#### 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 U_{I} 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