On $q-$Gevrey asymptotics for singularly perturbed $q-$difference-differential problems with an irregular singularity

We study a $q-$analog of a singularly perturbed Cauchy problem with irregular singularity in the complex domain which generalizes a previous result by S. Malek in \cite{malek}. First, we construct solutions defined in open $q-$spirals to the origin. By means of a $q-$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 $q-$Gevrey asymptotic expansion (of certain type) of the actual solutions.


Introduction
We study a family of q-difference-differential equations of the following form: where q ∈ C such that |q| > 1, m 0,k , m 1,k are positive integers, b k , z are polynomials in z with holomorphic coefficients in on some neighborhood of 0 in C, and σ q is the dilation operator given by σ q X , t, z X , qt, z . As in previous works 1-3 , the map t, z → q m 0,k t, zq −m 1,k is assumed to be a volume shrinking map, meaning that the modulus of the Jacobian determinant |q| m 0,k −m 1,k is less than 1, for every 0 ≤ k ≤ S − 1.
In 4 , the second author studies a similar singularly perturbed Cauchy problem. In this previous work, the polynomial b k , z : s∈I k b ks z s is such that, for all 0 ≤ k ≤ S−1, I k is a finite subset of N {0, 1, . . .} and b ks are bounded holomorphic functions on some disc D 0, r 0 in C which verify that the origin is a zero of order at least m 0,k . The main From these results, we get a sequence W I β β∈N consisting of holomorphic functions in the variable τ so that the q-Laplace transform can be applied to its elements. In addition, the function X I , t, z : β≥0 L λ I q;1 W I β , t z β β! 1.4 turns out to be a holomorphic function defined in U I q −N × T × C which is a solution of the initial problem. Here, T is an adequate open half q-spiral to 0 and λ I corresponds to certain q-directions for the q-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 q-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 X I I∈I of solutions of 4.22 and 4.23 . The main goal is to study its asymptotic behavior at the origin in some sense. Let ρ > 0. One observes Theorem 4.11 that whenever the intersection U I ∩ U I is not empty we have |X I , t, z − X I , t, z | ≤ C 1 e − 1/A log 2 | | 1.5 for positive constants C 1 , A and for every , t, z ∈ U I q −N ∩ U I q −N × T × D 0, ρ . Equation The main result heavily rests on a Malgrange-Sibuya-type theorem involving q-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][15][16] . It is also worth saying that, although q-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 q-spiral to the origin times a q-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 q-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 q-Gevrey version of the Malgrange-Sibuya theorem, developed in Section 5.

A Cauchy Problem in Weighted Banach Spaces of Taylor Series
M, A 1 , C > 0 are fixed positive real numbers throughout the whole paper.
Let U, V be nonempty bounded open sets in C : C \ {0}, and let q ∈ C such that |q| > 1. We define We assume there exists M 1 > 0 such that |τ 1| > M 1 for all τ ∈ V q R and also that the distance from the set V to the origin is positive.

2.7
Taking into account the definition of the norm · β, ,V q R , we get and 0 < C U : max{| | : ∈ U}. Moreover, for every β ∈ N. Regarding condition 2.5 we obtain the existence of C 1 > 0 such that Abstract and Applied Analysis for every τ ∈ V q R and β ∈ N. Inequality 2.6 follows from 2.7 , 2.8 , and 2.10 :

Lemma 2.3. Let F , τ be a holomorphic and bounded function defined on
for every ∈ Uq −N , every δ > 0, and all v ∈ H , δ, V q R .
Proof. Direct calculations regarding the definition of the elements in H , δ, V q R allow us to conclude when taking Let S ≥ 1 be an integer. For all 0 ≤ k ≤ S − 1, let m 0,k , m 1,k be positive integers and b k , z s∈I k b ks z s a polynomial in z, where I k is a finite subset of N and b ks are holomorphic bounded functions on D 0, r 0 . We assume Uq −N ⊆ D 0, r 0 . We consider the following functional equation: We make the following assumption.
Assumption A. For every 0 ≤ k ≤ S − 1 and s ∈ I k , we have for every ∈ Uq −N and τ ∈ V q R .
where w τ, z : In the following lemma, we show that the restriction of A to a neighborhood of the origin in H , δ, V q R is a Lipschitz shrinking map for an appropriate choice of δ > 0. Lemma 2.5. There exist R > 0 and δ > 0 (not depending on ) such that Proof. Let R > 0 and 0 < δ < 1.
For the first part we consider W τ, z ∈ B 0, R ⊆ H , δ, V q R . Lemmas 2.2 and 2.3 can be applied so that for a positive constant C 2 . We conclude this first part from an appropriate choice of R and δ > 0.
For the second part we take W 1 , W 2 ∈ B 0, R ⊆ H , δ, V q R . Similar arguments as before yield

2.22
An adequate choice for δ > 0 allows us to conclude the proof.
We choose constants R, δ 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 W τ, z ∈ H , δ, V q R which is a fixed point for A in B 0, R ; it is to say, W τ, z ,δ,V q R ≤ R and A W τ, z W τ, z . Let us define W τ, z : ∂ −S z W τ, z w τ, z .
Abstract and Applied Analysis 9 This is valid for every ∈ Uq −N . We define W , τ, z : W τ, z and W β , τ : W β, τ for every , τ ∈ Uq −N × V q R , z ∈ C and β ≥ S. From 2.23 , it is straightforward to prove that W , τ, z β≥0 W β , τ z β /β! is a solution of 2.13 and 2.14 . Moreover, holomorphy of W β in Uq −N × V q R for every β ≥ 0 can be deduced from the recursion formula verified by the coefficients: It only remiains to prove 2.17 . Upper and lower bounds for the modulus of the elements in Uq −N and V q R , respectively, and usual calculations lead us to assure the existence of a positive constant R 1 > 0 such that for every β ≥ S, and for every ∈ Uq −N and τ ∈ V q R . This concludes the proof for β ≥ S. Hypothesis 2.16 leads us to obtain 2.26 for 0 ≤ k ≤ S − 1.
Remark 2.6. If s > 0 for every s ∈ I k , 0 ≤ k ≤ S − 1, then, for every R > 0, there exists small enough δ > 0 in such a way that Lemma 2.5 holds.

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 1 < ρ 0 , and let U ⊆ C , a bounded and open set with positive distance to the origin. D ρ 0 stands for D 0, ρ 0 \ {0} in this section. M, A 1 , C remain the same positive constants as in the previous section.

Lemma 3.2.
Let s, k, m 1 , m 2 ∈ N, δ > 0 and ∈ D 0, r 0 \ {0}. One assumes that the following conditions hold: Then, there exists a constant The proof follows similar steps to those in Lemma 2.2. We have From the definition of the norm | · | β, ,Ḋ ρ 0 , we get Identical arguments to those in Lemma 2.2 allow us to conclude.

Lemma 3.3. Let F , τ be a holomorphic and bounded function defined on
Let S, r 0 , m 0,k , m 1,k and b k , as in Section 2 and ρ 0 > 0. One considers the Cauchy problem Abstract and Applied Analysis 11 with initial conditions Theorem 3.4. Let Assumption A be fulfilled. One makes the following assumption on the initial conditions 3.9 : there exist constants Δ > 0 and 0 < M < M such that Proof. The proof of Theorem 2.4 can be adapted here so details will be omitted. Let ∈ D 0, r 0 \ {0} and 0 < δ < 1. We consider the map A from O Ḋ ρ 0 z into itself defined as in 2.18 and construct w τ, z as above. From 3.10 we derive for a positive constant C 3 not depending on nor δ. Lemmas 3.2, 3.3, and 3.12 allow us to affirm that one can find R > 0 and δ > 0 such that the restriction of A to the disc D 0, R in H 2 , δ,Ḋ ρ 0 is a Lipschitz shrinking map.
Abstract and Applied Analysis The formal power series 14 turns out to be a solution of 3.8 and 3.9 verifying that W β , τ is a holomorphic function in D 0, r 0 \ {0} ×Ḋ ρ 0 and the estimates 3.11 hold for β ≥ 0.

A q-Analog of the Laplace Transform and q-Asymptotic Expansion
In this subsection, we recall the definition and several results related to the Jacobi Theta function and also a q-analog of the Laplace transform which was firstly developed by Zhang in 7 .
The Jacobi Theta function is defined in C by From the fact that the Jacobi Theta function satisfies the functional equation xqΘ x Θ qx , for x / 0, we have for every m ∈ Z. The following lower bounds for the Jacobi Theta function will be useful in the sequel.
Proof. Let δ > 0. From Lemma 5.1.6 in 17 we get the existence of a positive constant Let us fix |x|. The function Abstract and Applied Analysis 13 takes its maximum value at t 0 log |x|/ log |q| 1/2 with f t 0 C 2 exp log 2 |x|/2 log |q| |x| 1/2 , for certain C 2 > 0. Taking into account that one can conclude the result. Here · stands for the entire part.
From now on, H, · H stands for a complex Banach space. For any λ ∈ C and δ > 0 The following definition corresponds to a q-analog of the Laplace transform and can be found in 7 when working with sectors in the complex plane.
for positive constants C 1 > 0 and 0 < M < 1/2 log |q|. Let π q log q n≥0 1 − q −n−1 −1 , and put where the path 0, ∞λ is given by t ∈ −∞, ∞ → q t λ. Then, L λ q;1 F defines a holomorphic function in R λ,q,δ and it is known as the q-Laplace transform of f following direction λ .
Proof. Let K ⊆ R λ,q,δ be a compact set and z ∈ K. From the parametrization of the path 0, ∞λ we have Abstract and Applied Analysis Let 0 < ξ 1 < 1 such that 0 < M < ξ 1 /2 log |q|, and let t ∈ R. We have that w q t λ/z satisfies |1 q k w| > δ for every k ∈ Z. Corollary 4.2 and 4.9 yield for a positive constant L 1 . There exist 0 < A < B such that A ≤ |z| ≤ B for every z ∈ K, so that the last term in the chain of inequalities above is upper bounded by

4.13
The result follows from this last expression.
In the next proposition, we recall a commutation formula for the q-Laplace transform and the multiplication by a polynomial.
Proof. It is direct to prove that mφ is a holomorphic function in V q R ∪Ḋ ρ 0 and also that mφ verifies bounds as in 4.14 . From 4.
Abstract and Applied Analysis 15

Analytic Solutions in a Parameter of a Singularly Perturbed Cauchy Problem
The following definition of a good covering firstly appeared in 17 , p. 36. 4.17 Let I be a finite family of tuple I as above verifying 2 the open sets U I q −N , I ∈ I are four-by-four disjoint.
Then, we say that U I q −N I∈I is a good covering. We say that the family { V I I∈I , T} is associated to the good covering U I q −N I∈I . Let S ≥ 1 be an integer. For every 0 ≤ k ≤ S − 1, let m 0,k , m 1,k be positive integers and b k , z s∈I k b ks z s a polynomial in z, where I k is a subset of N and b ks are bounded holomorphic functions on some disc D 0, r 0 in C, 0 < r 0 ≤ 1. Let U I q −N I∈I be a good covering such that U I q −N ⊆ D 0, r 0 for every I ∈ I. for every , τ ∈ Uq −N × V q R ∪Ḋ ρ 0 . One says that the set {W, W UV , ρ 0 } is admissible. Let I be a finite family of indices. For every I ∈ I, we consider the following singularly perturbed Cauchy problem: with b k as in 2.13 , and with initial conditions where the functions φ I,j , t are constructed as follows. Let { V I I∈I , T} be a family of open sets associated to the good covering U I q −N I∈I . For every 0 ≤ j ≤ S − 1 and I ∈ I, let {W j , W U I ,V I ,j , ρ 0 } be an admissible set. Let λ I be a complex number in V I ∩ D 0, ρ 0 . We can assume that r 0 < 1 < |λ I |. If not, we diminish r 0 as desired. We put φ I,j , t : L λ I q;1 τ −→ W U I ,V I ,j , τ , t .

4.24
Lemma 4.9. The function , t → φ I,j , t , constructed as above, turns out to be holomorphic and bounded on U I q −N × T for every I ∈ I and all 0 ≤ j ≤ S − 1.
Proof. Let I ∈ I and 0 ≤ j ≤ S − for some C j > 0 which does not depend on nor t.
The following assumption is related to technical reasons appearing in the proof of Lemma 4.9 and Theorem 4.11.
Proof. Let δ > 0 and I ∈ I. We consider the Cauchy problem 3.8 with initial conditions ∂ j z W , τ, 0 W j , τ for 0 ≤ j ≤ S − 1. From Theorem 3.4 we obtain the existence of a unique formal solution W , τ, z β≥0 W β , τ z β /β ∈ O D 0, r 0 \ {0} ×Ḋ ρ 0 z and positive constants C 3 > 0 and 0 < δ 1 < 1 such that Moreover, from Theorem 2.4 we get that the coefficients W β , τ can be extended to holomorphic functions defined in U I q −N × V I q R and also the existence of positive constants C 2 and 0 < δ 2 < 1 such that for , τ ∈ U I q −N × V I q R . We choose λ I ∈ V I ∩ D 0, ρ 0 . In the following estimates we will make use of the fact that | | ≤ |λ I | for every ∈ D 0, r 0 \ {0} . Proposition 4.3 allows us to calculate the q-Laplace transform of W β with respect to τ for every β ≥ 0, L λ I q;1 W β , τ . It defines a holomorphic function in U I q −N × R λ I ,q,δ . From the fact that { V I I∈I , T} is chosen to be a family associated to the good covering U I q −N I∈I we derive that the function

4.36
We now establish bounds for both integrals:

4.38
Let a 1 , a 2 be as in Assumptions C.2 and C.3 . The previous integral is uniformly bounded for ∈ D 0, r 0 \ {0} and t ∈ T from the hypotheses made on these sets. The expression in 4.39 can be bounded by × e −ξlog 2 |t|/2 log |q| e ξ log |λ I / | log |t|/ log |q| ,

4.41
for an appropriate constant C 2 > 0. The function s → s γβ e −αlog 2 s takes its maximum at s e γβ/ 2α so each element in the image set is bounded by e γβ 2 / 4α . Taking this to the expression above we get C ξ e ξlog 2 |q s λ I / t|/2 log |q| ds.

4.45
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 z in the compact sets of C, for , t ∈ U I q −N × T. 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 z ∈ C. The boundness of the q-Laplace transform with respect to is guaranteed so the first part of the result is achieved.
Let I, I ∈ I such that U I q −N ∩ U I q −N / ∅ and ρ > 0. For every , t, z ∈ U I q −N ∩ U I q −N × T × D 0, ρ we have We can write
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 This yields

4.52
We choose ξ as in Assumption C.

22
Abstract and Applied Analysis The integral corresponding to the path γ 2 can be bounded following identical steps. We now give estimates concerning γ 3 − γ 4 . It is worth saying that the function in the integrand is well defined for , τ ∈ D 0, r 0 \ {0} ×Ḋ ρ 0 and does not depend on the index I ∈ I. This fact and the Cauchy theorem allow us to write for any n ∈ N where Γ n γ n,1 γ 5 − γ n,2 − γ n,3 is the closed path defined in the following way: s ∈ −n, 0 → γ n,1 s λ I q s ,γ 5 is the arc of circumference from λ I to λ I , s ∈ −n, 0 → γ n,2 s λ I q s , and γ n,3 is the arc of circumference from λ I q −n to λ I q −n . Taking n → ∞ we derive

4.58
Abstract and Applied Analysis 23 for adequate positive constants C 3 , C 3 . From the standard estimates we achieve

A q-Gevrey Malgrange-Sibuya-Type Theorem
In this section we obtain a q-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 I ∈ I, X I admits this formal solution as its q-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 qasymptotic expansion. This mentioned work heavily rests on the theory developed by Ramis et al. in 17 . In the present work, although q-Gevrey bounds are achieved, the q-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 H be a complex Banach space.  : t, z ∈ T × D 0, ρ −→ X I , t, z − X I , t, z 5.5 for I, I ∈ I is a q-Gevrey H T,ρ -cocycle of type A for every attached to the good covering U I q −N I∈I .
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 w : 0, ∞ → 0, ∞ is a weight function if it satisfies the following: The Young conjugate associated to φ, φ : 0, ∞ → R is defined by Definition 5.7. Let K be a nonempty compact set in R 2 . A jet on K is a family F f α α∈N 2 where f α : K → C is a continuous function on K for each α ∈ N 2 . Let w be a weight function. A jet F f α α∈N 2 on K is said to be a w-Whitney jet of Roumieu type on K if there exist m > 0 and M > 0 such that and for every l ∈ N, α ∈ N 2 with |α| ≤ l and x, y ∈ K one has where R l x F α y : f α y − |α β|≤l 1/β! f α β x y − x β .

26
Abstract and Applied Analysis The following result establishes conditions on a weight function so that a jet in E {w} K can be extended to an element in E {w} R 2 . Theorem 5.9 Corollary 3.10, 13 . For a given weight function w, the following statements are equivalent.
1 For every nonempty closed set K in R 2 the restriction map sending a function f ∈ E {w} R 2 to the family of derivatives of f in K, f α | K α∈N 2 ∈ E {w} K , is a surjective map.
2 w is a strong weight function, that is to say, Let k 1 1/4 log |q|. One considers the weight function defined by w 0 t k 1 log 2 t for t ≥ 1 and w 0 t 0 for 0 ≤ t ≤ 1. 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. w 0 is a weight function.
Under this definition of w 0 one has log q y 2 , y ≥ 0.

5.11
The spaces appearing in Definition 5.7 concerning this weight function are the following: for any nonempty compact set K ⊆ R 2 , E {w 0 } K is the set of w 0 -Whitney jets on K, which consists of every jet F f α α∈N 2 on K such that there exist m ∈ N, M > 0 with and such that for every l ∈ N and α ∈ N 2 with |α| ≤ l one has We derive that E {w 0 } K consists of the Whitney jets on K such that there exist C 1 , H > 0 with f α x ≤ C 1 H |α| q A |α| 2 /2 , x ∈ K, α ∈ N 2 , 5.14 Abstract and Applied Analysis 27 and for every x, y ∈ K and all l ∈ N, α ∈ N 2 with |α| ≤ l,

5.15
Theorem 5.11. w 0 is a strong weight function so that Theorem 5.9 holds.
Proof. We have Remark 5.12. A continuous function f which is w 0 − C ∞ in the sense of Whitney on a compact set K is indeed C ∞ in the usual sense in Int K and verifies q-Gevrey bounds of the same type. Moreover, we have f k x, y ∂ k 1 x ∂ k 2 y f x, y , 5.17 for every k k 1 , k 2 ∈ N 2 and x, y ∈ Int K . The next result is an adaptation of Lemma 4.1.2 in 17 . Here, we need to determine bounds in order to achieve a q-Gevrey-type result. of n-complex derivatives of f satisfies that, for every compact set K ⊆ U and k, m ∈ N with k ≤ m, there exist C 1 , H > 0 such that for every a , b ∈ Kq −N ∪ {0}. Here, one writes ∂ l f 0 l!a l for l ∈ N.
Proof. We will first state the result when b 0. Indeed, we prove in this first step that the family of functions with q-Gevrey asymptotic expansion of type A > 0 in a fixed q-spiral is closed under derivation. Proposition 5.2 turns out to be a particular case of this result.
Let m ∈ N, K be a compact set in U, and consider another compact set K 1 such that K ⊆ K 1 ⊆ U. We define Proof. We consider the set of functions φ k 1 ,k 2 k 1 ,k 2 ∈N 2 defined by φ k 1 ,k 2 : i k 2 ∂ k 1 k 2 f , k 1 , k 2 ∈ N 2 , 1 , 2 ∈ K .

5.24
From Lemma 5.13, function f satisfies bounds as in 5.18 . Written in terms of the elements in φ k 1 ,k 2 k 1 ,k 2 ∈N 2 we have the existence of C 1 , H > 0 such that for, every k 1 , k 2 ∈ N 2 , m ≥ 0, for x 1 , y 1 , x 2 , y 2 ∈ K . Expression 5.14 can be directly checked from 5.24 and 5.18 for b 0 and m k. This yields that the set φ k 1 ,k 2 k 1 ,k 2 ∈N 2 is an element in E {w 0 } K .
The next result allows us to glue together a finite number of jets in E {w 0 } K , for a given compact set K. Theorem 5. 16 19 . Theorem II.1.3 . Let K 1 , K 2 be compact sets in R 2 . The following statements are equivalent.
ii Let f 1 ∈ E {w 0 } K 1 and f 2 ∈ E {w 0 } K 2 be such that iii If K 1 ∩ K 2 / ∅, then there exist A 3 , A 4 > 0 such that for every x ∈ K 1 . Here, M denotes the function given by M 0 0 and M t inf n∈N t n M n for t > 0. dist x, K stands for the distance from x to the set K.