Abstract and Applied Analysis

Volume 2012 (2012), Article ID 930385, 86 pages

doi:10.1155/2012/930385

Research Article

On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

Stéphane Malek

UFR de Mathématiques, Université de Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received 27 March 2012; Accepted 8 July 2012

Academic Editor: Roman Dwilewicz

Copyright © 2012 Stéphane Malek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study a family of singularly perturbed linear partial differential equations with irregular type in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter converges near the origin in and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say , for some integer . The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued in the larger sectors , where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.

1. Introduction

We consider a family of singularly perturbed linear partial differential equations of the form

for given initial conditions

where is a complex perturbation parameter, is some positive integer, is some positive integer larger than 2, and is a finite subset of with the property that there exists an integer with

for all , and the coefficients belong to where denotes the space of holomorphic functions in near the origin in . In this work, we make the assumption that the coefficients of (1.1) factorize in the form where belong to . The initial data are assumed to be holomorphic functions on a product of two sectors , where is a fixed bounded sector centered at 0 and , , are sectors with opening larger than centered at the origin whose union form a covering of , where is some neighborhood of 0. For all , this family belongs to a class of partial differential equations which have a so-called irregular singularity at (in the sense of [1]).

In the previous work [2], we have given sufficient conditions on the initial data , for the existence of a formal series

solution of (1.1), with holomorphic coefficients on for some disc , with , such that, for all , the solution of the problem (1.1), (1.2) defines a holomorphic function on which is the 1-sum of on . In other words, for all fixed , the Borel transform of with respect to defined as is holomorphic on some disc and can be analytically continued (with exponential growth) to sectors , centered at 0, with infinite radius and with the bisecting direction of the sector . But in general, due to the fact that the functions do not coincide on the intersections (known as the Stokes phenomenon), the Borel transform cannot be analytically extended to the whole sectors for all , where by convention , , and .

In this work, we address the question of the possibility of analytic continuation, location of singularities, and behaviour near these singularities of the Borel transform within the sector . More precisely, our goal is to give stronger conditions on the initial data under which the Borel transform can be analytically continued to the full-punctured sector except a half lattice of points , , depending on and some well-chosen complex number and moreover develop logarithmic singularities at (Theorem 5.8).

In a recent paper of Fruchard and Schäfke, see [3], an analogous study has been performed for formal WKB solutions to the singularly perturbed Schrödinger equation

where is a formal series with holomorphic coefficients on some domain avoiding and . The authors show that the Borel transform of with respect to converges near the origin and can be analytically continued along any path avoiding some lattices of points depending on . We also mention that formal parametric Stokes phenomenon for 1-dimensional stationary linear Schrödinger equation , where is a polynomial, has been investigated by several other authors using WKB analysis, see [4–6]. In a more general framework, analytic continuation properties related with the Stokes phenomenon have been studied by several authors in different contexts. For nonlinear systems of ODEs with irregular singularity at of the form and for nonlinear systems of difference equations , under nonresonance conditions, we refer to [7, 8]. For linearizations procedures for holomorphic germs of in the resonant case, we make mention to [9, 10]. For analytic conjugation of vector fields in to normal forms, we indicate [11, 12]. For Hamiltonian nonlinear first-order partial differential equations, we notice [13].

In the proof of our main result, we will use a criterion for the analytic continuation of the Borel transform described by Fruchard and Schäfke in [3] (Theorem (FS) in Theorem 5.8). Following this criterion, in order to prove the analytic continuation of the Borel transform , say, on the sector , for any fixed , we need to have a complete description of the Stokes relation between the solutions and of the form

for all , for some integer , where is a set of aligned complex numbers such that with (for some ) and , , are the -sums of some formal series on . If the relation (1.6) holds, then can be analytically continued along any path in the punctured sector and has logarithmic growth as tends to in a sector. Actually, under suitable conditions on the initial data , we have shown that such a relation holds for , for some well-chosen and for all , see (5.145) in Theorem 5.8. In order to establish such a Stokes relation (1.6), we proceed in several steps.

In the first step, following the same strategy as in [2], using the linear map , we transform the problem (1.1) into an auxiliary regularly perturbed singular linear partial differential equation which has an irregular singularity at and whose coefficients have poles with respect to at the origin, see (4.9). Then, for , we construct a formal transseries expansion of the form

solution of the problem (4.9), (4.10), where each is a formal series in with coefficients , which are holomorphic on a punctured polydisc . We show that the Borel transform of each with respect to , defined by , satisfies an integrodifferential Cauchy problem with rational coefficients in , holomorphic with respect to near the origin and meromorphic in with a pole at zero, see (4.20), (4.21). For well-chosen and suitable initial data, we show that each defines a holomorphic function near the origin with respect to and on a punctured disc with respect to and can be analytically continued to functions defined on the products , where , are suitable open sectors with small opening and infinite radius. Moreover, the functions have exponential growth rate with respect to , namely, there exist such that

for all in their domain of definition and all (Proposition 4.12). In order to get these estimates, we use the Banach spaces depending on two parameters and with norms of functions bounded by for some bounded sequence already introduced in [2]. If one expands the functions with respect to , we show that the generating function can be majorized by a series which satisfies a Cauchy problem of Kowalevski type (4.47), (4.48) and is therefore convergent near the origin in .

We construct a sequence of actual functions , , as Laplace transform of the functions with respect to along a halfline . We show that the functions are holomorphic functions on the domain and that the functions are exponentially flat as tends to 0 on as -valued functions. In the proof, we use, as in [2], a deformation of the integration's path in and the estimates (1.8). Using the Ramis-Sibuya theorem (Theorem (RS) in Proposition 4.15), we deduce that each is the -sum of a formal series on , for (Proposition 4.15). We notice that the functions actually coincide with the functions mentioned above solving the problem (1.1), (1.2). We deduce that, for a suitable choice of , the function

solves (1.1) on the domain .

In the second part of the proof, we establish the connection formula which is exactly the Stokes relation (1.6) on (Proposition 5.2). The strategy we follow consists in expressing both functions and as Laplace transforms of objects that are no longer functions in general but distributions supported on which are called staircase distributions in the terminology of [8]. We stress the fact that such representations of transseries expansions as generalized Laplace transforms were introduced for the first time by Costin in the paper [8]. Notice that similar arguments have been used in the work [14] to study the Stokes phenomenon for sectorial holomorphic solutions to linear integro-differential equations with irregular singularity.

In Lemma 5.5, we show that can be written as a generalized Laplace transform in the direction of a staircase distribution , which is a convergent series in on with coefficients in some Banach spaces of staircase distributions on depending on the parameters and (see Definition 2.3). We observe that the distribution solves moreover an integro-differential Cauchy problem with rational coefficients in , holomorphic with respect to near the origin and meromorphic with respect to at zero, see (5.80), (5.81). The idea of proof consists in showing that each function can be expressed as a Laplace transform in a sequence of directions tending to of a sequence of staircase distributions (which are actually convergent series in with coefficients that are functions in on with exponential growth). Moreover, each distribution solves an integro-differential Cauchy problem (5.37), (5.38), whose coefficients tend to the coefficients of an integro-differential equation (5.39), (5.40), as tends to , having a unique staircase distribution solution . Under the hypothesis that the initial data (5.38) converge to (5.40) as , we show that the sequence converges to in the Banach space with precise norm estimates with respect to and (Lemma 5.3). In order to show this convergence, we use a majorazing series method together with a version of the classical Cauchy-Kowalevski theorem (Proposition 2.22) in some spaces of analytic functions near the origin in with dependence on initial conditions and coefficients applied to the auxiliary problem (5.66), (5.68). Using a continuity property of the Laplace transform (3.5), we show that each function can be actually expressed as the Laplace transform of in the direction and finally that itself is the Laplace transform of some staircase distribution solving (5.80), (5.81).

On the other hand, in Lemma 5.7, under suitable conditions on , , we can also write as a generalized Laplace transform in the direction of the staircase distribution mentioned above solving (5.80), (5.81). Therefore, the equality holds on . The method of proof consists again in showing that can be written as Laplace transform in a sequence of directions tending to of a sequence of staircase distributions (which are actually convergent series in with coefficients that are functions in on with exponential growth). Moreover, each distribution solves an integro-differential Cauchy problem (5.98), (5.99), whose coefficients tend to the coefficients of the integro-differential equation (5.80). Under the assumption that the initial data (5.99) converge to the initial data (5.81), we show that the sequence converges to the solution of (5.80), (5.81) (i.e., ) in the Banach space , as , see Lemma 5.6. This convergence result is obtained again by using a majorazing series technique which reduces the problem to the study of some linear differential equation (5.106), (5.109), whose coefficients and initial data tend to zero as . Finally, by continuity of the Laplace transform, can be written as the Laplace transform of in direction .

After Theorem 5.8, we give an application to the construction of solutions to some specific singular linear partial differential equations in having logarithmic singularities at the points , for . We show that under the hypothesis that the coefficients are polynomials in , the Borel transform turns out to solve the linear partial differential equation (5.149). We would like to mention that there exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in [15]. Several authors have considered Cauchy problems , where is a differential operator of some order , for initial data , . Under specific hypotheses on the symbol , precise descriptions of the solutions of these problems are given near the singular locus of the initial data . For meromorphic initial data, we may refer to [16–18] and for more general ramified multivalued initial data, we may cite [19–23].

The layout of this work is as follows.

In Section 2, we introduce Banach spaces of formal series whose coefficients belong to spaces of staircase distributions and we study continuity properties for the actions of multiplication by functions and integro-differential operators on these spaces. In this section, we also exhibit a Cauchy Kowalevski theorem for linear partial differential problems in some space of analytic functions near the origin in with dependence of their solutions on the coefficients and initial data which will be useful to show the connection formula (5.28) stated in Section 5.

In Section 3, we recall the definition of a Laplace transform of a staircase distribution as introduced in [8] and we give useful commutation formulas with respect to multiplication by polynomials, exponential functions, and derivation.

In Section 4, we construct formal and analytic transseries solutions to the singularly perturbed partial differential equation with irregular singularity (1.1).

In Section 5, we establish the crucial connection formula relying on the analytic transseries solution and the solution of (1.1). Finally, we state the main result of the paper which asserts that the Borel transform in the perturbation parameter of the formal solution of (1.1) can be analytically continued along any path in the punctured sector and has logarithmic growth as tends to in a sector for all .

2. Banach Spaces of Formal Series with Coefficients in Spaces of Staircase Distributions: A Cauchy Problem in Spaces of Analytic Functions

2.1. Weighted Banach Spaces of Distributions

We define to be the space of complex valued -functions with compact support in , where is the set of the positive real numbers . We also denote by the space of distributions on . For , we write the -derivative of in the sense of distribution, for , with the convention .

Definition 2.1. A distribution is called staircase if can be written in the form

for unique integrable functions such that the support of is in for all .

Remark 2.2. Given a compact set , a general distribution can always be written as a -derivative of a continuous function on restricted to the test functions with support in , where depends on , see [24].

Definition 2.3. Let be a real number, an integer and let for all integers . Let be an open sector centered at 0 and let . We denote by the vector space of all locally integrable functions such that

is finite. We denote by the vector space of staircase distributions such that

is finite.

Remark 2.4. Let such that . If , then for all and we have that is a decreasing sequence on . Likewise, if , then for all and we have that is a decreasing sequence on .

Let be the Heaviside one-step function defined by , if and , if . Let the operator defined on distributions by . For a subset , we denote by the function which is equal to 1 on and 0 elsewhere.

The proofs of the following Lemmas 2.5 and 2.6, Propositions 2.7, 2.8, 2.9, and Corollary 2.10 are given in the appendix of [25], see also [8].

Lemma 2.5. Let and , where and . Then is a staircase distribution and the decomposition of has the following terms , and for , where and .

Lemma 2.6. Let be as in Lemma 2.5 and such that . Then, one has

if and for ,

Proposition 2.7. Let and such that . Then belongs to and the decomposition (2.1) of has the following terms with and , for . Moreover, there exists a universal constant such that .

Proposition 2.8. The set of -functions with compact support in is dense in for all , and .

Proposition 2.9. Let such that . For all , we have . Moreover, there exists a universal constant such that

for all .

In this paper, for all integers , we will denote the convolution for all where stands for the convolution product of with itself times for and with the convention that . From Propositions 2.7 and 2.9, we deduce the following.

Corollary 2.10. Let be such that and let be an integer. For all , one has . Moreover, there exists a universal constant such that

for all .

In the next proposition, we study norm estimates for the multiplication by bounded analytic functions.

Proposition 2.11. Let and such that

and let be a -function on such that there exist constants and such that

for all . Then, for all , we have . Moreover, there exists a constant (depending on ) such that

for all .

Proof. The proof can be found in [14] and is inspired from [25, Lemma ], but for the sake of completeness, we sketch it below. Without loss of generality, we can assume that has the following form , where with , for . Put . Then, .

From the Leibniz formula, we get the identity

On the other hand, one can rewrite , where and denotes the th iteration of .

Due to Lemma 2.5, can be written , with and .

Therefore, we get the following identity

First of all, we have

where is given in (2.9). From Lemma 2.6, we have the estimates

for and all . Now, we give estimates for .

Using the Taylor formula with integral remainder and the hypothesis (2.9), we get

Hence, from the Fubini theorem and the identity

we deduce

and hence

From (2.14) and (2.18), we obtain

for , all , where

Now, we need to estimate . Due to the Stirling formula, as tends to infinity, there exists a universal constant such that

for all . Using the hypothesis , we have

Using again the Stirling formula, we get a constant (depending on ) such that

for all . Moreover,

Hence,

for all . Finally, we obtain a constant depending only on such that

for all . From (2.14) and (2.18), we have

where

Now, we show that , , is a bounded sequence. Again, by the Stirling formula, we get a universal constant such that

From the assumption (2.8) and the estimates that for all two real numbers, we have

we get a constant such that

for all .

Finally, from the equality (2.12) and estimates (2.13), (2.19), (2.26), (2.27) and (2.31), we get a constant depending only on such that for all . It remains to consider the case .

When , let , with . By definition, we can write

In the next proposition, we study norm estimates for the multiplication by polynomials.

Proposition 2.12. Let and such that

and let be integers. Then, for all , one has . Moreover, there exists a constant (depending on ,) such that

for all .

Proof. The proof is an adaptation of Proposition 2.11. Without loss of generality, we can assume that has the following form where with , for . We also put . Let . Then, . From the Leibniz formula, we get the identity

On the other hand, one can rewrite , where and denotes the th iteration of .

Due to Lemma 2.5, can be written , with , and . Therefore, we get the following identity as:

(1) We first give estimates for . We write

where

Now, we gives estimates for . We write

where for all . From the Taylor formula applied to on , we get that

for all . Now, we recall that for all two real numbers, we have

From (2.39), (2.40) and (2.41), we deduce that

for all . From (2.37) and (2.42), we deduce that

(2) We give estimates for for all and all . From Lemma 2.6, we have the estimates

for and all . Now, we give estimates for . Using the Taylor formula with integral remainder, we have that

and from the classical identity

we get from the Fubini theorem that

Again, we write

From the expression of , we have that

for all , if , and , if . Using (2.49) in the right-hand side of the equality (2.48), we deduce from (2.47) that

if , and if .

(3) We give estimates for , for . From the estimates (2.44) and (2.50), we get that

From (2.37) and (2.42), we deduce from (2.51) that

where

for all , and . Now, we show that , , is a bounded sequence. We have

for all . From the Stirling formula which asserts that as , we get a universal constant and a constant (depending on , ) such that

for all . From (2.54), (2.55), we get a constant (depending on ) such that

for all .

(4) We give estimates for . From the estimates (2.44) and (2.50), we get that

Again from (2.37) and (2.42), we deduce from (2.57) that

where and

for all . Now, we remind from (2.31) that is a bounded sequence.

Finally, from (2.31), (2.36), (2.43), (2.52), (2.56), and (2.58), we deduce a constant (depending on , ) such that

which gives the result. It remains to consider the case .

When , let , with . By definition, we can write

Using (2.41), we deduce from (2.61) that

Hence there exists a constant (depending on , ) such that

which yields the result.

Proposition 2.13. Let be real numbers such that

Let be a nonnegative integer. Then, for all , one has . Moreover, there exists a constant (depending on , , ) such that

for all .

Proof. The line of reasoning will follow the proof of Proposition 2.12. We start from the identity (2.36).

We first give estimates for . We write

where

Now, we give estimates for . We write

where , for all . From the Taylor formula applied to on , we get that

From (2.68), (2.69), and (2.41), we deduce that

From (2.66) and (2.70), we get that

We give estimates for , for all , all . We start from the formula (2.44) and (2.47). We write

We get that

if , and if .

We give estimates for , for . From the estimates (2.44) and (2.73), we get that

From (2.66) and (2.70), we deduce from (2.74) that

where is the bounded sequence given in the proof of Proposition 2.12.

We give estimates for . From the estimates (2.44) and (2.73), we get that

From (2.66) and (2.70), we deduce from (2.76), that

where is the bounded sequence given in the proof of Proposition 2.12.

Finally, from (2.31), (2.36), (2.56), (2.71), (2.75), and (2.77), we deduce a constant (depending on , , ) such that

which gives the result. It remains to consider the case .

When , let , with . By definition, we can write

Using (2.41), we deduce from (2.79) that

Hence, there exists a constant (depending on , , ) such that

which yields the result.

2.2. Banach Spaces of Formal Power Series with Coefficients in Spaces of Distributions

Definition 2.14. Let be a real number. We denote by the vector space of formal series such that for all and

is finite. One can check that the normed space is a Banach space.

In the next proposition, we study some parameter depending linear operators acting on the space .

Proposition 2.15. Let be positive integers. Assume that the condition

holds. Then, if

the operator is a bounded linear operator from the space into itself. Moreover, there exists a constant (depending on , , , ) such that

for all .

Proof. Let . By definition, we have

From Corollary 2.10 and Proposition 2.12, we get a constant (depending on ,) such that

From the assumptions (2.83), we get a constant (depending on , , ) such that

for all . Finally, from the estimates (2.87) and (2.88), we get the inequality (2.85).

In the next proposition, we study linear operators of multiplication by bounded holomorphic and functions.

Proposition 2.16. For all , let be a function with respect to on such that there exist with

for all . One consider the series

which is convergent for all , all . Let . Then, if

the linear operator of multiplication by is continuous from into itself. Moreover, there exists a constant (depending on , , ) such that

for all satisfying (2.91).

Proof. Let . By definition, we have that

From Proposition 2.11 and Remark 2.4, we deduce that there exists (depending on ) such that

for all such that . From (2.93) and (2.94), we deduce that

which yields (2.92).

2.3. Cauchy Problems in Analytic Functions Spaces with Dependence on Initial Data

In this section, we recall the well-know-Cauchy Kowalevski theorem in some spaces of analytic functions for which the dependence on the coefficients and initial data can be obtained.

The following Banach spaces were used in [26].

Definition 2.17. Let be real numbers such that . We define a vector space of holomorphic functions on a neighborhood of the origin in . A formal series ,

belongs to if the series

converges. We also define a norm on as

One can easily show that is a Banach space.

Remark 2.18. Let be in for given . Then, also belongs to the spaces for all and . Moreover, the maps and are increasing functions from (resp., ) into .

We depart from some preliminary lemma from [26]. In the following, for , we denote by the formal series .

Lemma 2.19. Let such that . The operator is a bounded linear operator from into itself. Moreover, there exists a universal constant such that the estimates

hold for all .

Lemma 2.20. Let be an analytic function on an open polydisc containing and let be in . Then, the product belongs to . Moreover,

where .

Proof. Let

We have

where

for all . By definition, we have

On the other side, the next inequalities are well known:

for all such that and such that .

Finally, from (2.105), we deduce that converges and that the estimates (2.100) hold.

Lemma 2.21. Let and let be in for given . Then, there exist small enough (depending on ) such that the formal series belongs to . Moreover, there exists a constant (depending on ) such that

for all .

Let be a finite subset of . For all , let be analytic functions on some polydisc containing the closed polydisc for some . As in Lemma 2.20, we define

which converges on . We also consider , for some . The following proposition holds.

Proposition 2.22. Let be an integer. One make the following assumptions. For all , one has

One consider the following Cauchy problem:

for the given initial conditions

which are analytic functions on some containing the closed . If , we define , which converges for all .

Then, there exist with (depending on ,,) and with (depending on , , , ) such that the problem (2.109), (2.110) has a unique formal solution . Moreover, there exist constants (depending on ,,,) such that

Proof. We denote by the linear operator from into itself defined by

and denotes the linear map from into itself:

for all . By construction, we have that , where represents the identity map from into itself.

Now, we show that for any given such that , there exists with (depending on , , , ) such that is an invertible map from into itself for all . Moreover, the following inequality

holds for all , for any . Indeed, from the assumption (2.108) and Lemmas 2.19 and 2.20, we get a universal constant such that

for all . Since , for all , for the given one can choose small enough such that . Therefore, the inequality (2.114) holds.

Let . From the hypothesis (2.110), we deduce that and belong to , for some (depending on , ). Indeed, from Lemmas 2.20 and 2.21 we get constants , (depending on , ) such that

Now, for this constructed satisfying (2.116) that we choose in such a way that also holds, we select such that . From the estimates (2.116) and Remark 2.4, we deduce that , , and belong to . From (2.114), we deduce the existence of a unique such that

Now, we put . By Lemma 2.19, we deduce that and solves the problem (2.109), (2.110). Moreover, from (2.114) and (2.116), we get constants (depending on , , , ) such that (2.111) holds, which yields the result.

3. Laplace Transform on the Spaces

We first introduce the definition of Laplace transform of a staircase distribution.

Proposition 3.1. (1) Let be an integer, a real number, and . Let

and choose . Then, there exist , such that the function

is holomorphic on the sector for all . Moreover, for all compacts , there exists (depending on and ) such that

for all .

(2) Let and let . We define the Laplace transform of in direction to be the function

which defines a holomorphic function on , for some , , for all . Moreover, for all compacts , there exists (depending on and ) such that

for all .

Proof. We prove part (1). The second part (2) is a direct application of (1). We have

We choose and such that for all . Moreover, we choose and such that

for all . Let an integer, for , we get that

We deduce that for ,

and for ,

From the estimates (3.9) and (3.10) we get the inequality (3.3).

In the next proposition, we show that if is a function, then the Laplace transform of introduced in Proposition 3.1 coincides with the classical one.

Proposition 3.2. Let . Then, from Proposition 2.7, one knows that . The Laplace transform coincides with the classical Laplace transform of in the direction defined by

for all .

Proof. From Proposition 2.7, the staircase decomposition of has the following form , with and for all . We have to compute the integrals

for all . For , we have that

For , by one integration by parts, we get that

and using successive integrations by parts, we get that

for all . On the other hand, from the hypothesis that and from the fact that for all , we have that the next telescopic sum

is convergent and equal to zero for all . Finally, we deduce that .

In the next proposition, we describe the action of multiplication by a polynomial and derivation on the Laplace transform.

Proposition 3.3. Let . Then, the following relations

hold for all . Let be two integers such that . Then, there exist a finite subset such that for all , and integers , for (depending on , ) such that

for all .

Proof. First of all, we have to check that the relations (3.17) and (3.18) hold when . Since is dense in , from the inequality (3.3) and with the help of Corollary 2.10 and Proposition 2.12, we will get that (3.17) and (3.18) hold for all . Now, let . The first relation of (3.17) is obtained by integrating once by parts and the second formula of (3.17) is a consequence of the equality

for all . To get the formula (3.18), we first show the following relation:

for all . Indeed, using one integration by parts, we get that

By a second integration by parts on the right-hand side of (3.21) and by comparison with (3.19), we get (3.20). Now, let be such that . Applying the first relation of (3.17) and (3.20), we get that

Now, we recall a variant of Lemmas 5 and 6 in [2].

Lemma  3.4. For all , there exist constants , such that

for all functions .

Lemma  3.5. Let be positive integers such that and . We put . Then, for all function , the function can be written in the form

where is a finite subset of such that for all , , , , and .

Finally, we observe that the relation (3.18) follows from (3.22) and Lemmas 3.4 and 3.5.

The next proposition can be found in [25, Appendix  A], see also [8].

Proposition 3.6. Let and with . Then, for every , the expression belongs to . Moreover, there exist a universal constant and (depending on , , ) such that

with when .

In the forthcoming proposition, we explain the action of multiplication by an exponential function on the Laplace transform.

Proposition 3.7. Let and with . From the latter proposition, one knows that belongs to . The following formula

holds for all .

Proof. Since is dense in , it is sufficient to prove that

for all , all . Then, we get the inequality (3.26) by using (3.3) and Proposition 3.6. Now, let . We write

where is an integer chosen such that . From our assumption, we have that belongs to and that . By Lemma 2.5, we deduce that is a staircase distribution where the functions are constructed as follows:

and for all , we have where

By definition, we have

Now, we will compute the integrals for all . By construction, we have that for all . For , we get

For , by one integration by parts, we get that

For , with , by successive integrations by parts, we get that

Since , for all , we deduce that the telescopic sum

is equal to 0. From the formula (3.32), (3.33), (3.34), and (3.35), we get that

From Proposition 3.3, we have that

Finally, from (3.36) and (3.37), we get the equality (3.27).

4. Formal and Analytic Transseries Solutions for a Singularly Perturbed Cauchy Problem

4.1. Laplace Transform and Asymptotic Expansions

We recall the definition of Borel summability of formal series with coefficients in a Banach space, see [27].

Definition 4.1. A formal series

with coefficients in a Banach space is said to be -summable with respect to in the direction if

(i)there exists such that the following formal series, called formal Borel transform of of order 1,

is absolutely convergent for ; (ii)there exists such that the series can be analytically continued with respect to in a sector . Moreover, there exist and such that

for all . We say that has exponential growth of order 1 on .

If this is so, the vector valued Laplace transform of order of in the direction is defined by

along a half-line , where depends on and is chosen in such a way that , for some fixed , for all in a sector

where and . The function is called the -sum of the formal series in the direction . The function is a holomorphic and a bounded function on the sector . Moreover, the function has the formal series as Gevrey asymptotic expansion of order 1 with respect to on . This means that for all , there exist such that

for all , all .

In the next proposition, we recall some well-known identities for the Borel transform that will be useful in the sequel.

Proposition 4.2. Let and be formal series in . One has the following equalities as formal series in :

4.2. Formal Transseries Solutions for an Auxiliary Singular Cauchy Problem

Let be an integer. Let be a finite subset of and let

be holomorphic and bounded functions on a polydisc , for some , with , for all . We consider the following singular Cauchy problems:

for given formal transseries initial conditions

where for all and .

Proposition 4.3. The problem (4.9), (4.10) has a formal transseries solutions

where the formal series , for all , all , satisfy the following singular Cauchy problems:

with initial conditions

for some real numbers , for and .

Proof. We have that

and from the Leibniz rule we also have

On the other hand, by the Faa Di Bruno formula we have, for all , that

where and , for all .

Using the expressions (4.14), (4.15), (4.16), by plugging the formal expansion into the problem (4.9), (4.10) and by identification of the coefficients of we get that satisfies the problem (4.12), (4.13).

4.3. Formal Solutions to a Sequence of Regular Cauchy Problems

Proposition 4.4. One makes the assumption that

for all . Then, the problem (4.12), (4.13) has a unique formal solution for all . Let

where , be the formal solution of (4.12), (4.13) for all . One denotes by

the formal Borel transform of with respect to . Then, for all , satisfies the problem

with initial data

where is a finite subset of such that implies and is a finite subset of such that implies , and , are integers.

Proof. The proof follows by direct computation on the problems (4.12) and (4.13), using Proposition 4.2 and the following two lemmas from [2].

Lemma  4.5.  For all , there exist constants , , such that

for all holomorphic functions on an open set .

Lemma  4.6.  Let be positive integers such that and . We put . Then, for all holomorphic functions , the function can be written in the form

where is a finite subset of such that for all , , , , and .

4.4. An Auxiliary Cauchy Problem

We denote by an open star-shaped domain in (meaning that is an open subset of such that for all , the segment belongs to ). Let be an open set in contained in the disc . We denote by . For any open set , we denote by the vector space of holomorphic functions on .

Definition 4.7. Let a real number and let for all integers . Let and be a real number. We denote by the vector space of all functions such that

is finite.

Proposition 4.8. One makes the assumption that

for all . Moreover, one makes the assumption that there exists such that

For all , all , the problem (4.20) with initial conditions

has a unique formal series

where satisfies the following recursion:

for all , all .

Proposition 4.9. One makes the assumption that

for all . Let also the assumption (4.26) holds. Let us assume that

Then, one has that for all , all , all . We put , for all , all , and all . Then, the following inequalities hold: there exist two constants (depending on ,,) such that

for all , all .

Proof. The proof follows by direct computation using the recursion (4.29) and the next lemma. We keep the notations of Proposition 4.8.

Lemma  4.10. There exists a constant (depending on , , , , ) such that

for all , and all , , all with .

Proof. We follow the proof of Lemma 1 from [2]. By definition, we have that for all . Using the parametrization with , we get that

where . More generally, for all , we have by definition:

for all . Using the parametrization , , with , for , we can write

where is a monomial in whose coefficient is equal to 1. Using these latter expressions, we now write

Therefore,

By construction of , we have

for all . From (4.38) and (4.39), we get that

for all . From (2.41), we deduce that

for all . From the estimates (4.40) and (4.41), we deduce the inequality (4.33).

Proposition 4.11. Assume that the conditions (4.26) and (4.31) hold. Assume moreover, that

for all and that the following sums converge near the origin in ,

One make also the hypothesis that for all , one can write

where is holomorphic for all on . Then, the problem (4.20) with initial data

has a unique solution which is holomorphic with respect to for all .

The constant is such that and depends on , (which denotes a common radius of absolute convergence of the series (4.43)), , , , , , , where and , are defined below.

Moreover, the following estimates hold: there exists a constant such that (depending on , , and , ) and a constant (depending on (where are defined above), , , , , , , , ) such that

for all , all , and all .

Proof. We consider the following Cauchy problem

for given initial data

where

are convergent series near the origin in with respect to . From the assumption (4.42) and the fact that , we also deduce that

for all and all . Since the initial data (4.48) and the coefficients (4.47) are analytic near the origin, we get that all the hypotheses of the classical Cauchy Kowalevski theorem from Proposition 2.22 are fulfilled. We deduce the existence of with , where denotes a common radius of absolute convergence for the series (4.48), which depends on , and , and with (depending on , , , , , , , , where ) such that there exist a unique formal series which solves the problem (4.47), (4.48).

Now, let be its Taylor expansion at . Then, by construction the sequence satisfies the following equalities:

for all and all , with

Using the inequality (4.32) and the equality (4.51), with the initial conditions (4.52), one gets that

for all , all . Using the fact that and the estimates (2.111), we deduce from (4.53) that there exist a constant (depending on ,, , , , , , , , ) such that

for all , all , all , and all .

4.5. Analytic Solutions for a Sequence of Singular Cauchy Problems

Assume that the conditions (4.42) and (4.44) hold. We consider the following problem:

with initial conditions

The initial conditions , are defined as follows. Let be an open sector centered at 0, with infinite radius and bisecting direction , an open disc centered at with radius , and an open sector centered at 0 contained in the disc . We make the assumption that the condition (4.26) holds for the set . We consider a set of functions for all such that

We also assume that for all and all , has an analytic continuation denoted by for all such that

Let

be the convergent Taylor expansion of with respect to on for all . We consider the formal series

for all . We define as the -sum (in the sense of Definition 4.1) of in the direction . From the hypotheses, we deduce that defines a holomorphic function for all , for all , where

for some and some constant (independent of ) for all .

Proposition 4.12. Assume that the conditions (4.26), (4.31), (4.42), and (4.44) hold.

Then, the problem (4.55), (4.56) has a solution which is holomorphic and bounded on the set , for some (independent of ), for all , where depends on , (which denotes a common radius of absolute convergence of the series (4.57), (4.58)), , , , , , , where .

The function can be written as the Laplace transform of order 1 in the direction (in the sense of Definition 4.1) of a function , which is holomorphic on the domain and satisfies the following estimates.

There exists a constant such that (depending on , and ,) and a constant (depending on (where are defined above), , , , , , , , ) such that

for all , all .

Moreover, the function is the analytic continuation of a function which is holomorphic on the punctured polydisc and verifies the following estimates.

There exists a constant (depending on (where are defined above), , , , , , , , ) such that

for all , all , all , and all .

Proof. From the hypotheses of Proposition 4.12, we deduce from Proposition 4.11 applied to the situation the existence of a holomorphic function satisfying the estimates (4.63), which is the solution of the problem (4.20) with initial conditions , , on the domain . Likewise, from Proposition 4.11 applied to the situation , we get the existence of a holomorphic function satisfying (4.62), which is the solution of the problem (4.20) with initial conditions , on the domain .

With Proposition 4.3, we deduce that the formal solution of the problem (4.12), (4.13) is -summable with respect to in the direction as series in the Banach space , for all . We denote by its -sum which is holomorphic with respect to on a domain due to Definition 4.1 and the estimates (4.62). Moreover, from the algebraic properties of the -summability procedure, see [27, Section 6.3], we deduce that is a solution of the problem (4.55), (4.56).

4.6. Summability in a Complex Parameter

We recall the definition of a good covering.

Definition 4.13. Let be an integer. For all , we consider open sectors centered at , with radius , bisecting direction and opening , with , such that for all (with the convention that and such that , where is some neighborhood of 0 in . Such a set of sectors is called a good covering in .

Definition 4.14. Let be a good covering in . Let be an open sector centered at 0 with radius and consider a family of open sectors

where , for , where , which satisfy the following properties:

(1) For all , all , .

(2) For all , for all , and all , we have that .

(3) (3.1) We assume that . We consider the two closed sectors

We make the assumption that there exist two constants with

for all and all .

(3.2) There exists such that for all and all .

We say that the family is associated to the good covering .

Now, we consider a set of functions for , , , constructed as follows. For all , let be an open sector of infinite radius centered at 0, with bisecting direction and with opening . The numbers and are chosen in such a way that for all and all . Now, we put

for all and all , where is given by the formula (4.56). Recalling how these functions are constructed, we consider a set of functions

for all such that

We also assume that for all , all , has an analytic continuation denoted by for all such that

Let

be the convergent Taylor expansion of with respect to on for all . We consider the formal series

for all . We define as the -sum (in the sense of Definition 4.1) of in the direction . We deduce that defines a holomorphic function for all and for all , where

for some and some constant (independent of ) for all .

From Proposition 4.12, for all , we consider the solution of the problem (4.55) with the initial conditions

which defines a bounded and holomorphic function on .

Proposition 4.15. The function defined by

is holomorphic and bounded on , for all , all , and for some .

Moreover, the functions from into the Banach space are the -sums on of a formal series . In other words, for all , there exists a function which is holomorphic on which admits for all , an analytic continuation which is holomorphic on , where is an open sector centered at 0 with infinite radius and bisecting direction such that

along a half-line .

Proof. The proof is based on a cohomological criterion for summability of formal series with coefficients in a Banach space, see [27, page 121], which is known as the Ramis-Sibuya theorem in the literature.

Theorem (RS).  Let be a Banach space over and a good covering in . For all , let be a holomorphic function from into the Banach space and let the cocycle be a holomorphic function from the sector into (with the convention that and ). We make the following assumptions.

(1)The functions are bounded as tends to the origin in for all .(2)The functions are exponentially flat of order 1 on for all . This means that there exist constants such that

for all all .

Then, for all , the functions are the 1 -sums on of a 1 -summable formal series in with coefficients in the Banach space .

By Definition 4.14 and the construction of in Proposition 4.12, we get that the function defines a bounded and holomorphic function on the domain for all all , where depends on , (which denotes a common radius of absolute convergence of the series (4.69), (4.70)), , , , , , , where . More precisely, we have the following.

Lemma  4.16.  Consider the following:

(1)There exist a constant , a constant such that (depending on , and , ), a constant such that (depending on , , , , , , , where ), and a constant (depending on (where are defined above), , , , , , , , ) such that

for all , for all , and all .(2)There exist a constant , a constant such that (depending on , and , ), a constant such that (depending on , , , , , , , , where ), a constant , a constant (depending on , for (where are defined above), , , , , , , , , ) such that

for all , for all , and all (where by convention ).

Proof. (1) Let be an integer such that . From Proposition 4.12, we can write

where and is a holomorphic function on for which the estimates (4.62) hold. By construction, the direction (which depends on ) is chosen in such a way that , for all , all , and for some fixed . From the estimates (4.62), we get

for all , with , for some , and for all .

(2) Let an integer such that . From Proposition 4.12, we can write again

where , , and (resp., ) is a holomorphic function on (resp., on ) for which the estimates (4.62) hold and which is moreover an analytic continuation of a function which satisfies the estimates (4.63).

From the fact that is holomorphic on for all , the integral of along the union of a segment starting from 0 to , an arc of circle with radius connecting and and a segment starting from to 0 is equal to zero. Therefore, we can rewrite the difference as a sum of three integrals:

where , , and is an arc of circle with radius connecting with with a well-chosen orientation.

We give estimates for . By construction, the direction (which depends on ) is chosen in such a way that , for all , all , and for some fixed . From the estimates (4.62), we get

for all , with , for some , and for all .

We give estimates for . By construction, the direction (which depends on ) is chosen in such a way that there exists a fixed with , for all and all . From the estimates (4.62), we deduce as before that

for all , with , for some , and for all .

Finally, we get estimates for . From the estimates (4.63), we have

By construction, the arc of circle is chosen in such a way that that for all (if ), (if ) for all , all . From (4.86), we deduce that

for all , with , for some , and for all . Using the inequality (4.87) and the estimates (2.41), we deduce that

for all , with , and for all .

Finally, collecting the inequalities (4.84), (4.85), and (4.88), we deduce from (4.83), that

for all , with , for some , for all , and for all . Hence, the estimates (4.79) hold.

Now, let us fix . For all , we define , which is, by Lemma  4.16, a holomorphic and bounded function from into the Banach space of holomorphic and bounded functions on the set equipped with the supremum norm. Therefore, the property of Theorem (RS) is satisfied for the functions , . From the estimates (4.79), we get that the cocycle is exponentially flat of order 1 on for all . We deduce that the property of Theorem (RS) is fulfilled for the functions , . From Theorem (RS), we get that are the -sums of a formal series with coefficients in . In particular, from Definition 4.1, we deduce the existence of the functions which satisfy the expression (4.76).

4.7. Analytic Transseries Solutions for a Singularly Perturbed Cauchy Problem

We keep the notations of the previous section.

Proposition 4.17. The following singularly perturbed Cauchy problem

for given initial data

which are holomorphic and bounded functions on , has a solution

which defines a holomorphic and bounded function on , for some .

Proof. Let and . By construction, we have that for all and all . From Lemma  4.16, (1), we get that there exist a constant , a constant such that (depending on , and , ), and a constant (depending on (where are defined above), , , , , , , , ) such that

for all , all , all . From (4.93) and from the property of Definition 4.14, we deduce the estimates

for all . This latter sum converges provided that is small enough. We deduce that defines a holomorphic and bounded function on .

Likewise, from (4.78) and from the property of Definition 4.14, we deduce that there exist a constant , a constant such that (depending on , and , ), a constant such that (depending on , , , , , , , , where ) and a constant (depending on (where are defined above), , , , , , , , ) such that

for all . Again, this latter sum converges if is small enough and if . We get that defines a holomorphic and bounded function on . By construction, we have that , for . Finally, from Proposition 4.3, we deduce that solves (4.90).

5. Parametric Stokes Relations and Analytic Continuation of the Borel Transform in the Perturbation Parameter

5.1. Assumptions on the Initial Data

We keep the notations of the previous section. Now, we make the following additional assumption that there exists a sequence of unbounded open sectors such that

for all and a sequence of real numbers , such that

with the property that for all , all , and for all (where and were introduced in Definition 4.14). We also make the assumption that for all , the function can be analytically continued to a holomorphic function on for all such that

with the property that

and has a common radius of absolute convergence (denoted by ) for all . From the assumption (5.4), we get a constant (depending on ) and a constant (depending on and ) such that

for all . We deduce that

for all , all , all , and all . In particular, we have that belongs to the space for . Moreover, from Proposition 2.7, we deduce that belongs to the space and that there exists a universal constant such that

for all , all , and all , all .

We make the crucial assumption that for all , there exists a sequence of distributions , for , a constant and a sequence with such that

for all and all . From the estimates (5.7) and (5.8), we deduce that

Lemma 5.1. Let . One can write the initial data in the form of a Laplace transform in direction ,

where and for all , and all , all .

Proof. For , from the definition of the initial data, we can write

for all , all , and all . Now, we can write

for all , all , all . From the continuity estimates (3.3) for the Laplace transform, we deduce that for given , , there exists a constant (depending on , ) such that

for all . By letting tend to in this latter inequality and using the hypothesis (5.8), we get that

for all , all , and all .

On the other hand, from Corollary 2.10, we have that for all , the distribution belongs to and that there exists a universal constant such that

for all , all . From (5.14) and using Propositions 3.3 and 3.7, we can write

where

with for all and all . From Proposition 3.6, we have a universal constant and a constant (depending on , , and , which tend to zero as ) such that

From the estimates (5.9) and using (5.15), (5.18), we deduce that the distribution

for all , if is chosen small enough. Finally, by the continuity estimates (3.3) for the Laplace transform and the formula (5.11), (5.16), we get the expression (5.10).

On the other hand, we assume the existence of a sequence of unbounded open sectors with

for all and a sequence of real numbers , such that

with the property that for all , all , and all (where and are introduced in Definition 4.14). We make the assumption that for all , the function can be analytically continued to a holomorphic function on for all such that

with the property that

and has a common radius of absolute convergence (defined by ) for all . From the assumption (5.23), we get a constant (depending on ) and a constant (depending on and ) such that

for all . We deduce that

for all , all , all , and all . In particular, we have that belongs to the space for . Moreover, from Proposition 2.7, we deduce that belongs to the space and that there exists a universal constant such that

for all , all , all , and all .

Now, we make the crucial assumption that for all , there exists a sequence with such that

for all , where are the distributions defined in Lemma 5.1.

5.2. The Stokes Relation and the Main Result

In the next proposition, we establish a connection formula for the two holomorphic solutions and of (4.90) constructed in Proposition 4.15.

Proposition 5.2. Let the assumptions (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold for the initial data. Then, there exists such that one can write the following connection formula:

for all , all , and all .

The proof of this proposition will need two long steps and will be the consequence of the formula (5.79) and (5.124) from Lemmas 5.5 and 5.7.

Step 1. In this step, we show that the function can be expressed as a Laplace transform of some staircase distribution in direction satisfying the problem (5.80), (5.81).

From the assumption (5.4), we deduce from Proposition 4.12 that the function constructed in (4.80) has an analytic continuation denoted by on the domain which satisfies estimates of the form (4.62) for all , where depends on , (which denotes a common radius of convergence of the series (5.4)), , , , , , , where . This constant is, therefore, independent of and . Now, one defines the functions

for all , all , all , and all .

Lemma 5.3. Let . Then, there exists (depending on , , , , uj, (introduced in (5.8)), , , , , , (introduced in Lemma  5.4)), there exist (depending on , , , , uj, for , , , , ), (depending on , , , , , , , (introduced in Lemma  5.4), , , uj for ) and a constant (depending on , , , , , , , , , for ) such that for all all , there exists a staircase distribution with

where is a positive sequence (converging to 0 as tends to ) introduced in the assumption (5.8) and is the positive sequence (tending to 0 as ) introduced in Lemma  5.4. Moreover, one has

Proof. From the estimates (4.54), we can write

where are holomorphic functions such that there exists a constant such that (depending on , , and , ), a constant such that (depending on , , , , , , , , where ), and a constant (depending on (where are defined in (5.4)), , , , , , , , ) with

for all , , all , all , and all . We deduce that

for all , all , all , all , and all . In particular, belongs to . From Proposition 2.7, we deduce that belongs to . From Proposition 2.7 and (5.34), we get a universal constant such that

for all , all , and all . From (5.35), we deduce that the distribution

for all , all , all , and all .

One gets from (4.20), (4.21) and the assumption (4.44) that the following problem holds:

with initial data

On the other hand, we consider the problem

with initial data

In the next lemma, we give estimates for the coefficients of (5.37) and (5.39).

Lemma  5.4.  Let

the convergent Taylor expansion of with respect to near 0. Let be a real number. Then, there exist positive constants , , , , , and a sequence such that with

for all , all , all , all , all , and all .

Proof. We first show (5.42). From the fact that is holomorphic near , we get from the Cauchy formula that there exist such that

for all , and all . On the other hand, from Definition 4.14(3.1), there exist such that for all , all , and all . Hence,

for all , and all , all , all . We deduce (5.42) from (5.44) and (5.45).

Now, we show (5.43). Using the classical identities and , we get the estimates

On the other hand, again from Definition 4.14 (3.1), there exist such that

for all , all , and all . Using (5.46), (5.47) and the fact that for all , we deduce the estimates (5.43).

In the first part of the proof of Lemma 5.3, we show the existence of a staircase distribution solution of the problem (5.39), (5.40), which satisfies the estimates (5.31). As a starting point, it is easy to check that the problem (5.39), (5.40) has a formal solution of the form

where are distributions on , for which the next recursion holds:

for all , , with initial conditions

Using Corollary 2.10, Propositions 2.11 and 2.12, the estimates (5.9), and Remark 2.4, we deduce that for all and that the following inequalities hold for the real numbers : there exist constants , (depending on , , , , ) with

for all , where are defined in Lemma  5.4. We define the following Cauchy problem:

for given initial data

From the assumption (4.42) and the fact that , we deduce that

for all and all . Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the Cauchy problem (5.52), (5.53). Since the initial data is an analytic function on a disc containing some closed disc , for and since the coefficients of (5.52) are analytic on , we deduce that all the hypotheses of Proposition 2.22 are fulfilled for the problem (5.52), (5.53). We deduce the existence of a formal solution , where (depending on ) and (depending on , , , , , , , , ).

Now, let be its Taylor expansion at the origin. Then, the sequence satisfies the next equalities:

for all , with

Gathering the inequalities (5.51), the equalities (5.55) with the initial conditions (5.56), one gets

for all . From (5.57) and the fact that , we get a constant such that

for all . From this last estimates (5.58), we deduce that for all , belongs to for and moreover that

holds. This yields the property (5.31).

In the second part of the proof, we show (5.30). One defines the distribution

for all , all , with , all . If one writes the Taylor expansion

for , then the coefficients satisfy the following recursion:

where

for all and all . Now, we put . Using Corollary 2.10, Propositions 2.11 and 2.12, and Lemma  5.4, we get that there exist constants (depending on , , , , ) such that the following inequalities:

hold for all , where are defined in Lemma  5.4 and is a sequence which satisfies the next estimates: there exist constants (depending on ,,,,) with

for all , where , is the sequence defined in Lemma  5.4.

We consider the following sequence of Cauchy problem:

where

and is already defined as the solution of the problem (5.52), (5.53), for given initial data

which are convergent near the origin with respect to due to the assumption (5.8) and Remark 2.4. Moreover, the initial data satisfy the estimates

for all , , all .

From the assumption (4.42) and the fact that , we deduce that

for all and all . Therefore, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.66), (5.68).

On the other hand, from Lemmas 2.20 and 2.21, there exist a constant (depending on ), a constant , and a constant such that

for all , where the constant is introduced in (5.58).

Since the initial data is an analytic function on some disc containing the closed disc , for and the coefficients of (5.66) are analytic on , we deduce that all the hypotheses of Proposition 2.22 for the problem (5.66), (5.68) are fulfilled. We deduce the existence of a formal solution of (5.66), (5.68), where (depending on ) and (depending on , , , , for , , , , , ).

Moreover, from (2.111) and (5.71), there exist constants (depending on , , , , for , , , , , ) and (depending on , for , , ) such that

for all . Now, let be its Taylor expansion at the origin. Then, the sequence satisfies the following equalities:

where

for all , with

Gathering the inequalities (5.64), (5.65) and the equalities (5.73), with the initial conditions (5.75), one gets that

for all and all .

From (5.76) and the estimates (5.72), we deduce that

for all , all . From (5.77), we get that

for all and all . This yields the estimates (5.30).

In the next lemma, we express as a Laplace transform of a staircase distribution.

Lemma 5.5. Let . Then, one can write the solution of (4.90), (4.91) in the form of a Laplace transform in direction

for all , where (with ) solves the following Cauchy problem:

where the sets and the integers are introduced in (4.20), with initial data

Proof. From Proposition 4.17, we can write the solution of (4.90), (4.91) in the form

for all and all . Now, we write

for all and all . Now, we define . From the continuity estimates (3.5) for the Laplace transform, we deduce that for given , , there exists a constant (depending on , ) such that

for all , all . By letting tend to in this latter inequality and using the estimates (5.30), we obtain

for all and all .

On the other hand, from Corollary 2.10, we have that for all , the distribution belongs to and that there exists a universal constant such that

for all .

From (5.85) and using Propositions 3.3 and 3.7, we can write

where

with , for all , all . From Proposition 3.6, we have a universal constant and a constant (depending on , , and , which tend to zero as ) such that

From the convergence of the series (5.31) near the origin and using (5.86), (5.89), we deduce the distribution

if is chosen small enough. Finally, by the continuity estimates (3.5) of the Laplace transform and the formula (5.82), (5.87), we get the expression (5.79). Moreover, from the formulas in Proposition 3.3, as solves the problem (4.90), (4.91), we deduce that the distribution solves the Cauchy problem (5.80), (5.81).

Step 2. In this step, we show that the function can be expressed as a Laplace transform of some staircase distribution in direction , satisfying the problem (5.80), (5.81).

From the assumption (5.23), we deduce from Proposition 4.12, that the function constructed in (4.80) has an analytic continuation denoted by on and satisfies estimates (4.62) for all , where depends on , (which denotes a common radius of absolute convergence of the series (5.23), , , , , , , where . This constant is, therefore, independent of . Now, one defines the functions

for all , all , and all .

Lemma 5.6. Let as in Lemma 5.3. Then, there exists (depending on and introduced in Lemma  5.7), there exist (depending on and introduced in Lemma  5.7) such that

for all , where is defined in Lemma 5.5 and solves the problem (5.80), (5.81) and is the sequence (which tends to zero as ) defined in Lemma  5.7.

Proof. From the estimates (4.54), we can write

where are holomorphic functions such that there exist a constant with (depending on , , and , ), a constant such that (depending on , , , , , , , , where ), and a constant (depending on (where are defined in (5.23)), , , , , , , , ) with

for all , , all , and all . We deduce that

for all , all , all , and all . In particular, belongs to . From Proposition 2.7, we deduce that belongs to . From Proposition 2.7 and (5.95), we get a universal constant such that

for all and all . From (5.96), we deduce that the distribution

for all , all , and all .

From (4.20), (4.21), we have that the distribution solves the following problem:

where is the set and are the integers from (5.80), with initial data

In the next lemma, we give estimates for the coefficients of (5.98) and (5.80). The proof is exactly the same as the one described for Lemma  5.4.

Lemma  5.7.  Let

be the convergent Taylor expansion of with respect to near 0. Then, there exist positive constants , , , , , and a sequence such that   with

for all , all , all , all and all .

Now, we consider the distribution

for all , all , with and defined in Lemma 5.5, for all . One writes the Taylor expansions as follows:

for ; then the coefficients satisfy the next recursion:

for all , all . We put . Using Corollary 2.10, Propositions 2.11 and 2.12 and Lemma  5.7, we get a constant (depending on ) and (depending on ) such that the next inequalities:

hold for all , where and the sequence , are defined in Lemma  5.7.

We consider the following sequence of Cauchy problems:

where

with

for given initial data

which are finite positive numbers due to the assumption (5.27) and Remark 2.4. Moreover, the initial data satisfy the estimates

for all and all .

On the other hand, we have that is convergent for all (where is chosen in (5.58)). Indeed, we know, from (5.90), that

is convergent for all , all , and all . From (5.86) and (5.89), we know that

for all and all . From (5.58) and (5.112), we deduce that

and this last sum is convergent provided that is small enough. We deduce that belongs to , for any . Let .

From Lemmas 2.20 and 2.21, we get constants (depending on ), , and such that

for all .

From the assumption (4.42) and the fact that , we deduce that

for all . Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.106), (5.109). Moreover, the initial data can be seen as constant functions (therefore analytic) with respect to a variable on the closed disc for any given and the coefficients of (5.106) are analytic with respect to on and constant (therefore analytic) with respect to on . We deduce that all the hypotheses of Proposition 2.22 for the problem (5.106), (5.109) are fulfilled. A direct computation shows that the problem (5.106), (5.109) has a unique formal solution , with . From Proposition 2.22, we deduce that where (depending on ) and (depending on ). Moreover, from (2.111) and (5.114), there exist constants (depending on ) and (depending on ) such that

for all .

Now, the coefficients satisfy the following equalities:

for all and all , with

Gathering the inequalities (5.105) and the equalities (5.117), with the initial data (5.118), one gets that

for all .

From (5.119) and the estimates (5.116), we deduce that

for all . From (5.120), we get that

for all and for all . This implies the estimates (5.92).

In the following lemma, we express the function as Laplace transform of a staircase distribution.

Lemma 5.7. Let as in Lemma 5.3. Then, one can write the function , which by construction of Proposition 4.12, solves the singularly perturbed Cauchy problem

for given initial data

in the form of a Laplace transform in direction

for all , where solves the Cauchy problem (5.80), (5.81).

Proof. From Proposition 4.12 and the assumption (5.23), we get that the function can be expressed as a Laplace transform in the direction ,

for all , all . Now, let , . For all , we can rewite as a Laplace transform in the direction as follows:

for all . Using the expression (5.126), we deduce that from the estimates (3.5), there exists a constant such that

for all and all . By letting tend to and using the estimates (5.92), we get the formula (5.124).

Now, we are in the position to state the main result of our work.

Theorem 5.8. Let the assumptions (4.42), (4.44), (4.67), (4.69), (4.70), (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold. Then, if one denote by (resp. ) the opening of the sector (resp. ), one has that for all , , the function (constructed in Proposition 4.15) can be analytically continued along any path in the punctured sector

as a function denoted by . Moreover, for all , and any path from 0 to a neighborhood of , there exists a constant such that

as tends to in a sector centered at .

Proof. The proof is based on the following version of a result on analytic continuation of Borel transforms obtained by Fruchard and Schäfke in [3]. This result extends a former statement obtained by the same authors in [28].

Theorem (FS).  Let and let be a holomorphic function that can be analytically continued as a function (resp., ) with exponential growth of order 1 on an unbounded sector (resp. ) centered at 0, with bisecting direction (resp. ) and opening (resp. ). Let be a real number and let be an integer. Let be a set of aligned points and let with , for all . For all integers , let be an unbounded open sector centered at , with bisecting direction which is parallel to , and opening such that the do not intersect for all .

Now, for all , let be a holomorphic and bounded function on a small neighborhood of 0 and with exponential growth of order 1 on the sector with bisecting direction . We consider the Laplace transforms

for all , where is the half-line starting from 0 in the direction and is the half-line starting from 0 in the direction . The function (resp. ) defines a holomorphic and bounded function on an open sector (resp. ) with finite radius, with bisecting direction (resp. ) and opening (resp. ). The sectors are chosen in such a way that is contained in a sector with direction and with opening less than . Assume that the following Stokes relation

holds for all , where is a holomorphic function on such that there exists a constant with

for all .

Then, the function can be analytically continued along any path in the punctured sector

Moreover, for all , and any path from 0 to a neighborhood of , if we denote by the analytic continuation of along , then there exists a constant such that

as tends to in a sector centered at .

Proof. For the sake of completeness, we give a sketch of proof of this theorem. In the first step, let us consider the following sums of Cauchy integrals

where is the segment starting from in the direction with length . The multivalued function can be analytically continued along any path in by deforming the path of integration in the sector and keeping the endpoints of the segment fixed for all . Moreover, let and , where denotes the half-line starting from in the direction . We denote by the analytic continuation of along a loop around constructed as follows: the loop follows a segment starting from in the direction then turns around along a circle of small radius positively oriented and then goes back to following the same segment. We have that

Indeed, by the Cauchy theorem, one can write as a Cauchy integral

where is a positively oriented closed curve enclosing starting from and containing the point . By the residue theorem, one gets that . From the relation (5.136), we also deduce the existence of a holomorphic function near such that

for all near , for a well-chosen determination of the logarithm .

In the second step, let us define the truncated Laplace transforms and Laplace transforms

where is the segment starting from 0 to and is the segment starting from 0 to , for any fixed . By the Cauchy formula, one can write the difference as the sum

where is the segment starting from to for any small enough. Due to the decomposition (5.138), is integrable at . By letting tending to 0 and tending to infinity, using the relation (5.136) in (5.140), ones gets that

where is the half-line starting from in the direction .

Now, one considers the differences and . From the Stokes relations (5.131) and (5.141), one deduces that

for all . Using a similar Borel transform integral representation as in the proof of Theorem 1 in [28], one can show that the difference , which is by construction analytic near the origin in , can be analytically continued to a function , which is holomorphic on the sector . Since can be analytically continued along any path in , one gets that the function can be analytically continued along any path in and from the decomposition (5.138) one deduces the estimates (5.134).

Now, we return to the proof of Theorem 5.8. From the formula (4.76) and Proposition 5.2, the following equality

holds for all , and all , all . Let and fixed. Let be an integer. From the estimates (4.78), we get that

for all . From (5.143) and (5.144), we deduce that the following Stokes relation

Holds, where is a holomorphic function on such that there exists a constant with

for all . We can apply Theorem (FS) with , for , to get that the function (constructed in Proposition 4.15) can be analytically continued along any path in the punctured sector

as a function denoted by . Moreover, for all , and any path from 0 to a neighborhood of , there exists a constant such that as tends to in a sector centered at . Since this result is true for all , Theorem 5.8 follows.

In the next result, Onee show that under the additional hypothesis that the coefficients of (4.90) are polynomials in the parameter , the function solves a singular linear partial differential equation in .

Corollary 5.9. Let the assumptions of Theorem 5.8 hold. We assume moreover that, for all tuple chosen in the set , the coefficients belong to with the following expansion in :

for some . Then, for all with and , the function (constructed in Proposition 4.15) satisfies the following singular linear partial differential equation

for all . From Theorem 5.8, for all , this solution can be analytically continued with respect to along any path in the punctured sector with logarithmic estimates (5.129) near the singular points for all .

Proof. From Proposition 4.15, we have that the function

solves (5.122) on . From the formulas in Proposition 4.2, we deduce that the function solves the singular integrodifferential equation

for all . Since is holomorphic on and has as analytic continuation on , we get that also solves (5.151). By differentiating times of each hand side of the equation with respect to , one gets that solves the partial differential equation (5.149).

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