Abstract and Applied AnalysisVolume 2012 (2012), Article ID 860716, 35 pagesdoi:10.1155/2012/860716

Research Article

On -Gevrey Asymptotics for Singularly Perturbed -Difference-Differential Problems with an Irregular Singularity

Alberto Lastra¹ and Stéphane Malek²

¹Facultad de Ciencias, Universidad de Valladolid, Calle del Doctor Mergelina s/n, 47011 Valladolid, Spain²UFR de Mathématiques, Université Lille 1, Cité Scientifique M2, 59655 Villeneuve d'Ascq Cedex, France

Received 30 September 2011; Revised 28 November 2011; Accepted 28 November 2011

Academic Editor: Allan C. Peterson

Copyright © 2012 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 -analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by Malek in (2011). First, we construct solutions defined in open -spirals to the origin. By means of a -Gevrey version of Malgrange-Sibuya theorem we show the existence of a formal power series in the perturbation parameter which turns out to be the -Gevrey asymptotic expansion (of certain type) of the actual solutions.

1. Introduction

We study a family of -difference-differential equations of the following form: where such that are positive integers, are polynomials in with holomorphic coefficients in on some neighborhood of 0 in , and is the dilation operator given by . As in previous works [1–3], the map is assumed to be a volume shrinking map, meaning that the modulus of the Jacobian determinant is less than 1, for every .

In [4], the second author studies a similar singularly perturbed Cauchy problem. In this previous work, the polynomial is such that, for all is a finite subset of and are bounded holomorphic functions on some disc in which verify that the origin is a zero of order at least . The main point on these flatness conditions on the coefficients in is that the method used by Canalis-Durand et al. in [5] could be adapted so that the initial singularly perturbed problem turns into an auxiliary regularly perturbed -difference-differential equation with an irregular singularity at , preserving holomorphic coefficients (we refer to [4] for the details). These constricting conditions on the flatness of is now omitted, so that previous result is generalized. In the present work we will make use not only of the procedure considered in [5] but also of the methodology followed in [6]. In that work, the second author considers a family of singularly perturbed nonlinear partial differential equations such that the coefficients appearing possess poles with respect to at the origin after the change of variable . This scenario fits our problem.

In both the present work and [6], the procedure for locating actual solutions relies on the research of certain appropriate Banach spaces. The ones appearing here may be regarded as -analogs of the ones in [6].

In order to fix ideas we first settle a brief summary of the procedure followed. We consider a finite family of discrete -spirals in such a way that it provides a good covering at 0 (Definition 4.6).

We depart from a finite family, with indices belonging to a set , of perturbed Cauchy problems (4.22) and (4.23). Let be fixed. Firstly, by means of a nondiscrete -analog of Laplace transform introduced by Zhang in [7] (for details on classical Laplace transform we refer to [8, 9]), we are able to transform our initial problem into auxiliary equation (2.13) (or (3.8)).

The transformed problem fits into a certain Cauchy auxiliary problem such as (2.13) and (2.14) which is considered in Section 2. Here, its solution is found in the space of formal power series in with coefficients belonging to the space of holomorphic functions defined in the product of discrete -spirals to the origin in the variable (this domain corresponds to in the auxiliary transformed problem) times a continuous -spiral to infinity in the variable ( for the auxiliary equation). Moreover, for any fixed and regarding our auxiliary equation, one can deduce that the coefficients, as functions in the variable , belong to the Banach space of holomorphic functions in subject to -Gevrey bounds for positive constants , where the index of the coefficient considered is (see Theorem 2.4).

Also, the transformed problem fits into the auxiliary problem (3.8) and (3.9), studied in detail in Section 3. In this case, the solution is found in the space of formal power series in with coefficients belonging to the space of holomorphic functions defined in the product of a punctured disc at 0 in the variable times a punctured disc at the origin in . For a fixed , the coefficients belong to the Banach space of holomorphic functions in such that for positive constants when is the index of the coefficient considered (see Theorem 3.4).

From these results, we get a sequence consisting of holomorphic functions in the variable so that the -Laplace transform can be applied to its elements. In addition, the function turns out to be a holomorphic function defined in which is a solution of the initial problem. Here, is an adequate open half -spiral to 0 and corresponds to certain -directions for the -Laplace transform (see Proposition 4.3). The way to proceed is also followed by the authors in [10, 11] when studying asymptotic properties of analytic solutions of -difference equations with irregular singularities.

It is worth pointing out that the choice of a continuous summation procedure unlike the discrete one in [4] is due to the requirement of the Cauchy theorem on the way.

At this point we own a finite family of solutions of (4.22) and (4.23). The main goal is to study its asymptotic behavior at the origin in some sense. Let . One observes (Theorem 4.11) that whenever the intersection is not empty we have for positive constants and for every . Equation (1.5) implies that the difference of two solutions of (4.22) and (4.23) admits -Gevrey null expansion of type at 0 in as a function with values in the Banach space of holomorphic bounded functions defined in endowed with the supremum norm. Flatness condition (1.5) allows us to establish the main result of the present work (Theorem 6.3): the existence of a formal power series formal solution of (1.1), such that, for every , each of the actual solutions (1.4) of the problem (4.22) and (4.23) admits as its -Gevrey expansion of a certain type in the corresponding domain of definition.

The main result heavily rests on a Malgrange-Sibuya-type theorem involving -Gevrey bounds, which generalizes a result in [4] where no precise bounds on the asymptotic appear. In this step, we make use of the Whitney-type extension results in the framework of ultradifferentiable functions. The Whitney-type extension theory is widely studied in literature under the framework of ultradifferentiable functions subject to bounds of their derivatives (see e.g., [12, 13]) and also it is a useful tool taken into account on the study of continuity of ultraholomorphic operators (see [14–16]). It is also worth saying that, although -Gevrey bounds have been achieved in the present work, the type involved might be increased when applying an extension result for ultradifferentiable functions from [13].

The paper is organized as follows.

In Sections 2 and 3, we introduce Banach spaces of formal power series and solve auxiliary Cauchy problems involving these spaces. In Section 2, this is done when the variables rely in a product of a discrete -spiral to the origin times a -spiral to infinity, while in Section 3 it is done when working on a product of a punctured disc at 0 times a disc at 0.

In Section 4 we first recall definitions and some properties related to -Laplace transform appearing in [7], firstly developed by Zhang. In this section we also find actual solutions of the main Cauchy problem (4.22) and (4.23) and settle a flatness condition on the difference of two of them so that, when regarding the difference of two solutions in the variable , we are able to give some information on its asymptotic behavior at 0. Finally, in Section 6 we conclude with the existence of a formal power series in with coefficients in an adequate Banach space of functions which solves in a formal sense the problem considered. The procedure heavily rests on a -Gevrey version of the Malgrange-Sibuya theorem, developed in Section 5.

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

are fixed positive real numbers throughout the whole paper.

Let be nonempty bounded open sets in , and let such that . We define We assume there exists such that for all and also that the distance from the set to the origin is positive.

Definition 2.1. Let and denotes the vector space of functions such that is finite.Let . denotes the complex vector space of all formal series belonging to such that It is straightforward to check that the pair is a Banach space.

We consider the formal integration operator defined on by

Lemma 2.2. Let . One assumes that the following conditions hold: Then, there exists a constant (not depending on nor ) such that for every .

Proof. Let . We have that Taking into account the definition of the norm , we get with . From (2.5) we derive for every and , where and . Moreover, for every . Regarding condition (2.5) we obtain the existence of such that for every and . Inequality (2.6) follows from (2.7), (2.8), and (2.10):

Lemma 2.3. Let be a holomorphic and bounded function defined on . Then, there exists a constant such that for every , every , and all .

Proof. Direct calculations regarding the definition of the elements in allow us to conclude when taking .

Let be an integer. For all , let be positive integers and a polynomial in , where is a finite subset of and are holomorphic bounded functions on . We assume .

We consider the following functional equation: with initial conditions where the functions belong to for every .

We make the following assumption.

Assumption A. For every and , we have

Theorem 2.4. Let Assumption A be fulfilled. One also makes the following assumption on the initial conditions in (2.14): there exist a constant and such that, for every for all . Then, there exists , solution of (2.13) and (2.14), such that, if , then there exist and such that for every and .

Proof. Let . We define the map from into itself by where . In the following lemma, we show that the restriction of to a neighborhood of the origin in is a Lipschitz shrinking map for an appropriate choice of .

Lemma 2.5. There exist and (not depending on ) such that (1) for every ; denotes the closed ball centered at 0 with radius in ;(2) for every .

Proof. Let and .For the first part we consider . Lemmas 2.2 and 2.3 can be applied so that with . Taking into account the definition of and (2.16) we have for a positive constant .We conclude this first part from an appropriate choice of and .For the second part we take . Similar arguments as before yield An adequate choice for allows us to conclude the proof.

We choose constants as in the previous lemma.

From Lemma 2.5 and taking into account the shrinking map theorem on complete metric spaces, we guarantee the existence of which is a fixed point for in ; it is to say, and .

Let us define If we write and , then we have that for and , .

From we arrive at for every . This implies for every and .

This is valid for every . We define and for every and . From (2.23), it is straightforward to prove that is a solution of (2.13) and (2.14).

Moreover, holomorphy of in for every can be deduced from the recursion formula verified by the coefficients: This implies that is holomorphic in for every .

It only remiains to prove (2.17). Upper and lower bounds for the modulus of the elements in and , respectively, and usual calculations lead us to assure the existence of a positive constant such that for every , and for every and . This concludes the proof for .

Hypothesis (2.16) leads us to obtain (2.26) for .

Remark 2.6. If for every , then, for every , there exists small enough in such a way that Lemma 2.5 holds.

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

This section is devoted to the study of the same equation as in the previous section when the initial conditions are of a different nature. Proofs will only be sketched not to repeat calculations.

Let , and let , a bounded and open set with positive distance to the origin. stands for in this section. remain the same positive constants as in the previous section.

Definition 3.1. Let , and . denotes the vector space of functions such that is finite. Let . stands for the vector space of all formal series belonging to such that It is straightforward to check that the pair is a Banach space.

Lemma 3.2. Let and . One assumes that the following conditions hold: Then, there exists a constant (not depending on nor ) such that for every .

Proof. Let . The proof follows similar steps to those in Lemma 2.2. We have From the definition of the norm , we get with . Identical arguments to those in Lemma 2.2 allow us to conclude.

Lemma 3.3. Let be a holomorphic and bounded function defined on . Then, there exists a constant such that for every , every , and every .

Let and , as in Section 2 and . One considers the Cauchy problem with initial conditions where the functions belong to for every .

Theorem 3.4. Let Assumption A be fulfilled. One makes the following assumption on the initial conditions (3.9): there exist constants and such that for every and . Then, there exists , solution of (3.8) and (3.9) such that, if , then there exist and such that for every and .

Proof. The proof of Theorem 2.4 can be adapted here so details will be omitted.Let and . We consider the map from into itself defined as in (2.18) and construct as above. From (3.10) we derive for a positive constant not depending on nor .Lemmas 3.2, 3.3, and (3.12) allow us to affirm that one can find and such that the restriction of to the disc in is a Lipschitz shrinking map. Moreover, there exists which is a fixed point for in .If we put , then one gets for . This implies The formal power series turns out to be a solution of (3.8) and (3.9) verifying that is a holomorphic function in and the estimates (3.11) hold for .

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

4.1. A -Analog of the Laplace Transform and -Asymptotic Expansion

In this subsection, we recall the definition and several results related to the Jacobi Theta function and also a -analog of the Laplace transform which was firstly developed by Zhang in [7].

Let such that .

The Jacobi Theta function is defined in by From the fact that the Jacobi Theta function satisfies the functional equation , for , we have for every . The following lower bounds for the Jacobi Theta function will be useful in the sequel.

Lemma 4.1. Let . There exists (not depending on ) such that for every such that for all .

Proof. Let . From Lemma  5.1.6 in [17] we get the existence of a positive constant such that for every such that for all . Now, Let us fix . The function takes its maximum value at with , for certain . Taking into account that one can conclude the result. Here stands for the entire part.

Corollary 4.2. Let . For any there exists such that for every such that , for all .

From now on, stands for a complex Banach space.

For any and

The following definition corresponds to a -analog of the Laplace transform and can be found in [7] when working with sectors in the complex plane.

Proposition 4.3. Let and . One fixes an open and bounded set in such that . Let , and, be a holomorphic function defined in with values in such that let can be extended to a function defined in and for positive constants and . Let , and put where the path is given by . Then, defines a holomorphic function in and it is known as the -Laplace transform of following direction .

Proof. Let be a compact set and . From the parametrization of the path we have Let such that , and let . We have that satisfies for every . Corollary 4.2 and (4.9) yield for a positive constant . There exist such that for every , so that the last term in the chain of inequalities above is upper bounded by The result follows from this last expression.

Remark 4.4. If we let , then will only remain holomorphic in for certain .In the next proposition, we recall a commutation formula for the -Laplace transform and the multiplication by a polynomial.

Proposition 4.5. Let be an open and bounded set in and such that . Let be a holomorphic function on with values in the Banach space which satisfies the following estimates: there exist and such that Then, the function is holomorphic on and satisfies estimates in the shape above. Let and . One has the following equality: for every .

Proof. It is direct to prove that is a holomorphic function in and also that verifies bounds as in (4.14). From (4.2) we have , , so for every .

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

The following definition of a good covering firstly appeared in [17], p. 36.

Definition 4.6. Let be a pair of open intervals in each one of length smaller than , and let UI be the corresponding open bounded set in defined by Let be a finite family of tuple as above verifying(1), where is a neighborhood of 0 in , (2)the open sets , are four-by-four disjoint. Then, we say that is a good covering.

Definition 4.7. Let be a good covering. Let . We consider a family of open bounded sets in such that(1)there exists with , for all ,(2)for every and ,(3)for every , and , we have for every ,(4) for every . We say that the family is associated to the good covering . Let be an integer. For every , let be positive integers and a polynomial in , where is a subset of and are bounded holomorphic functions on some disc in . Let be a good covering such that for every .

Assumption B. We have

Definition 4.8. Let such that . Let such that , and that be a bounded holomorphic function on verifying for every . Assume moreover that can be extended to an analytic function on and for every . One says that the set is admissible.Let be a finite family of indices. For every , we consider the following singularly perturbed Cauchy problem: with as in (2.13), and with initial conditions where the functions are constructed as follows. Let be a family of open sets associated to the good covering . For every and , let be an admissible set. Let be a complex number in . We can assume that . If not, we diminish as desired. We put

Lemma 4.9. The function , constructed as above, turns out to be holomorphic and bounded on for every and all .

Proof. Let and . From (4.21), one has for every . Let and . Then, (4.25) can be upper bounded by , for some . Estimates in (4.9) hold so that Proposition 4.3 can be applied here. The third item in Definition 4.7 derives the holomorphy of on .We now prove the boundness of in its domain of definition. One has for every , where We only give bounds for the first integral. The estimates for the second one can be deduced following a similar procedure.Let such that . From Corollary 4.2 and (4.21) we deduce for some which does not depend on nor .

The following assumption is related to technical reasons appearing in the proof of Lemma 4.9 and Theorem 4.11.

Assumption C. There exist such that(C.1),(C.2),(C.3).The next remark clarifies the availability of these constants for a posed problem.

Remark 4.10. Assumptions A, B, and C strongly depend on the choice of whose modulus must rest near 1. For example, these assumptions on the constants are verified when taking . Then, the next theorem provides a solution for the equation with being holomorphic functions near the origin.

Theorem 4.11. Let Assumption A be fulfilled by the integers , for and also Assumptions B and C for . One considers the problem (4.22) and (4.23) where the initial conditions are constructed as above. Then, for every , the problem (4.22) and (4.23) has a solution which is holomorphic and bounded in . Moreover, for every , if are such that , then there exists a positive constant such that with with chosen as in Assumption C.

Proof. Let and . We consider the Cauchy problem (3.8) with initial conditions for . From Theorem 3.4 we obtain the existence of a unique formal solution and positive constants and such that for .Moreover, from Theorem 2.4 we get that the coefficients can be extended to holomorphic functions defined in and also the existence of positive constants and such that for .We choose . In the following estimates we will make use of the fact that for every . Proposition 4.3 allows us to calculate the -Laplace transform of with respect to for every . It defines a holomorphic function in . From the fact that is chosen to be a family associated to the good covering we derive that the function is a holomorphic and bounded function defined in . We can define, at least formally, in . If were a holomorphic function in , then Proposition 4.5 would allow us to affirm that (4.34) is an actual solution of (4.22) and (4.23). In order to end the first part of the proof it remian to demonstrate that (4.34) defines in fact a bounded holomorphic function in . Let and . We have where We now establish bounds for both integrals: Let as in Assumption C. From (4.32) and (4.7), the previous integral is bounded by Let be as in Assumptions (C.2) and (C.3).From and (2.5) in Definition 4.7, the previous inequality is upper bounded by where and The previous integral is uniformly bounded for and from the hypotheses made on these sets. The expression in (4.39) can be bounded by for an appropriate constant .The function takes its maximum at so each element in the image set is bounded by . Taking this to the expression above we get for certain .Assumption (C.3) applied to the last term in the previous expression allows us to deduce that the sum converges in the variable uniformly in the compact sets of .We now study . We have From (3.11) and (4.7) the previous integral is bounded by Similar calculations to those in the first part of the proof resting on Assumption C can be made so that the series is uniformly convergent with respect to the variable in the compact sets of , for . We will not go into detail not to repeat calculations. The estimates (4.43) and (4.46) imply the convergence of the series in (4.34) for every . The boundness of the -Laplace transform with respect to is guaranteed so the first part of the result is achieved.Let such that and . For every we have We can write where the path is given by , by , by , and by . Without loss of generality, we can assume that . For the first integral we deduce Similar estimates as in the first part of the proof lead us to bound the right part of the previous inequality by for certain . For any we have This yields We choose as in Assumption C.The integral corresponding to the path can be bounded following identical steps.We now give estimates concerning . It is worth saying that the function in the integrand is well defined for and does not depend on the index . This fact and the Cauchy theorem allow us to write for any where is the closed path defined in the following way: , is the arc of circumference from to , , and is the arc of circumference from to . Taking we derive Usual estimates lead us to prove that Moreover, From (4.54), (4.55), and (4.56) we obtain where . Taking into account Definition 4.7 and (4.31) we derive that the modulus of the last term in the previous equality is bounded by for adequate positive constants . From the standard estimates we achieve From (4.47), (4.48), (4.52), (4.59) and Assumption (C.3) we conclude the existence of a positive constant such that for every , with .

5. A -Gevrey Malgrange-Sibuya-Type Theorem

In this section we obtain a -Gevrey version of the so-called Malgrange-Sibuya theorem which allows us to reach our final main achievement: the existence of a formal series solution of problem (4.22) and (4.23) which asymptotically represents the actual solutions obtained in Theorem 4.11, meaning that, for every admits this formal solution as its -Gevrey asymptotic expansion in the variable .

In [4], a Malgrange-Sibuya-type theorem appears with similar aims as in this work. We complete the information there giving bounds on the estimates appearing for the -asymptotic expansion. This mentioned work heavily rests on the theory developed by Ramis et al. in [17].

In the present work, although -Gevrey bounds are achieved, the -Gevrey type involved will not be preserved, suffering an increase on the way.

The nature of the proof relies on the one concerning the classical Malgrange-Sibuya theorem for Gevrey asymptotics which can be found in [18].

Let be a complex Banach space.

Definition 5.1. Let be a bounded open set in and . One says a holomorphic function admits as its -Gevrey asymptotic expansion of type in if for every compact set there exist such that for every .The following proposition can be found, under slight modifications in Section  4 of [17].

Proposition 5.2. Let and be an open and bounded set. Let be a holomorphic function that admits a formal power series as its -Gevrey asymptotic expansion of type in . Then, if stands for the th formal derivative of for every , one has that admits as its -Gevrey asymptotic expansion of type in .

Proposition 5.3. Let , and let a holomorphic function in . Then, (i)if admits as its -Gevrey expansion of type , then for every compact set there exists with for every and every ;(ii)if for every compact set there exists with for every , then admits as its -Gevrey asymptotic expansion of type in , for every .

Proof. Let and . The function reaches its minimum for at . We deduce both results from standard calculations.

Definition 5.4. Let be a good covering at 0 (see Definition 4.6) and a holomorphic function in for when the intersection is not empty. The family is a -Gevrey -cocycle of type attached to a good covering if the following properties are satisfied.(1) admits as its -Gevrey asymptotic expansion of type on for every .(2) for every , and .(3)One has for all , .

Let and be an open and bounded set. stands for the Banach space of holomorphic and bounded functions in with the supremum norm.

Proposition 5.5. Let . We consider the family constructed in Theorem 4.11. Then, the set of functions defined by for is a -Gevrey -cocycle of type for every attached to the good covering .

Proof. The first property in Definition 5.4 directly comes from Theorem 4.11 and Proposition 5.3. The other two are verified by the construction of the cocycle.

We recall several definitions and an extension result from [13] which will be crucial in our work.

Definition 5.6. A continuous increasing function is a weight function if it satisfies the following:() there exists with for all ,(), (), () is convex. The Young conjugate associated to is defined by

Definition 5.7. Let be a nonempty compact set in . A jet on is a family where is a continuous function on for each .Let be a weight function. A jet on is said to be a -Whitney jet (of Roumieu type) on if there exist and such that and for every with and one has where .

denotes the linear space of -Whitney jets on .

Definition 5.8. Let be a nonempty compact set and a weight function in . A continuous function is in the sense of Whitney in if there exists a -Whitney jet on , such that .For an open set we define.

The following result establishes conditions on a weight function so that a jet in can be extended to an element in .

Theorem 5.9 (Corollary  3.10, [13]). For a given weight function , the following statements are equivalent. (1)For every nonempty closed set in the restriction map sending a function to the family of derivatives of in , , is a surjective map.(2)w is a strong weight function, that is to say,

Let . One considers the weight function defined by for and for . As the authors write in [13], the value of a weight function near the origin is not relevant for the space of functions generated in the sequel.

The following lemma can be easily verified.

Lemma 5.10. is a weight function.

Under this definition of one has

The spaces appearing in Definition 5.7 concerning this weight function are the following: for any nonempty compact set is the set of -Whitney jets on , which consists of every jet on such that there exist with

and such that for every and with one has

We derive that consists of the Whitney jets on such that there exist with and for every and all with ,

Theorem 5.11. is a strong weight function so that Theorem 5.9 holds.

Proof. We have

Remark 5.12. A continuous function which is in the sense of Whitney on a compact set is indeed in the usual sense in and verifies -Gevrey bounds of the same type. Moreover, we have for every and .The next result is an adaptation of Lemma  4.1.2 in [17]. Here, we need to determine bounds in order to achieve a -Gevrey-type result.

Lemma 5.13. Let be an open set in and a holomorphic function with being its -Gevrey asymptotic expansion of type in . Then, for any , the family of -complex derivatives of satisfies that, for every compact set and with , there exist such that for every . Here, one writes  for .

Proof. We will first state the result when . Indeed, we prove in this first step that the family of functions with -Gevrey asymptotic expansion of type in a fixed -spiral is closed under derivation. Proposition 5.2 turns out to be a particular case of this result.Let , be a compact set in , and consider another compact set such that . We define where denotes the limit of for tending to 0. Then, we have that Moreover, from Definition 5.1, there exist such that for every .Lemma 5.14 (Lemma  4.4.1 [17]). There exists such that for every .The Cauchy’s integral formula and -Gevrey expansion of guarantee the existence of a positive constant such that This yields the existence of such that An induction reasoning is sufficient to conclude the proof for every .We now study the case where and only give details for . For one only has to take into account that the derivatives of also admit -Gevrey asymptotic expansion of type and consider the function .If we treat two cases:if , then is contained in and we conclude from the Cauchy integral formula.If , then we bear in mind that the result is obvious when is a polynomial and write where . So, it is sufficient to prove (5.18) when . The result follows from -Gevrey bounds for , and usual estimates.

The following lemma generalizes Lemma  6 in [4].

Lemma 5.15. Let be a holomorphic function having as its -Gevrey asymptotic expansion of type on . Let be a compact set. Then, the function is a function in the sense of Whitney on the compact set

Proof. We consider the set of functions defined by From Lemma 5.13, function satisfies bounds as in (5.18). Written in terms of the elements in we have the existence of such that for, every , , for . Expression (5.14) can be directly checked from (5.24) and (5.18) for and . This yields that the set is an element in .

The next result allows us to glue together a finite number of jets in , for a given compact set .

Theorem 5.16 ([19]. Theorem II.1.3). Let be compact sets in . The following statements are equivalent. (i)The sequence is exact. and . (ii)Let and be such that for every . The function defined by if and if belongs to .(iii)If , then there exist such that for every . Here, denotes the function given by and for . stands for the distance from to the set .

Corollary 5.17 ([17], Lemma  4.3.6). Given nonempty compact sets in , if one puts , then the previous theorem holds for and .

As the authors remark in [17], condition (iii) in the previous result is known as the transversality condition which is more constricting than Łojasiewicz’s condition (see [20]).

The next proposition is devoted to show that the cocycle constructed in Proposition 5.5 splits in the space of functions in the sense of Whitney. Whitney-type extension results on (Theorems 5.9 and 5.11) will play an important role in the following step.

Proposition 5.18. Let be a good covering, and let be the -Gevrey -cocycle of type constructed in Proposition 5.5. One chooses a family of compact sets for , with , in such a way that is , where is a neighborhood of 0 in . Then, for all , there exists a function in the sense of Whitney on the compact set , with values in the Banach space , such that for all such that and, for every .

Proof. The proof follows similar arguments to those Lemma  3.12 in [17] and it is an adaptation of Proposition  5 in [4] under -Gevrey settings.Let such that . From Lemma 5.15, we have that the function is a function in the sense of Whitney on . In the following we provide the construction of for verifying (5.28).Let us fix any . We consider any function in the sense of Whitney on . By definition of the good covering the following cases are possible.Case 1. If there is at least one , such that but for every with , then we define for every . is a function in the sense of Whitney in . From Theorems 5.9 and 5.11, we can extend to a function in the sense of Whitney on . This extension is called . We have Case 2. There exist two different with such that . We first construct a function in the sense of Whitney on , , verifying We define for every and whenever . From (5.30), we have for every . From this, we can define From Theorem 5.16 and Corollary 5.17 we deduce that can be extended to a function in the sense of Whitney in . It is straightforward to check, from the way was constructed, that when and also for .These two cases solve completely the problem since nonempty intersection of four different compacts in is not allowed when working with a good covering. The functions in satisfy (5.28).

6. Existence of Formal Series Solutions and -Gevrey Expansions

In the current section we set the main result in this work. We establish the existence of a formal power series with coefficients belonging to which asymptotically represents the actual solutions found in Theorem 4.11 for the problem (4.22) and (4.23). Moreover, each actual solution turns out to admit this formal power series as -Gevrey expansion of a certain type in the -spiral where the solution is defined.

The following lemma will be useful in the following. We only sketch its proof. For more details we refer to [21].

Lemma 6.1. Let be an open and bounded set in . We consider (in the classical sense) verifying bounds as in (5.14) and (5.15) for every . Let be the solution of the equation Then, also verifies bounds such as those in (5.14) and (5.15) for .

Proof. Let be any extension of the function to with compact support which preserves bounds in (5.14) and (5.15) in . We have that solves (6.1). Here, , and stands for the Lebesgue measure in -plane. Bounds in (5.14) for the function come out from for every and and from the fact that the function is Lebesgue integrable in any compact set containing 0.On the other hand, satisfies the estimates in (5.15) from the Taylor formula with integral remainder.

We now give a decomposition result of the functions constructed in Theorem 4.11. The procedure is adapted from [4] under -Gevrey settings. For every , we write for the holomorphic function given by .

Proposition 6.2. There exist a function and a holomorphic function defined on the neighborhood of 0 such that for every .

Proof. From the definition of the cocycle in Proposition 5.5 and from Proposition 5.18 we derive whenever and . The function given by is well defined on , where is a closed neighborhood of .For every is a holomorphic function on so that the Cauchy-Riemann equations hold: This yields for every and .We have that can be extended to a function in the sense of Whitney on . This yields that is in the sense of Whitney on . In fact, their -Gevrey types coincide.From this, we deduce that is a function in the sense of Whitney on for every and also that for every and every because is a holomorphic function on . The previous equality is also true for from the fact that is in the sense of Whitney on .From Theorem 5.16 and Corollary 5.17 we derive that is a function in the sense of Whitney on .Taking into account Lemma 6.1 we derive the existence of a function in the usual sense, defined in and verifying -Gevrey bounds of a certain positive type, such that From this last expression we have that defines a holomorphic function on .For every , is a bounded -function in , and so it is the function . The origin turns out to be a removable singularity so the function can be extended to a holomorphic function defined on . The result follows from here.

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

Theorem 6.3. Under the same hypotheses as in Theorem 4.11, there exists a formal power series formal solution of Moreover, let , and let be any compact subset of . There exists such that the function constructed in Theorem 4.11 admits as its -Gevrey asymptotic expansion of type in .

Proof. Let , and let be any compact subset of .From Proposition 6.2 we can extend to a function in the sense of Whitney on . Let us fix . We consider the family associated to by Definition 5.7. We have because is holomorphic on .We have that is continuous at for every so we can define whenever . The estimates held by any function in the sense of Whitney (see Definition 5.7 for ) lead us to the existence of positive constants such that for every and . As a matter of fact, this shows that admits as its -Gevrey expansion of type in .The formal power series does not depend on . Indeed, from Theorem 4.11 we have that admits both and as -asymptotic expansion on whenever this intersection is not empty. We put for any . The function does not depend on for every . We write for . admits as its -Gevrey asymptotic expansion of type in for all .In order to achieve the result, it only remains to prove that is a formal solution of (6.10). Let . If we derive times with respect to in (6.10), we get that is a solution of for every and . Letting tend to 0 in (6.14), we obtain for every . The holomorphy of with respect to at 0 implies for near 0 and for every . Statements (6.14) and (6.15) conclude that is a formal solution of (6.10).

Acknowledgments

A. Lastra is partially supported by Ministerio de Educación, Programa Nacional de Movilidad de Recursos Humanos, Plan Nacional I-D+i 2008–2011. S. Malek is partially supported by the french ANR-10-JCJC 0105 project.

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