixed type Gevrey and

q -

Gevrey, when modifying the norm set on the space of coefficients of the formal solution. In addition to this, Gevrey order expansions of order

1 / k

have been observed in both variables t and

ϵ

by setting appropriate norms on the spaces of holomorphic functions involved.

The paper is structured as follows. Section 2.1 is devoted to precise the main problem under study. In the next subsections, we provide different families of analytic solutions (Theorems 1 and 2) by fixing concise geometries in the problem. The first main results on the asymptotic behavior of the previous families of analytic solutions are stated in Section 3 (Theorems 3 and 4) by determining different norms in the space of coefficients of the formal solution. Symmetric results regarding the time variable (Theorems 5 and 6) are stated in Section 4. The work concludes with two annex which complete known facts about Laplace transform and its main properties and several versions of Ramis-Sibuya type theorems, appealed in the paper.

Notation:

We write

N : = {1, 2, 3, \dots}

and

N_{0} : = N \cup {0}

For every

r > 0

and

z_{0} \in C

, we write

D (z_{0}, r)

for the open disc centered at

z_{0}

and radius r, and for simplicity we denote

D_{r} : = D (0, r)

Given a nonempty open set

U \subseteq C

, and a complex Banach space

E

O_{b} (U, E)

stands for the set of bounded holomorphic functions

h : U \to E

which determines a Banach space with the norm of the supremum. For simplicity, we write

O_{b} (U)

instead of

O_{b} (U, C)

. We also denote the set of formal power series in the variable z and coefficients in

E

E [[z]]

2. Statement of the Main Problem and Analytic Solution

In this section, we state the main Cauchy problem under study in the present work and provide analytic solutions to it in adequate domains. The procedure heavily rests on the study developed in [11], and therefore most of the details are omitted, in order to enhance the main purpose of the present study, i.e. to focus on the appearance of different related asymptotic occurrence related to such analytic solutions.

2.1. Statement of the Main Problem

Let

k, S

be positive integers. Fix

q > 1

, and

ϵ_{0} > 0

Let

A \subseteq N_{0}^{4}

be a finite set, and

Δ_{\underset{̲}{ℓ}} \in N

for every

\underset{̲}{ℓ} \in A

. We assume that for every

\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A

ℓ_{2} < S, S \geq ℓ_{2} + ℓ_{3}, Δ_{\underset{̲}{ℓ}} \geq ℓ_{0},

(5)

and fix a polynomial

c_{\underset{̲}{ℓ}} (z, ϵ) = \sum_{h \in I_{\underset{̲}{ℓ}}} c_{\underset{̲}{ℓ}, h} (ϵ) z^{h} \in O_{b} (D_{ϵ_{0}}) [z]

, for some finite set

I_{\underset{̲}{ℓ}} \subseteq N

. We assume that

c_{\underset{̲}{ℓ}} (0, ϵ) \equiv 0

. The following hypotheses are also fulfilled: there exists

Δ \geq 1 / 2

such that

\begin{matrix} 2 (ℓ_{2} - h) Δ + ℓ_{0} + k ℓ_{1} - 2 (S - 1) Δ > 0, \\ Δ max {0, 2 (ℓ_{2} - h) - 1} - {(- h + ℓ_{2})}^{2} Δ < min_{a \in {S - 1, S}} a (ℓ_{0} + k ℓ_{1}) - a^{2} Δ, \\ - 2 Δ ℓ_{3} + ℓ_{0} + k ℓ_{1} > 0, \end{matrix}

(6)

for all

h \in I_{\underset{̲}{ℓ}}

and

\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A

Remark: Less restrictive conditions than those appearing in (6) can be assumed. We have decided to adopt (6) for the sake of simplicity. Observe moreover that the assumptions (5) and (6) are compatible whenever

ℓ_{0} + k ℓ_{1}

for

(ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A

is large enough, together with suitable choices of the remaining constants.

For every

0 \leq j \leq S - 1

, we consider a polynomial in time variable, say

φ_{j} (t, ϵ) \in (O_{b} (D_{ϵ_{0}})) [t]

, such that

φ_{j} (0, ϵ) \equiv 0

for

0 \leq j \leq S - 1

Let

P (τ) \in C [τ]

with

P (0) \neq 0

, and such that there exists

k_{1} > 0

with

k \deg (P) \geq k ℓ_{1} + ℓ_{0} + 2 k_{1} ℓ_{3} log (q), for \underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A .

(7)

The main problem under consideration is the following singularly perturbed linear Cauchy problem

P (ϵ^{k} t^{k + 1} \partial_{t}) \partial_{z}^{S} u (t, z, ϵ) = \sum_{\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A} ϵ^{Δ_{\underset{̲}{ℓ}}} c_{\underset{̲}{ℓ}} (z, ϵ) t^{ℓ_{0}} ({(ϵ^{k} t^{k + 1} \partial_{t})}^{ℓ_{1}} \partial_{z}^{ℓ_{2}} u) (q^{ℓ_{3}} t, z, ϵ),

(8)

with Cauchy data

(\partial_{z}^{j} u) (t, 0, ϵ) = φ_{j} (t, ϵ), 0 \leq j \leq S - 1 .

(9)

2.2. Construction of Analytic Solutions to the Main Problem

The strategy to find analytic solutions to (8), (9) is to search for functions in the form of a Laplace transform of order k along well chosen directions

γ \in R

to be determined. More precisely, we search for solutions

u (t, z, ϵ) = k \int_{L_{γ}} ω (u, z, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u},

for some suitable function

ω

under suitable growth properties at infinity regarding its first variable.

In view of the properties displayed in Propositions 9 and 10 on Laplace transform, we reduce the problem to solve the following auxiliary Cauchy problem

\begin{matrix} \partial_{z}^{S} ω (u, z, ϵ) = \sum_{\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A, ℓ_{0} = 0} ϵ^{Δ_{\underset{̲}{ℓ}}} \frac{c_{\underset{̲}{ℓ}} (z, ϵ)}{P (k u^{k})} {(k {(q^{ℓ_{3}} u)}^{k_{1}})}^{ℓ_{1}} (\partial_{z}^{ℓ_{2}} ω) (q^{ℓ_{3}} u, z, ϵ) \\ + \sum_{\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A, ℓ_{0} \geq 1} ϵ^{Δ_{\underset{̲}{ℓ}} - ℓ_{0}} \frac{c_{\underset{̲}{ℓ}} (z, ϵ)}{P (k u^{k})} \frac{u^{k}}{Γ (\frac{ℓ_{0}}{k})} \int_{0}^{u^{k}} {(u^{k} - s)}^{\frac{ℓ_{0}}{k} - 1} {(k {(q^{ℓ_{3}} s^{1 / k})}^{k})}^{ℓ_{1}} (\partial_{z}^{ℓ_{2}} ω) (q^{ℓ_{3}} s^{1 / k}, z, ϵ) \frac{d s}{s}, \end{matrix}

(10)

with Cauchy data

(\partial_{z}^{j} ω) (u, 0, ϵ) = P_{j} (u, ϵ), 0 \leq j \leq S - 1,

(11)

where

P_{j}

for

0 \leq j \leq S - 1

is determined from

φ_{j}

from the properties of Laplace transform (see Proposition 8 and Corollary 1). Indeed, one can express

φ_{j}

as the Laplace transform of order k of some polynomial

P_{j} (u, ϵ) \in (O_{b} (D_{ϵ_{0}})) [u]

. More precisely, one has that

P_{j} (u, ϵ) = \sum_{h \in J_{j}} p_{j, h} (ϵ) u^{h}

, for some finite subset

J_{j} \subseteq N

, and all

0 \leq j \leq S - 1

. In this situation,

φ_{j} (t, ϵ) = \sum_{h \in J_{j}} Γ (h / k) p_{j, h} (ϵ) t^{h}

, for all

0 \leq j \leq S - 1

, where

Γ (\cdot)

stands for Gamma function.

A finite family of analytic solutions to (10), (11) is now constructed from the following associated geometric configuration to the problem, on adequate domains.

The following result is a parametric version of Proposition 5 in [11], whose proof can be adapted without any difficulty to this setting. We only give a sketch of the points to be adapted in our framework.

Proposition 1.

Let

ς \geq 2

be an integer. We consider a set

\underset{̲}{U} = {(U_{p})}_{0 \leq p \leq ς - 1}

of unbounded sectors with vertex at the origin which satisfies that all the roots of the polynomial

u \mapsto P (k u^{k})

belong to

C ∖ (⋃_{p = 0}^{ς - 1} U_{p})

. Assume conditions (5), (6) and (7) hold. Then, there exists

R > 0

such that for every

0 \leq p \leq ς - 1

a series

ω_{p} (u, z, ϵ) = \sum_{n \geq 0} ω_{p, n} (u, ϵ) \frac{z^{n}}{n!}

(12)

can be constructed determining a holomorphic function on

U_{p} \times D_{R} \times D_{ϵ_{0}}

, which solves the auxiliary Cauchy problem (10), (11). Moreover, there exist

C_{3} > 0

u_{0} > 1

α \geq 0

and

k_{1} > 0

stated in (7) with

| ω_{p} (u, z, ϵ) | \leq 2 C_{3} | u | exp (k_{1} {log}^{2} (| u | + u_{0}) + α log (| u | + u_{0})),

(13)

for all

u \in U_{p}

z \in D_{R}

and

ϵ \in D_{ϵ_{0}}

. In addition to this, each function

ω_{p, n} (u, ϵ)

satisfies the following properties:

a.: For every $0 \leq p \leq ς - 1$ and $n \geq 0$ , the function $(u, ϵ) \mapsto ω_{p, n} (u, ϵ)$ belongs to $O (U_{p} \times D_{ϵ_{0}})$ with

$sup_{ϵ \in D_{ϵ_{0}}} | ω_{p, n} (u, ϵ) | \leq C_{3} \frac{1}{{(2 R)}^{n}} n! | u | exp (k_{1} {log}^{2} (| u | + u_{0}) + α log (| u | + u_{0})),$

(14)

for all $u \in U_{p}$ .
b.: For every $n \geq 0$ there exists an analytic function $ω_{n} (u, ϵ)$ , which is a common analytic extension of $(u, ϵ) \mapsto ω_{p, n} (u, ϵ)$ for all $0 \leq p \leq ς - 1$ . This analytic continuation is defined on $D_{R_{n}} \times D_{ϵ_{0}}$ with $R_{n} = R_{0} / q^{n}$ , for some small enough $R_{0} > 0$ such that

$D_{R_{0}} \cap {u \in C : P (k u^{k}) = 0} = \emptyset .$

Moreover, there exist $C_{1}, C_{2} > 0$ such that

$sup_{ϵ \in D_{ϵ_{0}}} | ω_{n} (u, ϵ) | \leq C_{1} {(C_{2})}^{n} \frac{n!}{q^{n^{2} Δ}} | u |,$

(15)

for $u \in D_{R_{n}}$ , where Δ is introduced in (6).
c.: For every $0 \leq p \leq ς - 1$ and $n \geq 0$ , the function $(u, ϵ) \mapsto ω_{p, n} (u, ϵ)$ is bounded holomorphic on the sectorial annulus

$A_{p, h} = \{u \in U_{p} : \frac{R_{0}}{q^{h + 1}} \leq | u | \leq \frac{R_{0}}{q^{h}}\}$

for all $0 \leq h \leq n - 1$ , existing $C_{5}, C_{6} > 0$ with

$sup_{ϵ \in D_{ϵ_{0}}} | ω_{p, n} (u, ϵ) | \leq C_{5} {(C_{6})}^{n} \frac{n!}{q^{h^{2} Δ}} | u |,$

(16)

for all $u \in A_{p, h}$ , where Δ is fixed in (6).

Proof.

The proof of Proposition 5 in [11] can be directly adapted to this parametric framework under the action of the perturbation parameter

ϵ

. More precisely, the third element in the assumption (5) guarantees that

sup_{ϵ \in D_{ϵ_{0}}} {| ϵ |}^{Δ_{\underset{̲}{ℓ}} - ℓ_{0}} \leq ϵ_{0}^{Δ_{\underset{̲}{ℓ}} - ℓ_{0}},

for every

\underset{̲}{ℓ} = (ℓ_{0}, ℓ_{1}, ℓ_{2}, ℓ_{3}) \in A

. The polynomial nature of the coefficients

c_{\underset{̲}{ℓ}} (z, ϵ) = \sum_{h \in I_{\underset{̲}{ℓ}}} c_{\underset{̲}{ℓ}, h} z^{h} \in O_{b} (D_{ϵ_{0}}) [z]

allows us to upper estimate

sup_{ϵ \in D_{ϵ_{0}}} | c_{\underset{̲}{ℓ}} (z, ϵ) | \leq \sum_{h \in I_{\underset{̲}{ℓ}}} (sup_{ϵ \in D_{ϵ_{0}}} | c_{\underset{̲}{ℓ}, h} (ϵ) |) {| z |}^{h} .

Taking the above facts into consideration, the proof follows directly from that of Proposition 5 in [11]. □

At this point, one can achieve the construction of actual solutions to the main Cauchy problem (8), (9). For that purpose, we introduce the following geometric definition.

Definition 1.

Let

ς \geq 2

be an integer. Let us consider a finite set of bounded sectors

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς - 1}

which conforms a good covering in

C^{★}

(see Definition 8). Associated to

\underset{̲}{E}

let us choose a bounded sector with vertex at the origin,

T \subseteq D_{r_{T}}

for some

r_{T} > 0

, and a family

\underset{̲}{U} = {(U_{p})}_{0 \leq p \leq ς - 1}

of unbounded sectors with vertex at the origin, under the following conditions:

The roots of $u \mapsto P (k u^{k})$ belong to $C ∖ (⋃_{p = 0}^{ς - 1} U_{p})$ ,
For every $0 \leq p \leq ς - 1$ , there exists $Δ_{p} > 0$ such that for all $t \in T$ and $ϵ \in E_{p}$ a direction $γ_{p} \in R$ exists (which depends on t and ϵ) such that

-

$L_{γ_{p}} = [0, \infty) e^{\sqrt{- 1} γ_{p}} \subseteq U_{p} \cup {0}$ , and

-

$cos (k (γ_{p} - \arg (ϵ t))) > Δ_{p}$ .

We say that the tuple

{T, \underset{̲}{U}}

is admissible with respect to the good covering

\underset{̲}{E}

Theorem 1.

Let

ς_{1} \geq 2

. Let

{T, \underset{̲}{U_{1}} = {(U_{p, 1})}_{0 \leq p \leq ς_{1} - 1}}

be an admissible set with respect to a given good covering

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς_{1} - 1}

(following Definition 1). Take for granted the assumptions made in Section 2.1 on the elements involved in the main Cauchy problem (8), (9).

Then, for every

0 \leq p \leq ς_{1} - 1

, there exists a solution

u_{p, 1}

of (8), (9) which is holomorphic and bounded on

T \times D_{R} \times E_{p}

. Such solution is of the form

u_{p, 1} (t, z, ϵ) = \sum_{n \geq 0} u_{p, 1, n} (t, ϵ) \frac{z^{n}}{n!}, (t, z, ϵ) \in T \times D_{R} \times E_{p},

(17)

with

u_{p, 1, n} (t, ϵ) = k \int_{L_{γ_{p}}} ω_{p, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u},

(18)

for every

n \geq 0

, and all

(t, ϵ) \in T \times E_{p}

. Here,

L_{γ_{p}} = [0, \infty) e^{\sqrt{- 1} γ_{p}}

, where

γ_{p}

is determined in Definition 1, and

ω_{p, 1, n}

represents the coefficient

ω_{p, n}

in (12) of Proposition 1 for the set

\underset{̲}{U} = \underset{̲}{U_{1}}

Proof.

Let

0 \leq p \leq ς_{1} - 1

, and consider the function

ω_{p}

constructed in Proposition 1 for the covering

\underset{̲}{U} = \underset{̲}{U_{1}}

, that we denote

ω_{p, 1}

with coefficients

ω_{p, 1, n}

. The function

ω_{p}

is a solution of the auxiliary problem (10), (11). We write

ω_{p}

in the form (12) and recall that its growth with respect to u at infinity, displayed in (14), allows us to apply Laplace transform of order k along direction

γ_{p}

. The construction of the admissible set with respect to the good covering

\underset{̲}{E}

guarantees that

u_{p, 1, n} (t, ϵ) = L_{k}^{γ_{p}} (ω_{p, 1, n} (u, ϵ)) (ϵ t, ϵ)

is holomorphic and bounded in

T \times E_{p}

. Moreover, from the bounds in (14) the expression in (17) determines a bounded holomorphic function in

T \times D_{R} \times E_{p}

. The properties of Laplace transform in Section 5 guarantee that (17) determines an actual solution of (8), (9). □

A symmetric construction can be followed by interchanging the role of the variables

ϵ

and t in the previous construction to achieve analytic solutions to the main problem in different families of domains. In this respect, Definition 1 reads as follows.

Definition 2.

Let

ς \geq 2

be an integer. Let us consider a finite set of bounded sectors

\underset{̲}{T} = {(T_{p})}_{0 \leq p \leq ς - 1}

which conforms a good covering in

C^{★}

. Associated to

\underset{̲}{T}

one chooses a bounded sector with vertex at the origin,

E \subseteq D_{ϵ_{0}}

, and a family

\underset{̲}{U} = {(U_{p})}_{0 \leq p \leq ς - 1}

of unbounded sectors with vertex at the origin, under the following conditions:

The roots of $u \mapsto P (k u^{k})$ belong to $C ∖ (⋃_{p = 0}^{ς - 1} U_{p})$ ,
For every $0 \leq p \leq ς - 1$ , there exists ${\hat{Δ}}_{p} > 0$ such that for all $ϵ \in E$ and $t \in T_{p}$ a direction ${\hat{γ}}_{p} \in R$ exists (which depends on t and ϵ) such that

-

$L_{{\hat{γ}}_{p}} = [0, \infty) e^{\sqrt{- 1} {\hat{γ}}_{p}} \subseteq U_{p} \cup {0}$ , and

-

$cos (k ({\hat{γ}}_{p} - \arg (ϵ t))) > {\hat{Δ}}_{p}$ .

We say that the tuple

{E, \underset{̲}{U}}

is admissible with respect to the good covering

\underset{̲}{T}

The following symmetric result to Theorem 1 is also valid, following the same arguments as above for its proof.

Theorem 2.

Let

ς_{2} \geq 2

. Let

{E, \underset{̲}{U_{2}} = {(U_{p, 2})}_{0 \leq p \leq ς_{2} - 1}}

be an admissible set with respect to a given good covering

\underset{̲}{T} = {(T_{p})}_{0 \leq p \leq ς_{2} - 1}

(in the sense of Definition 2). Assume the hypotheses made in Section 2.1 on the elements involved in the construction of the main Cauchy problem (8), (9) hold. Then, for every

0 \leq p \leq ς_{2} - 1

, there exists a solution

u_{p, 2}

of (8), (9) which is holomorphic and bounded on

T_{p} \times D_{R} \times E

. Such solution is of the form

u_{p, 2} (t, z, ϵ) = \sum_{n \geq 0} u_{p, 2, n} (t, ϵ) \frac{z^{n}}{n!}, (t, z, ϵ) \in T_{p} \times D_{R} \times E,

(19)

with

u_{p, 2, n} (t, ϵ) = k \int_{L_{{\hat{γ}}_{p}}} ω_{p, 2, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u},

(20)

for every

n \geq 0

, and all

(t, ϵ) \in T_{p} \times E

. Here,

L_{{\hat{γ}}_{p}} = [0, \infty) e^{\sqrt{- 1} {\hat{γ}}_{p}}

, is determined in Definition 2 and where

ω_{p, 2, n}

represents the coefficient

ω_{p, n}

in (12) of Proposition 1 for the set

\underset{̲}{U} = \underset{̲}{U_{2}}

3. Asymptotic Behavior of the Analytic Solutions, I

In this section, we provide two results regarding the asymptotic expansions of the previous analytic solutions determined in Theorem 1 with respect to the perturbation parameter regarding each of the elements in a good covering

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς_{1} - 1}

, for some fixed

ς_{1} \geq 2

. The main first result of the present work determines classical Gevrey asymptotic expansions by introducing

L_{1} - q -

relative-sup-norms on the partial functions

(t, z) \mapsto u_{p, 1} (t, z, ϵ)

, constructed in Theorem 1, for each fixed

ϵ \in E_{p}

, and all

0 \leq p \leq ς_{1} - 1

, whereas the second result deals with uniform sup-norms on the partial functions

(t, z)

Definition 3.

let

q > 1

be a real number. We fix a bounded sector with vertex at the origin

T

. Let us consider the set

O_{b}^{q} (T) [[z]]

of formal power series h of the form

h (t, z) = \sum_{n \geq 0} h_{n} (t) \frac{z^{n}}{n!},

where

h_{n} \in O_{b} (T \cap D_{1 / q^{n}})

for every

n \geq 0

Let

R_{1} > 0

be a real number.

We denote by

E_{L_{1}; q; R_{1}}

the vector space of formal power series

h \in O_{b}^{q} (T) [[z]]

of the form

h (t, z) = \sum_{n \geq 0} h_{n} (t) \frac{z^{n}}{n!}

, where

h_{n} \in O_{b} (T \cap D_{1 / q^{n}})

for every

n \geq 0

, such that the

L_{1} - q -

relative-sup-norm of h, defined by

{∥h (t, z)∥}_{L_{1}; q; R_{1}} : = \sum_{n \geq 0} sup_{t \in T \cap D_{1 / q^{n}}} | h_{n} (t) | \frac{R_{1}^{n}}{n!},

is finite.

Proposition 2.

The pair

(E_{L_{1}; q; R_{1}}, {∥\cdot∥}_{L_{1}; q; R_{1}})

is a complex Banach space. The vector space

O_{b} (T \times D_{R})

is contained in

E_{L_{1}; q; R_{1}}

provided that

R > R_{1}

Proof.

First, observe that any formal power series

h (t, z) = \sum_{n \geq 0} h_{n} (t) \frac{z^{n}}{n!} \in E_{L_{1}; q; R_{1}}

such that

{∥h (t, z)∥}_{L_{1}; q; R_{1}} = 0

entails that

h_{n} \equiv 0

T \cap D_{1 / q^{n}}

Let

{(h^{p})}_{p \geq 1}

be a Cauchy sequence in

E_{L_{1}; q; R_{1}}

. Let us write

h^{p} (t, z) = \sum_{n \geq 0} h_{n}^{p} (t) \frac{z^{n}}{n!}

, for every

p \in N

. It is clear that for every

n \in N_{0}

, the sequence

{(h_{n}^{p})}_{p \geq 1}

is a Cauchy sequence in the Banach space

(O_{b} (T \cap D_{1 / q^{n}}), {∥\cdot∥}_{\infty})

, where

{∥\cdot∥}_{\infty}

stands for the supremum norm with

{∥h_{n}^{p} - h_{n}^{q}∥}_{\infty} < \frac{ϵ n!}{R_{1}^{n}}

(21)

for all

p, q \in N

with

p, q \geq p_{0}

for some

p_{0} \in N

whenever

{∥h^{p} - h^{q}∥}_{L_{1}; q; R_{1}} < ϵ

. Therefore, for every

n \in N_{0}

there exists

H_{n} \in O_{b} (T \cap D_{1 / q^{n}})

such that

{(h_{n}^{p})}_{n \geq 0}

converges in sup norm to

H_{n}

. By considering the formal power series

H (t, z) = \sum_{n \geq 0} H_{n} (t) \frac{z^{n}}{n!}

it is direct to check that H is an element of

E_{L_{1}; q; R_{1}}

, with H being the limit of

{(h^{p})}_{p \geq 0}

in such space. □

From now on, we resume the assumptions and constructions related to the main problem (8), (9) in Section 2.1. In addition to this, we consider

ς_{1} \geq 2

and an admissible set

{T, \underset{̲}{U_{1}} = {(U_{p, 1})}_{0 \leq p \leq ς_{1} - 1}}

with respect to a given good covering

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς_{1} - 1}

(in the sense of Definition 1). Theorem 1 guarantees the existence of a finite family of solutions

u_{p, 1} (t, z, ϵ)

to (8), (9), for

0 \leq p \leq ς_{1} - 1

, holomorphic and bounded in

T \times D_{R} \times E_{p}

, of the form (17) with (18), for some

R > 0

. Let us fix

0 < R_{1} < R

Proposition 3.

Under the assumptions of Theorem 1, for every

0 \leq p \leq ς_{1} - 1

there exist

A_{p}, B_{p} > 0

such that

{∥u_{p + 1, 1} (t, z, ϵ) - u_{p, 1} (t, z, ϵ)∥}_{L_{1}; q; R_{1}} \leq A_{p} exp (- \frac{B_{p}}{{| ϵ |}^{k}}),

(22)

valid for every

ϵ \in E_{p} \cap E_{p + 1}

, where we have identified

u_{ς_{1}, 1} : = u_{0, 1}

and

E_{ς_{1}} : = E_{0}

, provided that

0 < R_{1} < R

is small enough.

Proof.

Let

0 \leq p \leq ς_{1} - 1

and fix

ϵ \in E_{p} \cap E_{p + 1}

. In view of (17) and (18), one can write

u_{p + 1, 1} (t, z, ϵ) - u_{p, 1} (t, z, ϵ)

in the form

\sum_{n \geq 0} (k \int_{L_{γ_{p + 1}}} ω_{p + 1, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u} - k \int_{L_{γ_{p}}} ω_{p, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u}) \frac{z^{n}}{n!} .

(23)

One can apply b. in Proposition 1 taking

ς_{1}

in place of

ς

and

\underset{̲}{U_{1}} = {(U_{p, 1})}_{0 \leq p \leq ς_{1} - 1}

in place of

\underset{̲}{U} = {(U_{p})}_{0 \leq p \leq ς - 1}

. Then, for all

n \geq 0

, the functions

u \mapsto ω_{p, 1, n} (u, ϵ)

and

u \mapsto ω_{p + 1, 1, n} (u, ϵ)

have a common analytic continuation, say

u \mapsto ω_{n} (u, ϵ)

D_{R_{n}}

, with

R_{n} = R_{0} / q^{n}

, for some

R_{0} > 0

. This entails that the integration path appearing in the previous difference can be deformed by the application of Cauchy theorem. Hence,

u_{p + 1, 1} (t, z, ϵ) - u_{p, 1} (t, z, ϵ)

equals

\begin{matrix} = \sum_{n \geq 0} (k \int_{L_{γ_{p + 1}, R_{n + 1}}} ω_{p + 1, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u} - k \int_{L_{γ_{p}, R_{n + 1}}} ω_{p, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u} \\ + k \int_{C_{γ_{p}, γ_{p + 1}, R_{n + 1}}} ω_{n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u}) \frac{z^{n}}{n!}, \end{matrix}

(24)

where

L_{γ_{j}, R_{n + 1}} = [R_{n + 1}, \infty) e^{\sqrt{- 1} γ_{j}}

for

j = p, p + 1

and

C_{γ_{p}, γ_{p + 1}, R_{n + 1}}

stands for the oriented arc of circle centered at the origin, of radius

R_{n + 1}

, from the point

R_{n + 1} e^{\sqrt{- 1} γ_{p}}

R_{n + 1} e^{\sqrt{- 1} γ_{p + 1}}

. Let us provide upper bounds for the previous elements. Let us define

I_{1} (γ_{p + 1}) : = |k \int_{L_{γ_{p + 1}, R_{n + 1}}} ω_{p + 1, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u}| .

In view of statement a. in Proposition 1 and the fact that

{T, \underset{̲}{U_{1}}}

is admissible with respect to

\underset{̲}{E}

, one has that

I_{1} (γ_{p + 1})

is upper bounded by

\begin{matrix} k C_{3} \frac{1}{R^{n}} n! \int_{R_{n + 1}}^{\infty} exp (k_{1} {log}^{2} (r + u_{0}) + α log (r + u_{0})) exp (- {(\frac{r}{| ϵ t |})}^{k} cos (k (γ_{p + 1} - \arg (ϵ t)))) d r \\ \leq k C_{3} \frac{1}{R^{n}} n! \int_{R_{n + 1}}^{\infty} exp (k_{1} {log}^{2} (r + u_{0}) + α log (r + u_{0})) exp (- \frac{1}{2} {(\frac{r}{| ϵ t |})}^{k} Δ_{p + 1}) \\ \times exp (- \frac{1}{2} {(\frac{r}{| ϵ t |})}^{k} Δ_{p + 1}) d r \\ \leq k C_{3} \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{n + 1}}{| ϵ t |})}^{k} Δ_{p + 1}) \int_{R_{n + 1}}^{\infty} exp (k_{1} {log}^{2} (r + u_{0}) + α log (r + u_{0})) \\ \times exp (- \frac{1}{2} {(\frac{r}{| ϵ t |})}^{k} Δ_{p + 1}) d r \\ \leq k C_{3} J_{1} (Δ_{p + 1}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{n + 1}}{| ϵ t |})}^{k} Δ_{p + 1}), \end{matrix}

with

J_{1} (Δ_{p + 1}) = \int_{0}^{\infty} exp (k_{1} {log}^{2} (r + u_{0}) + α log (r + u_{0})) exp (- \frac{1}{2} {(\frac{r}{ϵ_{0} r_{T}})}^{k} Δ_{p + 1}) d r < \infty .

Taking into account that

R_{n + 1} = R_{0} / q^{n + 1}

we conclude from the previous expression that

sup_{t \in T \cap D_{1 / q^{n}}} I_{1} (γ_{p + 1}) \leq k C_{3} J_{1} (Δ_{p + 1}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{0}}{q | ϵ |})}^{k} Δ_{p + 1}) .

(25)

An analogous reasoning yields

I_{2} (γ_{p}) : = |k \int_{L_{γ_{p}, R_{n + 1}}} ω_{p, 1, n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u}|

is upper estimated

sup_{t \in T \cap D_{1 / q^{n}}} I_{2} (γ_{p}) \leq k C_{3} J_{1} (Δ_{p}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{0}}{q | ϵ |})}^{k} Δ_{p}) .

(26)

Finally, we consider

I_{3} (γ_{p}, γ_{p + 1}) : = |k \int_{C_{γ_{p}, γ_{p + 1}, R_{n + 1}}} ω_{n} (u, ϵ) exp (- {(\frac{u}{ϵ t})}^{k}) \frac{d u}{u}| .

Regarding statement b. in Proposition 1 and the parametrization

u (θ) = R_{0} / q^{n + 1} e^{i θ}

for

θ

between

γ_{p}

and

γ_{p + 1}

, one arrives at the existence of

{\tilde{Δ}}_{p, p + 1} > 0

such that

\begin{matrix} I_{3} (γ_{p}, γ_{p + 1}) \leq k \int_{γ_{p}}^{γ_{p + 1}} C_{1} {(C_{2})}^{n} \frac{n!}{q^{n^{2} Δ}} \frac{R_{0}}{q^{n + 1}} exp (- {(\frac{R_{0} / q^{n + 1}}{| ϵ t |})}^{k} cos (k (θ - \arg (ϵ t)))) d θ \\ \leq k | γ_{p + 1} - γ_{p} | C_{1} {(C_{2})}^{n} \frac{n! R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0} / q^{n + 1}}{| ϵ t |})}^{k} {\tilde{Δ}}_{p, p + 1}) . \end{matrix}

(27)

This entails

sup_{t \in T \cap D_{1 / q^{n}}} I_{3} (γ_{p}, γ_{p + 1}) \leq k | γ_{p + 1} - γ_{p} | C_{1} {(C_{2})}^{n} \frac{n! R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0}}{q | ϵ |})}^{k} {\tilde{Δ}}_{p, p + 1}) .

(28)

Collecting the information in (25), (26) and (28), one arrives at

\begin{matrix} {∥u_{p + 1, 1} (t, z, ϵ) - u_{p, 1} (t, z, ϵ)∥}_{L_{1}; q; R_{1}} \\ \leq \sum_{n \geq 0} (2 k C_{3} max {J_{1} (Δ_{p}), J_{1} (Δ_{p + 1})} \frac{1}{R^{n}} exp (- \frac{1}{2} {(\frac{R_{0}}{q | ϵ |})}^{k} min {Δ_{p}, Δ_{p + 1}}) \\ + k | γ_{p + 1} - γ_{p} | C_{1} {(C_{2})}^{n} \frac{R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0}}{q | ϵ |})}^{k} {\tilde{Δ}}_{p, p + 1})) R_{1}^{n} . \end{matrix}

The choice of

R_{1} < R

yields (22) with

A_{p} : = (\sum_{n \geq 0} {(\frac{R_{1}}{R})}^{n} + \sum_{n \geq 0} \frac{{(C_{2} R_{1})}^{n}}{q^{n^{2} Δ + n + 1}}) (2 k C_{3} max {J_{1} (Δ_{p}), J_{1} (Δ_{p + 1})} + k | γ_{p + 1} - γ_{p} | C_{1} R_{0}),

and

B_{p} : = \frac{R_{0}^{k}}{q^{k}} min \{\frac{1}{2} min {Δ_{p}, Δ_{p + 1}}, {\tilde{Δ}}_{p, p + 1}\} .

□

Proposition 3 guarantees that Theorem 7 (RS) (see Section 6) can be applied when considering the next Banach space of functions.

Theorem 3.

Under the assumptions of Theorem 1, there exists a formal power series

{\hat{u}}_{1} (t, z, ϵ) = \sum_{n \geq 0} u_{1, n} (t, z) \frac{ϵ^{n}}{n!} \in E_{L_{1}; q; R_{1}} [[ϵ]]

satisfying that there exist

C, M > 0

such that for all

N \geq 0

one has

{∥u_{p, 1} (t, z, ϵ) - \sum_{n = 0}^{N} u_{1, n} (t, z) \frac{ϵ^{n}}{n!}∥}_{L_{1}; q; R_{1}} \leq C M^{N + 1} Γ (\frac{N + 1}{k}) {| ϵ |}^{N + 1},

(29)

for all

0 \leq p \leq ς_{1} - 1

and all

ϵ \in E_{p}

. In other words,

ϵ \mapsto {\hat{u}}_{1} (t, z, ϵ)

is the common Gevrey asymptotic expansion of order

1 / k

of the analytic solution

ϵ \mapsto u_{p, 1} (t, z, ϵ)

, as a function with values in

E_{L_{1}; q; R_{1}}

E_{p}

, for all

0 \leq p \leq ς_{1} - 1

Proof.

For all

0 \leq p \leq ς_{1} - 1

, consider the function

u_{p, 1} (t, z, ϵ)

constructed in Theorem 1, and define the map

G_{p} (ϵ)

ϵ \mapsto u_{p, 1} (t, z, ϵ)

for

ϵ \in E_{p}

, which is viewed as a function with values in the Banach space

(E_{L_{1}; q; R_{1}}, {∥\cdot∥}_{L_{1}; q; R_{1}})

. Observe that one can apply Theorem 7 (RS) in view of Proposition 3, in order to achieve the existence of

{\hat{u}}_{1} (t, z, ϵ) \in E_{L_{1}; q; R_{1}} [[ϵ]]

with the required properties. □

In the second part of this section we provide a different asymptotic expansion with respect to the perturbation parameter, under the action of a different norm compared to that in the first part. The main result of this section guarantees q-Gevrey asymptotic expansions relating the formal and the analytic solutions by incorporating the classical

L_{1}

-sup norm as follows. The procedure rests on the approach in [11].

Definition 4.

let

q > 1

be a real number. We fix a bounded sector with vertex at the origin

T

. Let us consider the set

O_{b} (T) [[z]]

of formal power series h of the form

h (t, z) = \sum_{n \geq 0} h_{n} (t) \frac{z^{n}}{n!},

where

h_{n} \in O_{b} (T)

for every

n \geq 0

Let

R_{1} > 0

be a real number.

We denote by

E_{L_{1}; R_{1}}

the vector space of formal power series

h \in O_{b} (T) [[z]]

of the form

h (t, z) = \sum_{n \geq 0} h_{n} (t) \frac{z^{n}}{n!}

, where

h_{n} \in O_{b} (T)

for every

n \geq 0

, such that the

L_{1} -

sup-norm of h, defined by

{∥h (t, z)∥}_{L_{1}; R_{1}} : = \sum_{n \geq 0} sup_{t \in T} | h_{n} (t) | \frac{R_{1}^{n}}{n!},

is finite.

The proof of the next result is analogous to that of Proposition 2, so it is omitted.

Proposition 4.

The pair

(E_{L_{1}; R_{1}}, {∥\cdot∥}_{L_{1}; R_{1}})

is a complex Banach space. The vector space

O_{b} (T \times D_{R})

is contained in

E_{L_{1}; R_{1}}

provided that

R > R_{1}

Remark: Observe that

O_{b} (T) [[z]] \subseteq O_{b}^{q} (T) [[z]]

for all

q > 1

. Furthermore,

E_{L_{1}; R_{1}} \subseteq E_{L_{1}; q; R_{1}}

for all

q > 1

and we have the next inequality

{∥h∥}_{L_{1}; q; R_{1}} \leq {∥h∥}_{L_{1}; R_{1}}

for all

h \in E_{L_{1}; R_{1}}

In this section, we resume the assumptions and constructions associated to (8), (9) of Section 2.1. We also fix a good covering

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς_{1} - 1}

for some

ς_{1} \geq 2

, and an admissible set (in the sense of Definition 1)

{T, \underset{̲}{U_{1}} = {(U_{p, 1})}_{0 \leq p \leq ς_{1} - 1}}

associated to the previous good covering. Let

u_{p, 1} (t, z, ϵ) \in O_{b} (T \times D_{R} \times E_{p})

be the analytic solution to (8), (9) for

0 \leq p \leq ς_{1} - 1

, for some fixed

R > 0

. We also choose

0 < R_{1} < R

Proposition 5.

Under the assumptions of Theorem 1, for every

0 \leq p \leq ς_{1} - 1

and

0 < R_{1} < R

, there exist

{\tilde{A}}_{p}, {\tilde{B}}_{p} > 0

such that

{∥u_{p + 1, 1} (t, z, ϵ) - u_{p, 1} (t, z, ϵ)∥}_{L_{1}; R_{1}} \leq {\tilde{A}}_{p} {({\tilde{B}}_{p})}^{N} Γ (\frac{N}{k}) q^{N^{2} / 2} {| ϵ |}^{N},

for every

ϵ \in E_{p} \cap E_{p + 1}

, where we have identified

u_{ς_{1}, 1} : = u_{0, 1}

and

E_{ς_{1}} : = E_{0}

Proof.

The proof of Theorem 1 [11] can be followed point by point, by considering uniform bounds with respect to

t \in T

, and by assuming that

ϵ

plays the role of t variable in Theorem 1 of [11], together with the estimates in Proposition 1. □

From the previous result, one achieves the following second asymptotic relation of Gevrey mixed order

(1 / k; (q, 1))

Theorem 4.

Under the assumptions of Theorem 1, there exists a formal power series

{\hat{\tilde{u}}}_{1} (t, z, ϵ) = \sum_{n \geq 0} {\tilde{u}}_{1, n} (t, z) \frac{ϵ^{n}}{n!} \in E_{L_{1}; R_{1}} [[ϵ]]

satisfying that there exist

C, M > 0

such that for all

N \geq 0

one has

{∥u_{p, 1} (t, z, ϵ) - \sum_{n = 0}^{N} {\tilde{u}}_{1, n} (t, z) \frac{ϵ^{n}}{n!}∥}_{L_{1}; R_{1}} \leq C M^{N + 1} Γ (\frac{N + 1}{k}) q^{\frac{{(N + 1)}^{2}}{2}} {| ϵ |}^{N + 1},

(30)

for all

0 \leq p \leq ς_{1} - 1

and all

ϵ \in E_{p}

Proof.

One can define the map

G_{p} (ϵ)

for

0 \leq p \leq ς_{1} - 1

as in the proof of Theorem 3 and apply Theorem 8 (q-RS) (see Section 6) in virtue of Proposition 5. □

4. Asymptotic Behavior of the Analytic Solutions, II

In this section, we describe the asymptotic behavior of the analytic solutions of (8), (9) with respect to the time variable, near the origin. We proceed in a similar manner to Section 3, by fixing an element in a good covering

\underset{̲}{T}

, and providing the asymptotic properties regarding different norms on the partial sums

(z, ϵ) \mapsto u_{p, 2} (t, z, ϵ)

, for each

t \in T_{p}

and all

0 \leq p \leq ς_{2} - 1

: an

L_{1} - q -

relative-sup norm, and the uniform sup-norm on the partial functions

(z, ϵ)

In accordance to Definition 3, one can state the following symmetric definition.

Definition 5.

let

q > 1

be a real number. We fix a bounded sector with vertex at the origin

E

. Let us consider the set

O_{b}^{q} (E) [[z]]

of formal power series g of the form

g (z, ϵ) = \sum_{n \geq 0} g_{n} (ϵ) \frac{z^{n}}{n!},

where

g_{n} \in O_{b} (E \cap D_{1 / q^{n}})

for every

n \geq 0

Let

R_{1} > 0

be a real number.

We denote by

F_{L_{1}; q; R_{1}}

the vector space of formal power series

g \in O_{b}^{q} (E) [[z]]

of the form

g (z, ϵ) = \sum_{n \geq 0} g_{n} (ϵ) \frac{z^{n}}{n!}

, where

g_{n} \in O_{b} (E \cap D_{1 / q^{n}})

for every

n \geq 0

, such that the

L_{1} - q -

relative-sup-norm of g, defined by

{∥g (z, ϵ)∥}_{L_{1}; q; R_{1}} : = \sum_{n \geq 0} sup_{ϵ \in E \cap D_{1 / q^{n}}} | g_{n} (ϵ) | \frac{R_{1}^{n}}{n!},

is finite.

Remark: We adopt the same notation for the norm in Definition 3 for simplicity. The pair

(F_{L_{1}; q; R_{1}}, {∥\cdot∥}_{L_{1}; q; R_{1}})

is a complex Banach space. Besides, the vector space

O_{b} (D_{R} \times E)

is contained in

F_{L_{1}; q; R_{1}}

provided that

R > R_{1}

Returning to the main problem (8), (9), we take for granted the assumptions made on it in Section 2.1. Let us first fix

ς_{2} \geq 2

and a family of admissible domains which determines the geometry of the problem. We choose a finite family of sectors

\underset{̲}{T} = {(T_{p})}_{0 \leq p \leq ς_{2} - 1}

, which determines a good covering in

C^{★}

(see Definition 8). Associated to such good covering, we consider a tuple

{E, \underset{̲}{U_{2}} = {(U_{p, 2})}_{0 \leq p \leq ς_{2} - 1}}

which is admissible with respect to the good covering

\underset{̲}{T}

(see Definition 2). We assume

E \subseteq D_{ϵ_{0}}

. Theorem 2 provides with a family of holomorphic solutions

u_{p, 2} (t, z, ϵ)

, for

0 \leq p \leq ς_{2} - 1

, of the main problem (8), (9), which are holomorphic and bounded in

T_{p} \times D_{R} \times E

, for some

R > 0

. In addition to this, the form of such solutions is determined by (19) with (20) for all

n \geq 0

. Let

0 < R_{1} < R

Proposition 6.

We take for granted the assumptions made in Theorem 2. Then, for every

0 \leq p \leq ς_{2} - 1

there exist

{\tilde{A}}_{p}, {\tilde{B}}_{p} > 0

such that

{∥u_{p + 1, 2} (t, z, ϵ) - u_{p, 2} (t, z, ϵ)∥}_{L_{1}; q; R_{1}} \leq {\tilde{A}}_{p} exp (- \frac{{\tilde{B}}_{p}}{{| t |}^{k}}),

for every

t \in T_{p} \cap T_{p + 1}

, where

u_{ς_{2}, 2}

and

T_{ς_{2}}

stand for

u_{0, 2}

and

T_{0}

, respectively, provided that

R_{1} > 0

is small enough.

Proof.

The proof of this result heavily rests on that of Proposition 3. We only give details on the steps in which this proof differs from that.

Let us fix

0 \leq p \leq ς_{2} - 1

and

t \in T_{p} \cap T_{p + 1}

. We write the difference of two consecutive solutions

u_{p + 1, 2} (t, z, ϵ) - u_{p, 2} (t, z, ϵ)

in the form (23), with

L_{{\hat{γ}}_{j}}

in place of

L_{γ_{j}}

for

j \in {p, p + 1}

. Once again, one can apply b. in Proposition 1, by substituting

ς

ς_{2}

and

\underset{̲}{U_{2}} = {(U_{p, 2})}_{0 \leq p \leq ς_{2} - 1}

replacing

\underset{̲}{U} = {(U_{p})}_{0 \leq p \leq ς - 1}

. For each

n \geq 0

, an analogous analytic continuation

u \mapsto ω_{n} (u, ϵ)

of the functions

u \mapsto ω_{p, 2, n} (u, ϵ)

and

u \mapsto ω_{p + 1, 2, n} (u, ϵ)

for all

ϵ \in E

allows us to rewrite the difference of the consecutive solutions in the form of (24), with

L_{{\hat{γ}}_{j}, R_{n + 1}}

and

C_{{\hat{γ}}_{p}, {\hat{γ}}_{p + 1}, R_{n + 1}}

instead of

L_{γ_{j}, R_{n + 1}}

and

C_{γ_{p}, γ_{p + 1}, R_{n + 1}}

, for

j \in {p, p + 1}

. From the proof of Proposition 3, we derive

I_{1} ({\hat{γ}}_{p + 1}) \leq k C_{3} J_{1} ({\hat{Δ}}_{p + 1}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{n + 1}}{| ϵ t |})}^{k} {\hat{Δ}}_{p + 1}),

leading to

sup_{ϵ \in E \cap D_{1 / q^{n}}} I_{1} ({\hat{γ}}_{p + 1}) \leq k C_{3} J_{1} ({\hat{Δ}}_{p + 1}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{0}}{q | t |})}^{k} {\hat{Δ}}_{p + 1}) .

(31)

It is straight to achieve the bounds

sup_{ϵ \in E \cap D_{1 / q^{n}}} I_{2} ({\hat{γ}}_{p}) \leq k C_{3} J_{1} ({\hat{Δ}}_{p}) \frac{1}{R^{n}} n! exp (- \frac{1}{2} {(\frac{R_{0}}{q | t |})}^{k} {\hat{Δ}}_{p}) .

(32)

Finally, taking (27) into account, we arrive at

I_{3} ({\hat{γ}}_{p}, {\hat{γ}}_{p + 1}) \leq k | {\hat{γ}}_{p + 1} - {\hat{γ}}_{p} | C_{1} {(C_{2})}^{n} \frac{n! R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0} / q^{n + 1}}{| ϵ t |})}^{k} {\tilde{\hat{Δ}}}_{p, p + 1}),

for some

{\tilde{\hat{Δ}}}_{p, p + 1} > 0

, which leads us to

sup_{ϵ \in E \cap D_{1 / q^{n}}} I_{3} ({\hat{γ}}_{p}, {\hat{γ}}_{p + 1}) \leq k | {\hat{γ}}_{p + 1} - {\hat{γ}}_{p} | C_{1} {(C_{2})}^{n} \frac{n! R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0}}{q | t |})}^{k} {\tilde{\hat{Δ}}}_{p, p + 1}) .

(33)

In view of (31), (32) and (33) we have

\begin{matrix} {∥u_{p + 1, 2} (t, z, ϵ) - u_{p, 2} (t, z, ϵ)∥}_{L_{1}; q; R_{1}} \\ \leq \sum_{n \geq 0} (2 k C_{3} max {J_{1} ({\hat{Δ}}_{p}), J_{1} ({\hat{Δ}}_{p + 1})} \frac{1}{R^{n}} exp (- \frac{1}{2} {(\frac{R_{0}}{q | t |})}^{k} min {{\hat{Δ}}_{p}, {\hat{Δ}}_{p + 1}}) \\ + k | {\hat{γ}}_{p + 1} - {\hat{γ}}_{p} | C_{1} {(C_{2})}^{n} \frac{R_{0}}{q^{n^{2} Δ + n + 1}} exp (- {(\frac{R_{0}}{q | t |})}^{k} {\tilde{\hat{Δ}}}_{p, p + 1})) R_{1}^{n} . \end{matrix}

The choice of

R_{1} < R

and

{\tilde{A}}_{p} : = (\sum_{n \geq 0} {(\frac{R_{1}}{R})}^{n} + \sum_{n \geq 0} \frac{{(C_{2} R_{1})}^{n}}{q^{n^{2} Δ + n + 1}}) (2 k C_{3} max {J_{1} ({\hat{Δ}}_{p}), J_{1} ({\hat{Δ}}_{p + 1})} + k | {\hat{γ}}_{p + 1} - {\hat{γ}}_{p} | C_{1} R_{0}),

and

{\tilde{B}}_{p} : = \frac{R_{0}^{k}}{q^{k}} min \{\frac{1}{2} min {{\hat{Δ}}_{p}, {\hat{Δ}}_{p + 1}}, {\tilde{\hat{Δ}}}_{p, p + 1}\}

allows us to conclude the result. □

The following result is a consequence of the classical Ramis-Sibuya Theorem 7 (RS), whose proof can be adapted from that of Theorem 3 to the good covering

\underset{̲}{T}

Theorem 5.

Under the assumptions of Theorem 2, there exists a formal power series

{\hat{u}}_{2} (t, z, ϵ) = \sum_{n \geq 0} u_{2, n} (z, ϵ) \frac{t^{n}}{n!} \in F_{L_{1}; q; R_{1}} [[t]]

satisfying that there exist

\tilde{C}, \tilde{M} > 0

such that for all

N \geq 0

one has

{∥u_{p, 2} (t, z, ϵ) - \sum_{n = 0}^{N} u_{2, n} (z, ϵ) \frac{t^{n}}{n!}∥}_{L_{1}; q; R_{1}} \leq \tilde{C} {\tilde{M}}^{N + 1} Γ (\frac{N + 1}{k}) {| t |}^{N + 1},

(34)

for all

0 \leq p \leq ς_{2} - 1

and all

t \in T_{p}

. Equivalently, the formal power series

t \mapsto {\hat{u}}_{2} (t, z, ϵ)

is the common Gevrey asymptotic expansion of order

1 / k

of the analytic solution

t \mapsto u_{p, 2} (t, z, ϵ)

, as a function with values in

F_{L_{1}; q; R_{1}}

T_{p}

, for all

0 \leq p \leq ς_{2} - 1

As a final step in the asymptotic study of the solutions to the main problem, we describe the results obtained when considering the classical

L_{1}

-sup norm, in accordance with [11]. The symmetric version of Definition 4 reads as follows.

Definition 6.

let

q > 1

be a real number. We fix a bounded sector with vertex at the origin

E

. Let us consider the set

O_{b} (E) [[z]]

of formal power series g of the form

g (z, ϵ) = \sum_{n \geq 0} g_{n} (ϵ) \frac{z^{n}}{n!},

where

g_{n} \in O_{b} (E)

for every

n \geq 0

Let

R_{1} > 0

be a real number.

We denote by

F_{L_{1}; R_{1}}

the vector space of formal power series

g \in O_{b} (E) [[z]]

of the form

g (z, ϵ) = \sum_{n \geq 0} g_{n} (ϵ) \frac{z^{n}}{n!}

, where

g_{n} \in O_{b} (E)

for every

n \geq 0

, such that the

L_{1}

-sup-norm of g, defined by

{∥g (z, ϵ)∥}_{L_{1}; R_{1}} : = \sum_{n \geq 0} sup_{ϵ \in E} | g_{n} (ϵ) | \frac{R_{1}^{n}}{n!},

is finite.

As a result, one can follow the proof of Theorem 1 [11], assuming uniform bounds with respect to

ϵ \in E

, to arrive at the following result.

Proposition 7.

Under the assumptions of Theorem 2, for every

0 \leq p \leq ς_{2} - 1

and

0 < R_{1} < R

, there exist

{\tilde{A}}_{p}, {\tilde{B}}_{p} > 0

such that

{∥u_{p + 1, 2} (t, z, ϵ) - u_{p, 2} (t, z, ϵ)∥}_{L_{1}; R_{1}} \leq {\tilde{A}}_{p} {({\tilde{B}}_{p})}^{N} Γ (\frac{N}{k}) q^{N^{2} / 2} {| t |}^{N},

for every

t \in T_{p} \cap T_{p + 1}

, with

u_{ς_{2}, 2} : = u_{0, 2}

and

T_{ς_{2}} : = T_{0}

Finally, a direct application of Theorem 8 (q-RS) to Proposition 7 leads us to the last asymptotic result on the behavior of the analytic solutions of the main problem.

Theorem 6.

Under the assumptions of Theorem 2, there exists a formal power series

{\hat{\tilde{u}}}_{2} (t, z, ϵ) = \sum_{n \geq 0} {\tilde{u}}_{2, n} (z, ϵ) \frac{t^{n}}{n!} \in F_{L_{1}; R_{1}} [[t]]

satisfying that there exist

\tilde{C}, \tilde{M} > 0

such that for all

N \geq 0

one has

{∥u_{p, 2} (t, z, ϵ) - \sum_{n = 0}^{N} {\tilde{u}}_{2, n} (z, ϵ) \frac{t^{n}}{n!}∥}_{L_{1}; R_{1}} \leq \tilde{C} {\tilde{M}}^{N + 1} Γ (\frac{N + 1}{k}) q^{\frac{{(N + 1)}^{2}}{2}} {| t |}^{N + 1},

(35)

for all

0 \leq p \leq ς_{2} - 1

and all

t \in T_{p}

5. Annex I: Laplace Transform

In this annex, we briefly describe the definition of Laplace transform and its main properties considered in the present work. We omit its proofs which can be found in previous research by the authors such as [7].

In the whole section,

(E, {∥\cdot∥}_{E})

stands for a complex Banach space.

Definition 7.

Let

k \in N

and choose an unbounded sector

S_{d, δ} : = {z \in C^{★} : | \arg (u) - d | < δ}

, for some

d \in R

and

δ > 0

. Let

f \in O (S_{d, δ}, E)

, continuous in

S_{d, δ} \cup {0}

, such that two constants

C, K > 0

exist with

{∥f (u)∥}_{E} \leq C | u | exp ({K | u |}^{k}), u \in S_{d, δ} .

(36)

The Laplace transform of order k of f along direction d is

L_{k}^{d} (f (u)) (t) = k \int_{L_{γ}} f (u) exp (- {(\frac{u}{t})}^{k}) \frac{d u}{u},

where

L_{γ} = [0, \infty) e^{i γ} \subseteq S_{d, δ} \cup {0}

and the value of

γ \in R

depends on t, with

cos (k (γ - \arg (t))) \geq δ_{1}

for some

δ_{1} > 0

L_{k}^{d} (f (u)) (t)

defines a holomorphic and bounded function in

O (S_{d, θ, R^{1 / k}}, E)

, with

S_{d, θ, R^{1 / k}} = D (0, R^{1 / k}) \cap S_{d, θ / 2}

, where

\frac{π}{k} < θ < \frac{π}{k} + 2 δ

and

0 < R < δ_{1} / K

Proposition 8.

Let

k \in N

and

d \in R

. Let

f \in O (C, E)

such that there exist

C, K > 0

with

{∥f (u)∥}_{E} \leq C | u | exp ({K | u |}^{k}), u \in C .

Then,

L_{k}^{d} (f (u)) : D \to E

defines a holomorphic function on some neighborhood of the origin, D, and it holds that

{(L_{k}^{d} (f (u)))}^{(p)} (0) = Γ (\frac{n}{k}) f^{(p)} (0),

for every

p \in N_{0}

Corollary 1.

In the situation of the previous result, one has that Laplace transform of order k of the polynomial

p (u) = \sum_{h = 0}^{m} a_{h} u^{h} \in E [u]

is the polynomial

L_{k}^{d} (p (u)) (t) = \sum_{h = 0}^{m} Γ (\frac{h}{k}) a_{h} t^{h} \in E [t]

Proposition 9.

Under the hypotheses of Definition 7, the function

u \mapsto k u^{k} f (u)

and, for all

m \in N

, the convolution product

u \mapsto u^{m} ★_{k} f (u)

given by

u^{m} ★_{k} f (u) = \frac{u^{k}}{Γ (\frac{m}{k})} \int_{0}^{u^{k}} {(u^{k} - s)}^{\frac{m}{k} - 1} f (s^{\frac{1}{k}}) \frac{d s}{s},

admit Laplace transform of order k along direction d with

L_{k}^{d} (k u^{k} f (u)) (t) = t^{k + 1} \partial_{t} (L_{k}^{d} (f (u)) (t))

and

L_{k}^{d} (u \mapsto (u^{m} ★_{k} f (u))) (t) = t^{m} L_{k}^{d} (f (u)) (t),

for all

t \in S_{d, θ, R^{1 / k}}

and

0 < R < δ_{1} / K

Proposition 10.

Let

q > 1

and

δ \geq 1

. Under the hypotheses of Definition 7, the function

u \mapsto f (q^{δ} u)

admits Laplace transform of order k along direction d and one has that

L_{k}^{d} (f (q^{δ} u)) (t) = L_{k}^{d} (f (u)) (q^{δ} t),

for every

t \in S_{d, θ, R^{1 / k}}

and

0 < R < δ_{1} / (K q^{k δ})

6. Annex II: Ramis-Sibuya Type Theorems

In this section, we recall different versions of Ramis-Sibuya theorem (RS theorem), whose classical statement can be found in [4], Lemma XI-2-6. Such type of results have been previously successfully applied in the asymptotic study of analytic solutions to many functional problems: a classical RS theorem in the asymptotic study of the solutions to partial differential equations in the complex domain in [2,5] and also a close version to the previous one adapted to multi-index sectors [6], another RS-type theorem adapted to the more general framework of strongly regular sequences in [9], also a q-Gevrey version of RS theorem in the framework of q-difference-differential equations in [7], a mixed (Gevrey and q-Gevrey) version of RS theorem in the study of q-difference-differential problems in [11].

Let

ς \geq 2

be an integer. In the whole section,

(E, {∥\cdot∥}_{E})

stands for a complex Banach space.

Definition 8.

A finite family

\underset{̲}{E} = {(E_{p})}_{0 \leq p \leq ς - 1}

consisting of bounded sectors with vertex at the origin is said to be a good covering in

C^{★}

whenever the following conditions are satisfied:

$E_{j} \cap E_{j + 1} \neq \emptyset$ for every $0 \leq j \leq ς - 1$ (by convention, we define $E_{ς} : = E_{0}$ ).
For every three indices $p_{1}, p_{2}, p_{3} \in {0, \dots, ς - 1}$ such that $p_{i} \neq p_{j}$ for $i, j \in {1, 2, 3}$ with $i \neq j$ , then one has $E_{p_{1}} \cap E_{p_{2}} \cap E_{p_{3}} = \emptyset$ .
There exists a neighborhood of $0 \in C$ , say D, such that $⋃_{p = 0}^{ς - 1} E_{p} = D ∖ {0}$ .

We recall the classical Ramis-Sibuya theorem.

Theorem 7

(RS). Let

{(E_{p})}_{0 \leq p \leq ς - 1}

be a good covering in

C^{★}

. Let

G_{p} : E_{p} \mapsto E

be a holomorphic map for all

0 \leq p \leq ς - 1

such that the following conditions hold:

$G_{p} \in O_{b} (E_{p}, E)$ for $0 \leq p \leq ς - 1$ .
Given $0 \leq p \leq ς - 1$ , the cocycle $Θ_{p} (ϵ) : = G_{p + 1} (ϵ) - G_{p} (ϵ)$ is exponentially flat of order k on $Z_{p} : = E_{p} \cap E_{p + 1}$ (we write $G_{ς} : = G_{0}$ ), i.e. there exist $A_{p}, B_{p} > 0$ such that

${∥Θ (ϵ)∥}_{E} \leq A_{p} exp (- \frac{B_{p}}{{| ϵ |}^{k}}), ϵ \in E_{p} \cap E_{p + 1} .$

Then, there exists a formal power series, common for every

0 \leq p \leq ς - 1

\hat{G} (ϵ) = \sum_{n \geq 0} G_{n} ϵ^{n} \in E [[ϵ]]

, such that

G_{p}

admits

\hat{G}

as its Gevrey asymptotic expansion of order

1 / k

E_{p}

, for all

0 \leq p \leq ς - 1

, meaning that there exist

C, M > 0

with

{∥G_{p} (ϵ) - \sum_{n = 0}^{N} G_{n} ϵ^{n}∥}_{E} \leq C M^{N + 1} Γ (\frac{N + 1}{k}) {| ϵ |}^{N + 1},

for all

N \geq 0

and

ϵ \in E_{p}

A mixed Gevrey and

q -

Gevrey version of Ramis-Sibuya theorem is also available.

Theorem 8

(

q -

RS). Let

{(E_{p})}_{0 \leq p \leq ς - 1}

be a good covering in

C^{★}

. Let

G_{p} : E_{p} \mapsto E

be a holomorphic map for all

0 \leq p \leq ς - 1

such that the following conditions hold:

$G_{p} \in O_{b} (E_{p}, E)$ for $0 \leq p \leq ς - 1$ .
Given $0 \leq p \leq ς - 1$ , the cocycle $Θ_{p} (ϵ) : = G_{p + 1} (ϵ) - G_{p} (ϵ)$ satisfies that there exist $A_{p}, B_{p} > 0$ with

${∥Θ (ϵ)∥}_{E} \leq A_{p} {(B_{p})}^{N} Γ (\frac{N}{k}) q^{\frac{N^{2}}{2}} {| ϵ |}^{N}, ϵ \in E_{p} \cap E_{p + 1},$

valid for all $N \geq 1$ .

Then, there exists a formal power series, common for every

0 \leq p \leq ς - 1

\hat{G} (ϵ) = \sum_{n \geq 0} G_{n} ϵ^{n} \in E [[ϵ]]

, such that

G_{p}

admits

\hat{G}

as its Gevrey asymptotic expansion of mixed order

(1 / k; (q, 1))

E_{p}

, for all

0 \leq p \leq ς - 1

, meaning that there exist

C, M > 0

with

{∥G_{p} (ϵ) - \sum_{n = 0}^{N} G_{n} ϵ^{n}∥}_{E} \leq C M^{N + 1} Γ (\frac{N + 1}{k}) q^{\frac{{(N + 1)}^{2}}{2}} {| ϵ |}^{N + 1},

for all

N \geq 0

and

ϵ \in E_{p}

Acknowledgments

Both authors are partially supported by the project PID2019-105621GB-I00 of Ministerio de Ciencia e Innovación, Spain and by Dirección General de Investigación e Innovación. The first author is partially supported by Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of Comunidad de Madrid, Universidad de Alcalá under grant CM/JIN/2021-014, and by Ministerio de Ciencia e Innovación-Agencia Estatal de Investigación MCIN/AEI/10.13039/501100011033 and the European Union “NextGenerationEU”/ PRTR, under grant TED2021-129813A-I00. This work was completed during the authors’ stay in Tokyo participating in workshops at Josai University, Shibaura Institute of Technology and Chiba University. The authors thank professor Haraoka for his kind invitation.

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Gevrey versus q−Gevrey Asymptotic Expansions for Some Linear q−Difference-Differential Cauchy Problem

Abstract

1. Introduction

2. Statement of the Main Problem and Analytic Solution

2.1. Statement of the Main Problem

2.2. Construction of Analytic Solutions to the Main Problem

3. Asymptotic Behavior of the Analytic Solutions, I

4. Asymptotic Behavior of the Analytic Solutions, II

5. Annex I: Laplace Transform

6. Annex II: Ramis-Sibuya Type Theorems

Acknowledgments

References

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