#### 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