Abstract and Applied Analysis

Volume 2013 (2013), Article ID 723040, 18 pages

http://dx.doi.org/10.1155/2013/723040

Research Article

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

Alberto Lastra¹ and Stéphane Malek²

¹Departamento de Física y Matemática, University of Alcalá, Apartado de Correos 20, Alcalá de Henares, 28871 Madrid, Spain

²Laboratoire Paul Painlevé, University of Lille 1, 59655 Villeneuve d'Ascq Cedex, France

Received 17 July 2013; Accepted 16 September 2013

Academic Editor: Alexander Domoshnitsky

Copyright © 2013 Alberto Lastra and 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 -difference-differential equations in the complex domain. We provide sectorial holomorphic solutions in the perturbation parameter . Moreover, we achieve the existence of a common formal power series in which represents each actual solution and establish -Gevrey estimates involved in this representation. The proof of the main result rests on a new version of the so-called Malgrange-Sibuya theorem regarding -Gevrey asymptotics. A particular Dirichlet like series is studied on the way.

1. Introduction

We study a family of difference-differential equations of the form

under appropriate initial conditions

Here, is an integer with and . stands for a finite subset of , where is the set of nonnegative integers. For every , turns out to be a polynomial in the variable with holomorphic and bounded coefficients in a neighborhood of the origin in the parameter and .

From now on, stands for a fixed real number with .

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

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

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

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

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

In the context of differential equations most of the statements in the literature are dedicated to linear problems of the form

where are vector valued polynomial functions in and linear in its other arguments, where , for are real numbers and concern the study of asymptotic behaviour of some of their solutions as tends to infinity for given initial data . When is real or matrix valued and with constant coefficients, we quote [11–14]. For polynomial in , we notice [15, 16]. For studies in a complex variable , we refer to [17, 18]. For more general delay functional equations, we indicate [19].

In the framework of linear partial differential equations, we mention a series of papers devoted to general results on the existence and unicity of holomorphic solutions to generalized Cauchy-Kowalevski type problems with shrinkings of the form

for some integer , a finite set , and where is analytic or of Gevrey type function and such that the functions and satisfy the shrinking constraints and for given initial data , that belong to some functional space. We refer to [20–22]. For partial differential problems with contractions dealing with less regular solution spaces like Sobolev spaces, we quote [23], for instance.

Let us briefly reproduce the strategy followed. We consider a finite family of sectors with vertex at the origin which provides a good covering at 0 in the variable (see Definition 19). Let . One can consider an auxiliary Cauchy problem as follows:

with initial conditions , . We assume is a holomorphic function in for some , for every , which is upper bounded in terms of Gevrey bounds (see (61)). Moreover, we assume each can be extended to , where is a sector with vertex at the origin and verifying Gevrey bounds in with (see (31)). Under these hypotheses, one can construct a formal solution to the auxiliary Cauchy problem, , where turns out to be a holomorphic function in . Here, is a disc centered at the origin with radius decreasing to 0 whenever tends to infinity and reproducing Gevrey bounds given by the initial conditions (see (62)). Moreover, each can be extended to under Gevrey bounds (see (32)), where , with being a sequence of positive numbers that decrease to 0. We assume for every . The decrease rate of both and the radius of has to be chosen adequately, in accordance to the elements of a Gevrey sequence such as for some .

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

General Dirichlet series of the form

have been throughly studied in the case when is an increasing sequence of real numbers to (see [25–27]) or a sequence of complex numbers with (see [28]). This theory has also been developed when working with almost periodic functions, introduced by Bohr (see [29–31]), which are the uniform limits in of exponential polynomials , where the values belong to the so-called spectrum . However, we are more interested in the behaviour of the sum when tends to in the positive imaginary axis. Our technique rests on the Euler-Mac-Laurin formula, Watson's lemma, and the equivalence between null -Gevrey asymptotics. The characterization of —exponentially at functions is also considered on the way.

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

The growth properties of for allow us to apply a Laplace like transform on each of them with respect to the variable in order to provide a holomorphic solution of the main problem, defined in , for some appropriate unbounded open set . In addition to this, one has null Gevrey asymptotic bounds for the difference of and when the domain of the variable is restricted to a bounded set, meaning that for every , there exist such that

for every .

Finally, a novel version regarding Gevrey asymptotics of Malgrange-Sibuya theorem (Theorem 25) leads us to the main result in the present work (Theorem 27), where we guarantee the existence of a formal power series in as follows:

with coefficients in the Banach space of bounded holomorphic functions defined in , which is common for every and such that admits as its Gevrey asymptotic expansion of some positive type in the variable (see (135)).

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

The paper is organized as follows.

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

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

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

, , , and are fixed positive real numbers throughout the present work. Let with and let be a sequence of positive real numbers.

We consider an open and bounded sector with vertex at the origin and we fix an open and unbounded sector with vertex at the origin having positive distance to a fixed complex number , it is to say, there exists such that for every . We write for the subset of defined by

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

Definition 1. Let and . denotes the vector space of functions such that

is finite.

Let . denotes the complex vector space of all formal power series with for every and such that

It is straightforward to check that the pair is a Banach space.

For our purposes, the elements in the sequence are chosen to be related to the ones in a Gevrey sequence. This choice would provide that tends to when .

Let be a family of complex functional Banach spaces. For every , we consider the formal integration operator defined on by

Lemma 2. Let , , , , and , , and . We assume that

In addition to this, we consider the elements in are such that

for every . Moreover, we assume there exist constants such that

for every . In addition to this, we assume

Under the previous assumptions, there exists a positive constant , which does not depend on nor such that

for every .

Proof. Let . We have

From (14), one derives that for every , is well defined and the function is holomorphic in for every . The expression in (18) equals

Let . From the definition of the norm , we get

Direct calculations allow us to obtain the following estimates

for some positive constants and only depending on , , and . Moreover,

for some constant depending on , , , and  . This last equality and (13) yield

for some positive constants and depending on , , , , , , and . From the hypothesis (15) on , the last expression is upper bounded by

for some positive constant only depending on , , , , , , , and . Now, from (16) one gets that is upper bounded by a constant which does not depend on , where

Taking into account all these computations, one achieves that (20) can be upper bounded by

The lemma follows bearing in mind (14) and the definition of the norms in and of .

Remark 3. The hypotheses made in (14), (15), and (16) are verified if one departs from for small enough positive and any provided (13) is satisfied and .

Lemma 4. Let be a holomorphic and bounded function defined on . Then, there exists a constant such that

for every , every and all .

Proof. Direct calculations on the definition of the norms in the space allow us to conclude when taking .

Let and let be a finite subset of . We also fix , where stands for the set .

For every , let , be nonnegative integers and , where is such that . We write , where is a finite subset of for every . We assume that for every .

We consider the functional equation

with initial conditions

where the function is an element in for every .

We make the following assumptions.

Assumption A. For every and every , we assume

Assumption B. and there exist , with , for every and every .

Theorem 5. Let Assumption A and Assumption B be fulfilled. We assume that the initial conditions in (29) verify there exist and such that for every

for every , , where . Then, there exists , formal solution of (28)+(29), where .

Then, there exist positive constants and (only depending on , , , , , , and  ) and such that

for every , all and every .

Proof. Let . We put and define the map from into itself by

where . For an appropriate choice of , the map turns out to be a Lipschitz shrinking map.

Lemma 6. There exist (not depending on ) such that

(1) for every . denotes the closed ball centered at 0 with radius in . (2)

for every , .

Proof. Let and . In order to prove the first enunciate, we take . From Lemmas 2 and 4 we deduce that

with for every and .

Let us fix and . Taking into account the definition of , we derive

for some which only depends on the parameters defining (28). The terms of the form in the previous expression can be upper bounded by an adequate constant. Taking into account (31), usual estimates in (36) derive

for some depending on the parameters defining the equation and such that it tends to 0 whenever both and tend to 0. An appropriate choice for these constants allows us to conclude the first part of the proof.

The second part of the lemma follows similar arguments as before. Let , . One has

The result is achieved with an adequate choice of .

Let , , and let be as in the previous lemma. Bearing in mind Lemma 6 one can apply the shrinking map theorem on complete metric spaces to guarantee the existence of a fixed point for in , say , which verifies , and . Let us define

We put , and . Then, can be written as a formal power series in as

where for every .

From the construction of , we have is a formal solution of (28)+(29). Moreover, from the domain of holomorphy of the initial conditions in (29) and the recursion formula satisfied by the coefficients in , we get

We can conclude the function for every .

Finally, the estimates in (32) are obtained for every from the fact that . The definition of the elements in lead us to

so that

for every . In addition to this, Assumption B and usual estimates allow us to refine the previous estimates leading to

for some constants and which only depend on , , , , , , and . This is valid for every and . The hypothesis (31) in the enunciate allows us to affirm that (32) is also valid for .

Remark 7. One derives holomorphy of in the variable in the whole sector and not only in for every whilst the estimates are only given for . It is also worth saying that can be arbitrarily chosen whenever for every , .

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

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

For every , stands for the set . We preserve the value of the positive constants , and from the previous section. Let with and let be a sequence of positive real numbers.

Definition 8. Let . For and ; stands for the vector space of functions such that

is finite. Let . We write for the vector space of all formal power such that with

The pair is a Banach space.

Lemma 9. Let , , , , and , , and . We assume that

Moreover, we assume that the elements of the sequence are such that

for every .

Under the previous assumptions, there exists a positive constant which depends on , , , , , , , , , , and (not depending on nor ) such that

for every .

Proof. Let be an element of . We have

From (48), one derives that for every , is well defined. In addition to this, the function is holomorphic in for every . The expression in (50) equals

Let . From the definition of the norm , we get

with .

The result follows provided that one is able to estimate the expression

From the first of the hypotheses made in (47), is upper bounded by a constant. Also, taking into account (48), there exists such that for every , so that

for some positive constant . The result immediately follows from (47) that guarantees that is bounded from above.

Let be as in the proof of the previous lemma, that is, for every .

Lemma 10. Let be a holomorphic and bounded function defined on .

Then, there exists a constant such that

for every , every and all .

Proof. Direct calculations on the definition of the norms in the space allow us to conclude when taking .

Let and let be a finite subset of . We also fix such that , with as before.

Let , , and let be as in Section 2, for every .

We consider the functional equation

with initial conditions

where the function is an element in for every .

We make the following assumptions.

Assumption A ′. For every and every , we assume

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

Assumption B ′. We assume for every , every , and every .

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

Lemma 12. Let , and .

We put for , and for every . Let us assume that (56)+(57) has a formal solution in , . Then, there exists such that for every , the function belongs to for every and all .

Proof. Let be a formal power series in of the form . One can plug the formal power series into (56) to obtain the recursion formula in (41) for the coefficients . From this recurrence, one derives that the domain of holomorphy for in the variable depends on the domain of holomorphy on of and also on for every and every such that .

The initial conditions are holomorphic functions in .

Lemma 13. For every the coefficients turn out to be holomorphic functions in , for and .

Proof. We prove it by recurrence on and regarding the recursion formula (41).

Let . One has for any as in (41) if and only if for every , it is to say, if and only if . In this case, only depends on the initial conditions . Moreover,

and the dilation on the variable allows us to obtain that ,…, are holomorphic functions in .

The proof can be followed recursively for every by considering analogous blocks of indices as before.

Regarding Lemma 13, the proof of Lemma 12 is concluded if one can check that for every , whenever

Let and , with . Let . We have if and only if . The result follows for any .

Lemma 14. Let be defined as in Lemma 12. Then, satisfies Assumption B′.

Proof. From the definition of , the lemma follows when taking for every and every .

Assumption . We assume for and for any , with as in Lemma 12.

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

Theorem 15. Let Assumption A′ and Assumption B′′ be fulfilled. We also make the next assumption on the initial conditions (57); there exist and such that

for every , and , where . Then, there exists a formal power series , with , which provides a formal solution of (56)+(57). Moreover, there exist positive constants and (only depending on , , , , , , , and ) and such that

for every , and for every .

Proof. The proof follows analogous steps as the one of Theorem 5, so we do not enter into details as to not to repeat arguments.

Let and . The set is taken to be . We consider the map from into itself defined in the same way as in (33).

From Lemma 12 and Assumption B′′, the unique formal solution of (56)+(57) determined by the recursion formula (41), , is such that for every .

Regarding the initial conditions of the Cauchy problem, one can reduce , if necessary, so that and so the map is well defined, for every , every , and . Moreover, from (61), the expression

can be estimated in an analogous manner as in the corresponding step of the proof of Theorem 5, for every and all .

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

4.1. Laplace Transform and Gevrey Asymptotic Expansion

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

Lemma 16. Let and let be a holomorphic function in an unbounded sector such that there exist with

for every . Let be an unbounded sector with vertex at 0 which verifies that

for some and . Then,

is a holomorphic and bounded function defined for . Moreover, the following identities hold:

where , for all .

In the sequel, we work with functions which satisfy more restrictive bounds that the ones in (64). Indeed, we deal with bounds of the form , for some . This alters the asymptotic behaviour of the Laplace transform and cause the appearance of Gevrey asymptotic expansions associated with estimates related to the sequence .

For any open sector in the complex plane with vertex at 0 with finite or infinite and , we say the finite sector with vertex at the origin is a proper subsector of , and we denote it as , if for some and some , .

stands for a complex Banach space.

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

Definition 17. Let be a sector in with vertex at the origin, and . We say a holomorphic function admits the formal power series as its Gevrey asymptotic expansion of type in if for every there exist such that

for every .

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

Proposition 18. Let and a holomorphic function in a sector with vertex at the origin. The following holds.

(i)If admits the power series with null coefficients, which is denoted by , as its Gevrey asymptotic expansion of type , then for every there exists with

for every and every . (ii)If for every there exists with

for every then admits as its Gevrey asymptotic expansion of type in , for every .

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

We recall the definition of a good covering.

Definition 19. Let be a finite family of open sectors with vertex at the origin and finite radius . We assume that for (we put ) and also that for some . Then, the family is known as a good covering in .

Definition 20. Let be a good covering in . We consider a family such that the following holds.

(1)There exist , such that

for every . (2) is an unbounded subset of an open sector with vertex at the origin. We assume for every . (3)For every and , there exists such that . (4)For every , , and , one has .

Under the previous settings, we say the family is associated to the good covering .

Let us consider a good covering in , .

Let and . We consider a finite subset of , . For every , let and let be a holomorphic and bounded function on , for some . For each , we consider the following main Cauchy problem in the present work:

with initial conditions

where the functions are constructed as follows. Let be a family of open sets associated with the good covering .

From now on, we assume the values of and are those in the preceeding sections. If necessary, one can adjust the values of , , , and so that for every , so that for every and every . Here, we have put

For every , we assume that is a bounded and holomorphic function on verifying

for every . Here , , , and are the constants provided in Theorem 5. Assume that can be extended to an analytic function defined on and

for every .

Take such that . We put

for every . One can check that is well defined and holomorphic in . Indeed, there exists such that for every . Moreover, from the growth properties of , one deduces

which is convergent for every .

Theorem 21. Let Assumptions A, B, and B′′ be fulfilled. For every , we consider the problem (72)+(73) with initial conditions constructed as above. Then, the problem (72)+(73) admits a solution which is holomorphic and bounded in .

Moreover, for every and for every there exists (not depending on ) such that

for every (where, by convention, ).

Proof. Let and . We consider the Cauchy problem (28) with initial conditions given by

Theorem 5 shows that the problem (31)+(80) has a formal solution , with for every . Moreover, for every one has

for every , where , and are positive constants provided in the proof of Theorem 5. In a parallel direction, one can consider the same Cauchy problem with initial conditions given by

where are as previously shown.

From Theorem 15, one concludes that the formal power series is such that can be extended to a holomorphic function defined in , for every . We preserve notation for these extensions. Moreover, for every one has

for every , and some positive constants and determined in the proof of Theorem 15.

We put , where

We fist check that is, at least formally, a solution of (72)+(73). From (67), one can check by inserting the formal power series in (72), that it turns out to be a formal solution in the variable of (72)+(73) if and only if is a formal solution of (28)+(29) and (56)+(57).

Bearing in mind that verifies (82) and (83), one derives is well defined in , for every . We now state a proof for the fact that is indeed a holomorphic solution of (72)+(73) in . Let , , and . One has

where , , and . We only give details on the first and second integrals appearing on the right-hand side of the previous inequality. The first integral on the right-hand side of (85) can be upper bounded by means of (83) and the choice of direction .

Consider

One has

for some . Now, the function attains its maximum at . One can reduce , if necessary, to conclude that this function is increasing for . The expression in (86) is upper bounded by

This yields

for some constants only depending on , , , , , , , , , , and . We now consider the second integral appearing on the right-hand side of (85). From (81) and similar estimates as before we get

The function , is such that for all , where , for some positive constant , not depending on . attains its maximum value at so that , for every . This implies

From (90) we derive

for some .

From (89) and (92), we lead to the existence of positive constants , not depending on , such that

for every . This allows us to conclude the first part of the proof.

Let and . For every we have

We can write

where stands for the path consisting of two parts: the first one going from to 0 along the segment and the path going from 0 to following direction .

This integral has already been estimated in (92), for the first part of the proof, so we omit the details. We also omit the details on the integral concerning the path which is analogous.

In order to estimate the integral along the path , one can observe that the function involved in the integrand does not depend on the index considered, for this function is well defined for . One can apply Cauchy Theorem to derive

where is the closed path with . Moreover, equals

for some . It only rests to take into account that the function is monotonely increasing in , so that can be included in the constants and .

From (92) and (97) one gets the existence of positive constants such that

for every . Taking this last estimate into the expression of one can conclude that

for every . The proof of the second statement in the theorem leans on the incoming lemma whose proof is left until the end of the current section. It provides information on the estimates for a Dirichlet type series. A similar argument concerning a Dirichlet series of different nature can be found in [2], Lemma 16, when dealing with Gevrey asymptotic expansions.

Lemma 22. Let , , , and let   be positive constants, with . Then, for every there exist and such that

for every .

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

Lemma 23. Let and a continuous function having the formal expansion as its q-asymptotic expansion of type at 0, meaning there exist such that

for every and , for some .

Then, the function

admits the formal power series as its Gevrey asymptotic expansion of type at 0. It is to say, there exist such that

for every and for some .

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

Lemma 24. Let , and let be a continuous function. The following holds.

(1)If there exist such that , for every , and , then for every there exists such that

for every . (2)If there exists such that , for every , and , then for every there exists such that

for every and for every .

Proof of Lemma 22. Let be a function. For every , one can apply the Euler-Mac-Laurin formula

where is the Bernoulli polynomial and stands for the floor function to . One has

Taking the limit when tends to infinity in the previous expression we arrive at the following equality for a convergent series:

Let and .

From the fact that for every and the change of variable , one gets

with , , and , for every . Bearing in mind that for and from usual estimates we derive for some . The proof is complete if one can estimate , , and appropriately. The first expression is clearly upper bounded according to (100).

From usual estimates we arrive at

for some . From Lemma 24, the function defined by (extended by continuity to ) is such that for every there exist with

for every and for every . From Lemma 23, the functions and admit the series with null coefficients as asymptotic expansion of type . Again, from Lemma 24, one can conclude that for every there exists such that both and are upper bounded by , for every , for some .

4.3. Existence of Formal Series Solutions in the Complex Parameter

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

Given with and a nonempty open subset , the discrete spiral associated with and consists of the products of an element in and , for some . For our purpose, is a real number and is chosen in such a way that the discrete spiral turns out to be a sector with vertex at the origin.

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

Theorem 25 (-MS). Let be a Banach space over and let be 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) has a exponential decreasing of some type , for every , meaning there exists such that

Then, there exists a formal power series such that admits as its Gevrey asymptotic expansion of type on , for every .

Proof. We first state an auxiliary result.

Lemma 26. For all , there exist bounded holomorphic functions such that

for all , where by convention . Moreover, there exist , , such that for each any and every , there exist with

for all and all .

Proof. We follow analogous arguments as in Lemma XI-2-6 from [34] with appropriate modifications in the asymptotic expansions of the functions constructed with the help of the Cauchy-Heine transform.

For all , we choose a segment

These segments divide the open punctured disc into open sectors , where

where by convention . Let

for all , for , be defined as a sum of Cauchy-Heine transforms of the functions . By deformation of the paths and without moving their endpoints and letting the other paths , untouched (with the convention that ), one can continue analytically the function onto . Therefore, defines a holomorphic function on , for all .

Now, take . In order to compute , we write

where the paths and are obtained by deforming the same path without moving its endpoints in such a way that

(a) and ,(b) is a simple closed curve with positive orientation whose interior contains .

Therefore, due to the residue formula, we can write

for all and for all (with the convention that ).

In a second step, we derive asymptotic properties of . We fix an and a proper closed sector contained in . Let (resp., ) be a path obtained by deforming (resp., ) without moving the endpoints so that is contained in the interior of the simple closed curve (which is itself contained in ), where is a circular arc joining the two points and . We get the representation

for all . One assumes that the path is given as the union of a segment , where and and a curve such that , , and for all . We also assume that there exists a positive number with for all . By construction of the path , we get that the function defines an analytic function on the open disc .

It remains to give estimates for the integral . Let be an integer. From the usual geometric series expansion, one can write

where

for all .

Gathering (112) and (122), we get

The changes of variable first, and afterwards, transform the right-hand side of (123) into

The application of

for every , which can be found in [35] (Chapter 10, page 498), leads us to

Moreover, as previously described, one can choose a positive number (depending on ) such that for all and all . Again by (112) and (122) and following analogous calculations as before, we obtain

for all . Using comparable arguments, one can give analogous estimates when estimating the other integrals

for all .

As a consequence, for any , there exist , for all and a constant such that

for all and all .

Taking into account Proposition 18, we deduce that for every and for every , the function has the formal series as Gevrey asymptotic expansion of type in . From the unicity of the asymptotic expansions on sectors, we deduce that all the formal series , , are equal to some formal series denoted .

We consider now the bounded holomorphic functions

for all and all . By definition, for any , we have that

for all . Therefore, each is the restriction on of a holomorphic function on . Since is bounded on , the origin turns out to be a removable singularity for which, as a consequence, defines a convergent power series on .

Finally, one can write

for all , all . Moreover, is a convergent power series, and for every , has the series as Gevrey asymptotic expansion of type on , for all .

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

Theorem 27. Let . Under the same hypotheses as in Theorem 21, we denote the Banach space of holomorphic and bounded functions in with the supremum norm. Then, there exists a formal power series

which is formal solution of

Moreover, for every and every , the function constructed in Theorem 21 admits as its Gevrey asymptotic expansion of type in , meaning that for every and , there exist such that

for every and all .

Proof. Let us consider the family constructed in Theorem 21. For every , we define the function , which belongs to the space . From (79), we derive the cocycle verifies (112) in Theorem 25, with for some fixed . Theorem 25 guarantees the existence of a formal power series , such that for every , admits as its Gevrey asymptotic expansion of type on . This is valid for every . This concludes the second part of the result.

It only rests to verify that is a formal solution of (72)+(73).

If we write , we have

for every and all .

Let . By construction, satisfies (72)+(73). We differentiate in the equality (72) times with respect to . By Leibniz’s rule, we deduce that satisfies

for every . Let in the previous expression. From (136) we obtain

is holomorphic wih respect to for every . This entails , for every in a neighborhood of the origin in . From this and (138), we deduce is a formal solution of (72)+(73).

Conflict of Interests

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

Acknowledgments

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

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