On a Partial q-Analog of a Singularly Perturbed Problem with Fuchsian and Irregular Time Singularities

A family of linear singularly perturbed difference differential equations is examined. )ese equations stand for an analog of singularly perturbed PDEs with irregular and Fuchsian singularities in the complex domain recently investigated by A. Lastra and the author. A finite set of sectorial holomorphic solutions is constructed by means of an enhanced version of a classical multisummability procedure due to W. Balser. )ese functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairs q-Gevrey and Gevrey bounds.


Introduction
In this work, we focus on singularly perturbed linear partial qdifference differential equations which couple two categories of operators acting both on the time variable, so-called qdifference operators of irregular type and Fuchsian differential operators.As a seminal reference concerning analytic and algebraic aspects of q-difference equations with irregular type, refer [1], and for a far reaching investigation of Fuchsian ordinary and partial differential equations, refer [2].
Our equations are presented in the following manner + P t, z, ϵ, t k σ q;t , tz t , z z  u(t, z, ϵ) for vanishing initial data u(0, z, ϵ) ≡ 0, where k, δ D , m D ≥ 1 are integers, σ q;t represents the dilation map t ⟶ qt acting on time t for some real number q > 1, and Q(X), R D (X) stand for polynomials in C[X].e main block P(t, z, ϵ, V 1 , V 2 , V 3 ) is polynomial in the arguments t, V 1 , V 2 , V 3 , holomorphic in the perturbation parameter ϵ on a disc D(0, ϵ 0 ) ⊂ C centered at 0 and in the space variable z on a horizontal strip of the form H β � z ∈ C/|Im(z)| < β  , for some β > 0. e forcing term f(t, z, ϵ) is analytic relatively to (z, ϵ) ∈ H β × D(0, ϵ 0 ) and defines an entire function w.r.t t in C with (at most) q-exponential growth (see (60), for precise bounds).
is paper is a natural continuation of the study [3] by Lastra and Malek and will share the same spine structure.Indeed, in [3], we aimed attention at the next problem + H z, ϵ, t k+1 z t , tz t , z z  y(t, z, ϵ) for vanishing initial data y(0, z, ϵ) ≡ 0, where Q(X), R D (X), H(z, ϵ, V 1 , V 2 , V 3 ) stand for polynomials in their arguments X, V 1 , V 2 , V 3 as above and h(t, z, ϵ) is like the forcing term f(t, z, ϵ) but with (at most) exponential growth in t.Under convenient conditions put on the shape of (2), we are able to construct a set of genuine bounded holomorphic solutions expressed as a Laplace transform of order k along a halfline and Fourier inverse integral in space z: where the Borel/Fourier map V p (τ, m, ϵ) is itself set forth as a Laplace transform of order k ′ � kδ D /m D : where W p (u, m, ϵ) has (at most) exponential growth along L c p and exponential decay in phase m on R. e resulting maps y p (t, z, ϵ) are therefore expressed as iterated Laplace transforms following a so-called multisummability procedure introduced by Balser, see [4].ese functions define bounded holomorphic functions on domains T × H β × E p for a well-selected bounded sector T at 0 and where E � E p   0≤p≤ς− 1 is a set of sectors which covers a full neighborhood of 0 and is called a good covering (cf.Definition 6).Additionally, the partial maps ϵ ⟼ y p (t, z, ϵ) share on E p a common asymptotic expansion  y(t, z, ϵ) �  n≥0 y n (t, z)ϵ n with bounded holomorphic coefficients y n (t, z) on T × H β .
is asymptotic expansion turns out to be (at most) of Gevrey order 1/κ with κ � kk ′ /(k + k ′ ), meaning that we can single out two constants C p , M p > 0 such that sup t∈T,z∈H β y p (t, z, ϵ) −  n− 1 m�0 y m (t, z)ϵ m for all n ≥ 1, all ϵ ∈ E p .We plan to obtain a similar statement for the problem under study (1).Namely, we will construct a set of genuine sectorial solutions to (1) and describe their asymptotic expansions as ϵ borders the origin.We first notice that our main problem (1) can be seen as a q-analog of (2), where the irregular differential operator t k+1 z t is replaced by the discrete operator t k σ q;t . is terminology originates from the basic observation that the expression f(qt) − f(t)/(qt − t) approaches the derivative f ′ (t) as q tends to 1. Here, as mentioned in the title, we qualify the q-analogy as partial since the Fuchsian operator tz t is not discretized in the process.is suggests that, in the building procedure of the solutions (that will follow the same guideline as in [3]), the classical Laplace transform of order k shall be supplanted by a q-Laplace transform of order k as it was the case in the previous work [5] of the author where a similar problem was handled.However, due to presence of the Fuchsian operator tz t , we will see that a single q-Laplace transform is not enough to construct true solutions and that a new mechanism of iterated q-Laplace and classical Laplace transforms is required.Furthermore, we witness that this enhanced multisummability procedure has a forthright effect on their asymptotic expression w.r.t ϵ.Namely, the expansions in the perturbation parameter are neither of classical Gevrey order as displayed in (3) nor of q-Gevrey order 1/k as in [5] (meaning that Γ(1 + (n/κ)) has to be replaced by q n 2 /2k in the control term of (3)).e asymptotic expansions we exhibit present a double scale structure which has a q-Gevrey leading part with order 1/k and a subdominant tail of Gevrey order 1/kδ D , that we call Gevrey asymptotic expansion of mixed order (1/kδ D ; (q, 1/k)) (cf.Definition 8).Such a coupled asymptotic structure has already been observed in another setting by Lastra et al. in [6].Indeed, we considered linear q-difference differential Cauchy problems with the shape tσ q;t   r 2 zz z  r 1 z S z X(t, z) � B z, tσ q;t , σ q − 1 ;z , z z  X(t, z), (6) for suitably chosen analytic Cauchy data and properly selected complex number q ∈ C * with |q| > 1, where r 1 ≥ 0 and r 2 , S ≥ 1 are integers and B stands for a polynomial.When r 1 ≥ 1, the Fuchsian operator (zz z ) r 1 is responsible of the classical Gevrey part of the asymptotic expansion  X(t, z) �  n≥0 X n (z)t n of the true solution X(t, z) which is shown to be of mixed order (r 1 /r 2 ; (q, 1)) (in the sense of Definition 8) outside some q-spiral λq Z for some λ ∈ C * w.r.t t near 0, uniformly in z in the vicinity of the origin.Here, the solutions are expressed through a single q-Laplace transform and the Γ((r 1 /r 2 )n) contribution in the asymptotics emerges from a discrete set of singularities that accumulates at 0 in the Borel plane.
It is worthwhile mentioning that the approach which consists in building solutions by means of iterated q-Laplace and Laplace transforms stems from a new work by Yamazawa.In [7], he examines linear q-difference differential equations of the form L t, σ q;t , z x  u(t, x) � f(t, x), (8) for the given holomorphic forcing term f(t, x) near the origin and where L(t, V 1 , V 2 ) is a polynomial in V 1 , V 2 with holomorphic coefficients w.r.t t near 0. Under special conditions on the structure of (4), he is able to construct a genuine solution u(t, x) obtained as a small perturbation of iterated truncated q-Laplace and Laplace of order 1 transforms of the iterated Borel and q-Borel transforms of a formal solution  u(t, x) �  k≥1 u k (x)t k of (4).Furthermore, he gets in particular that u(t, x) has  u(t, x) as asymptotic expansion of mixed order (1; (q, 1)) w.r.t t, uniformly in x near 0.
Notice that in our paper, the solutions are built up as complete iterated q-Laplace and classical Laplace transforms that are shown to be exact solutions of our problem (1). is is why the process we follow can actually be understood as an enhanced version of the multisummation mechanism introduced by Balser, see [4].
In a larger framework, this work is a contribution to the promising and fruitful realm of research in q-difference and q-difference-differential equations in the complex domain.For recent important advances in this area, we mention in particular the works by Tahara and Yamazawa [8][9][10].Notice that the fields of applications of q-difference equations have also encountered a rapid growth in the last years.

2
Abstract and Applied Analysis Some forefront studies in this respect are given, for instance, by [11][12][13] and references therein.Now, we describe a little more precisely our main results obtained in eorems 1 and 3. Namely, under convenient restrictions on the shape of (1) detailed in the statement of eorem 1, we can manufacture a family of bounded holomorphic solutions u p (t, z, ϵ) on domains T × H β × E p for a suitable bounded sector T at 0, H β a strip of width β > 0, and E p belonging to a good covering in C * , which can be displayed as a q-Laplace transform of order k along a halfline L c p � R + exp( �� � − 1 √ c p ) and Fourier integral: e q-Borel/Fourier map W d p (u, m, ϵ) is itself shaped as a classical Laplace transform of order kδ D along L c p : where w d p (h, m, ϵ) has (at most) q-exponential growth of some order 0 < k 1 < k along L c p (see ( 180)) and exponential decay in phase m ∈ R. In eorem 3, we explain the reason for which all the partial functions ϵ ⟼ u p (t, z, ϵ) share a common asymptotic expansion  u(t, z, ϵ) �  m≥0 h m (t, z)ϵ m on E p with bounded holomorphic coefficients h m (t, z) on T × H β , which turns out to be of mixed order (1/kδ D ; (q, 1/k)).
is last result leans on a new version of the classical Ramis-Sibuya theorem fitting the above asymptotics, which is fully expounded in eorem 2.
Our paper is arranged as follows.
In Section 2, we recall the definition of the classical Laplace transform and its q-analog.We also put forward some classical identities for the Fourier transform acting on functions spaces with exponential decay.
In Section 3, we set forth our main problem (33) and we discuss the formal steps leading to its resolution.Namely, a first part is devoted to the inquiry of solutions among q-Laplace transforms of order k and Fourier inverse integrals of Borel maps W with q-exponential growth on unbounded sectors and exponential decay in phase leading to the first main integrodifferential q-difference equation (69) that W is asked to fulfill.A second undertaking suggests to seek for W, as a classical Laplace transform of suitable order kδ D of a second Borel map w with again appropriate behaviour.e expression w is then contrived to solve a second principal integro q-difference equation (81).
In Section 4, bounds for linear convolution and q-difference operators acting on Banach spaces of functions with q-exponential growth are displayed.e second key equation (81) is then solved within these spaces at the hand of a fixed point argument.
In Section 5, genuine holomorphic solutions W of the first principal auxiliary equation (69) are built up and sharp estimates for their growth are provided (cf.( 146) and (147)).
In Section 6, we achieve our goal in finding a set of true holomorphic solutions (176) to our initial problem (33).
In Section 7, the existence of a common asymptotic expansion of the Gevrey type with mixed order (1/kδ D ; (q, 1/k)) is established for the solutions set up in Section 6. e decisive technical tool for its construction is detailed in eorem 2.

Transforms of Order k, and Fourier Inverse Maps
Let k ′ ≥ 1 be an integer.We remind the reader the definition of the Laplace transform of order k ′ as introduced in [14].
as some unbounded sector with bisecting direction d ∈ R and aperture 2δ > 0 and D(0, ρ) as a disc centered at 0 with radius ρ > 0. Consider a holomorphic function w: S d,δ ∪ D(0, ρ) ⟶ C that vanishes at 0 and withstands the bounds.
ere exist C > 0 and K > 0 such that for all τ ∈ S d,δ .We define the Laplace transform of w of order k ′ in the direction d as the integral transform { }, where c depends on Tand is chosen in such a way that cos(k ′ (c − arg(T))) ≥δ 1 , for some fixed real number δ 1 > 0. e function L d k′ (w)(T) is well defined, holomorphic, and bounded on any sector: where 0 < θ < (π/k ′ ) + 2δ and 0 < R < δ 1 /K.If one sets w(τ) �  n≥1 w n τ n , the Taylor expansion of w, which converges on the disc D(0, ρ/2), the Laplace transform as Gevrey asymptotic expansion of order 1/k ′ . is means that for all 0 < θ 1 < θ, two constants C, M > 0 can be selected with the bounds:  (11), its Laplace transform L d k′ (w)(T) does not depend on the direction d in R and represents a Abstract and Applied Analysis bounded holomorphic function on D(0, R 1/k′ ), whose Taylor expansion is represented by the convergent series X(T) �  n≥1 w n Γ(n/k ′ )T n on D(0, R 1/k′ ).
Let k ≥ 1 be an integer and q > 1 be a real number.At the next stage, we display the definition of the q-Laplace transform of order k which was used in a former work of Malek [5].
Let us first recall some essential properties of the Jacobi eta function of order k defined as the Laurent series: for all x ∈ C * . is analytic function can be factorized as a product known as Jacobi's triple product formula: for all x ∈ C * , from which we deduce that its zeros are the set of real numbers − q m/k /m ∈ Z  .We recall the next lower bound estimates on a domain, bypassing the set of zeroes of Θ q 1/k (x), from [5] Lemma 3, which are crucial in the sequel.
ere exists a constant C q,k > 0 depending on q, k and independent of Δ such that Definition 2. Let ρ > 0 be a real number and S d be an unbounded sector centered at 0 with bisecting direction d ∈ R.
Let f: D(0, ρ) ∪ S d ⟶ C be a holomorphic function, continuous on the adherence D(0, ρ), such that there exist constants K, α > 0 and δ > 1 with where c is a halfline in the direction c. e following lemma is a slightly modified version of Lemma 4 from [5].
Lemma 2. Let Δ > 0 chosen as in Lemma 1 above.e integral transform L c q;1/k (f(x))(T) defines a bounded holomorphic function on the domain R c,Δ ∩ D(0, r 1 ) for any radius 0 < r 1 ≤ q − (1/k)(α+1) /2, where Notice that the value due to the Cauchy formula.e next lemma describes conditions under which the q-Laplace transform defines a convergent series near the origin.Lemma 3. Let f: C ⟶ C be an entire function with Taylor expansion f(x) �  n≥1 f n x n fulfilling bound (19) for all x ∈ C.
en, its q-Laplace transform of order k, L d q;1/k (f)(T), does not depend on the direction d ∈ R and represents a bounded holomorphic function on D(0, r 1 ) with the restriction 0 < r 1 ≤ q − (1/k)(α+1) /2 whose Taylor expansion is given by the convergent series Y(T) �  n≥1 f n q n(n− 1)/2k T n . Proof.
e proof is a direct consequence of the next formulas: , where the last equality follows (for instance) from identity (4.7) from [15], for all n ≥ 1.
We restate the definition of some family of Banach spaces mentioned in [14].
We set E (β,μ) as the vector space of continuous functions h: R ⟶ C such that is finite.e space E (β,μ) endowed with the norm ‖.‖ (β,μ) becomes a Banach space.Finally, we remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in [14].Definition 4. Let f ∈ E (β,μ) with β > 0 and μ > 1. e inverse Fourier transform f is given by for all x ∈ R. e function F − 1 (f) extends to an analytic bounded function on the strips for all given 0 < β ′ < β.
(a) Define the function m ⟼ ϕ(m) � imf(m) which belongs to the space E (β,μ− 1) .en, the next identity occurs.(b) Take g ∈ E (β,μ) and set 4 Abstract and Applied Analysis as the convolution product of f and g.

Layout of the Principal Initial Value Problem and Associated Auxiliary Problems
We set k ≥ 1 as an integer.Let m D , δ D ≥ 1 be integers.We set We consider a finite set I of N 3 that fulfills the next feature, whenever (l 0 , l 1 , l 2 ) ∈ I and we set nonnegative integers Δ l ≥ 0 with and l ∈ I be polynomials such that for all m ∈ R, all l ∈ I.
We consider a family of linear singularly perturbed initial value problems for vanishing initial data u(0, z, ϵ) ≡ 0. Here, q > 1 stands for a real number and the operator σ q;t is defined as the dilation by q acting on the variable t through σ q;t u(t, z, ϵ) � u(qt, z, ϵ). e coefficients c l (z, ϵ) are built in the following manner.For each l ∈ I, we consider a function m ⟼ C l (m, ϵ) that belongs to the Banach space E (β,μ) for some β, μ > 0, which depends holomorphically on the parameter ϵ on some disc D(0, ϵ 0 ) with radius ϵ 0 > 0 and for which one can find a constant C l > 0 with sup ϵ∈D 0,ϵ 0 ( ) We construct as the inverse Fourier transform of the map C l (m, ϵ) for all l ∈ I.As a result, c l (z, ϵ) is bounded holomorphically w.r.t ϵ on D(0, ϵ 0 ) and w.r.t z on any strip H β′ for 0 < β ′ < β in view of Definition 4.
e presentation of the forcing term requires some preliminary groundwork.We consider a sequence of functions m ⟼ ψ n (m, ϵ), for n ≥ 1, that belongs to the Banach space E (β,μ) with the parameters β, μ > 0 given above and which relies analytically and is bounded w.r.t ϵ on the disc D(0, ϵ 0 ).We assume that the next bounds, sup ϵ∈D 0,ϵ 0 ( ) hold for all n ≥ 1 and given constants K 0 , T 0 > 0. We define the formal series for some real number 0 < k 1 < k.We introduce the next Banach space.
Definition 5. Let k 1 , β, μ, r, α > 0 and q, δ > 1 be real numbers.Let U d be an open unbounded sector with bisecting direction d ∈ R centered at 0 in C. We denote Exp the vector space of complex valued continuous functions (u, m) ⟼ h(u, m) on the adherence U d ∪ D(0, r) × R, which are holomorphic w.r.t u on U d ∪ D(0, r) and such that the norm is finite.One can check that the normed space (Exp q (k 1 ,β,μ,α,r) , ‖.‖ (k 1 ,β,μ,α,r) ) represents a Banach space.
Remark 1. e spaces above are faint modifications of the Banach spaces already introduced in the works of Dreyfus and Lastra [16][17][18].
e next lemma is a proper adjustment of Lemma 5 out of [5] to the new Banach spaces from Definition 5. Lemma 4. Let T 0 be fixed as in (36).We take a number α > 0 such that Let k 1 , β, μ be chosen as above.en, the function (u, m) ⟼ ψ(u, m, ϵ) belongs to the Banach space Exp q (k 1 ,β,μ,α,r) for any unbounded sector U d , any disc D(0, r).Moreover, one can find a constant C 1 > 0 (depending on q, k 1 , α, T 0 ) with Proof.ound (36) implies that Abstract and Applied Analysis (41) According to the elementary fact that the polynomial h(x) � x(n − 1 − α) − (k 1 /2)(x 2 /log(q)) admits its maximum value (log(q)/2k 1 )(n − 1 − α) 2 at x � (log(q)/k 1 ) (n − 1 − α), we deduce by means of the change of variable for all n ≥ 1. erefore, we deduce that which converges, provided that (39) holds, whenever ϵ ∈ D(0, ϵ 0 ).We define as the Laplace transform of ψ(u, m, ϵ) w.r.t u of order k ′ in direction d ∈ R. Notice that two constants K 1 , K 2 > 0 (depending on k 1 , q, α, δ, k ′ ) can be found such that for all u ∈ C. As a result, owing to bound (40) and the last part of Definition 1, we deduce that Ψ d does not depend on the direction d and can be written as a convergent series: w.r.t τ near the origin.Now, we fix some real number k 2 such that 0 for all n ≥ 1. is inequality is a consequence of the Stirling formula, which states that as x tends to +∞ and from the existence of a constant K 3 > 0 (depending on q, k 1 , k 2 , k ′ ) with for all n ≥ 1.Consequently, it turns out that Ψ d (τ, m, ϵ) represents an entire function w.r.t τ such that for all τ ∈ C. Furthermore, owing to bound (42), we know that for all n ≥ 1, all τ ∈ C. Henceforth, we get the next global bounds provided that for all τ ∈ C, m ∈ R, and ϵ ∈ D(0, ϵ 0 ).Next, we set as the q-Laplace transform of Ψ d (u, m, ϵ) w.r.t u of order k in direction d and Fourier inverse integral w.r.t m.We put for all n ≥ 1.We first provide bounds for this sequence of functions.Namely, we can get a constant C μ,β,β′ > 0 (relying on μ, β, β ′ ) with 6 Abstract and Applied Analysis for all n ≥ 1, whenever ϵ ∈ D(0, ϵ 0 ) and z belongs to the horizontal strip H β′ for some 0 < β ′ < β (see Definition 4).Owing to Lemma 3, we deduce that the function F d (T, z, ϵ) converges near the origin w.r.t T, where it carries the next Taylor expansion: for all ϵ ∈ D(0, ϵ 0 ) and z ∈ H β′ .In particular, the function F d is independent of the direction d chosen.
We now show that F d (T, z, ϵ) represents an entire function w.r.t T and supply explicit upper bounds.Namely, in accordance with (47), we obtain Again, estimate (42) yields for all T ∈ C, all n ≥ 1, where κ 2 > 0 is defined by 1/κ 2 � (1/k 2 ) − (1/k).By gathering the two last above inequalities, the next global estimates can be figured out: for all T ∈ C, all z ∈ H β′ and ϵ ∈ D(0, ϵ 0 ), provided that Lastly, we define the forcing term f as a time rescaled version of F d , that represents a bounded holomorphic function w.r.t z ∈ H β′ and ϵ ∈ D(0, ϵ 0 ) and an entire function w.r.t t with q-exponential growth of order κ 2 .
roughout this paper, we are looking for time rescaled solutions of (33) of the form u(t, z, ϵ) � U(ϵt, z, ϵ). (63) As a consequence, the expression U(T, z, ϵ), through the change of variable T � ϵt, is asked to solve the next singular problem: At the onset, we seek for a solution U(T, z, ϵ) that can be expressed as an integral representation via a q-Laplace transform of order k and Fourier inverse integral: where the inner integration is performed along a halfline Overall this section, we assume that the partial functions u ⟼ W(u, m, ϵ) have at most qexponential growth of order k on some unbounded sector S d centered 0 with bisecting direction d and m ⟼ W(u, m, ϵ) belong to the Banach space E (β,μ) mentioned in Definition 3, whenever ϵ ∈ D(0, ϵ 0 ).Precise bounds will be given later in Section 5. Here, we assume that L c ⊂ S d ∪ 0 { }.Our aim is now the presentation of a related problem fulfilled by the expression W(u, m, ϵ).We first need to state two identities which concern the action of q-difference and Fuchsian operators on q-Laplace tranforms.

□ Lemma 5.
e actions of the q-difference operators T l 0 σ l 1 q;T for integers l 0 , l 1 ≥ 0 and the Fuchsian differential operator Tz T are given by Proof. e first identity is a direct consequence of the commutation formula (76) displayed in Proposition 6 from [5].For the second, a derivation under the integral followed by an integration by parts implies the sequence of equalities: Abstract and Applied Analysis from which the forecast formula follows since the map u ⟼ W(u, m, ϵ) is assumed to possess a growth of q-exponential order k and vanishes at u � 0.
e application of the above identities (66) and (67) in a row with ( 26) and ( 28) leads to the first integrodifferential qdifference equation fulfilled by the expression W(u, m, ϵ) as long as U c (T, z, ϵ) solves (64): We turn now to the second stage of the procedure.Solutions of this latter equation are expected to be found in the class of Laplace transforms of order k ′ since by construction Ψ d (τ, m, ϵ) owns this structure after (44).Namely, we take for granted that where { } where U d represents an unbounded sector centered at 0 with bisecting direction d.Within this step, we assume that the expression (u, m) ⟼ w(u, m, ϵ) belongs to the Banach space Exp q (k 1 ,β,μ,α,r) introduced in Definition 5, for all ϵ ∈ D(0, ϵ 0 ), where the constants k 1 , β, μ, and α are selected accordingly to the construction of the forcing term f(t, z, ϵ).
e next lemma has already been stated in our previous work [3].□ Lemma 6.For all integers l ≥ 1, positive integers a q,l ≥ 1 and 1 ≤ q ≤ l can be found such that With the help of this last expansion, equation ( 69) can be recast in the form Abstract and Applied Analysis is last prepared shape allows us to apply the next lemma that repharses formula (8.7) p. 3630 from [19], in order to express all differential operators appearing in (72) in terms of the most basic one τ k′+1 z τ .
By convention, we take for granted that the above sum Indeed, by construction of the finite set I, we can represent the next integers in a specific way: where e h,l 0 � l 0 − hk ′ ≥ 1 for all (l 0 , l 1 , l 2 ) ∈ I and 1 As a consequence, we can further expand the next piece of (72) in its final convenient form: and we can remodel equation ( 72) in such a way that it contains only primitive building blocs: Similarly to our previous technical Lemma 5, we disclose some useful commutation formulas dealing with the actions of the basic irregular operator τ k′+1 z τ , multiplication by monomials τ m′ , and of the q-difference operator σ q;τ , Lemma 8 (1) e action of the differential operators τ k′+1 z τ on W(τ, m, ϵ) is given by (2) Let m ′ ≥ 1 be an integer.e action of the multiplication by τ m′ on W(τ, m, ϵ) is described through the next formula: (3) Let c ∈ Z be an integer.e action of the operator σ c q;τ is represented through the following integral transform: Proof. e first two formulas have already been given in our previous works [3,20].We focus on the third equality.By definition, Abstract and Applied Analysis and if one deforms the path of integration L c through u � q c v which keeps the path invariant since q c ∈ R + , we get formula (79).Departing from the arranged equation ( 76) with the help of Lemma 8, we can exhibit an ancillary problem satisfied by the expression w(u, m, ϵ), □

An Integral q-Difference Equation with
Complex Parameter e objective of this section is the construction of a unique solution of equation (81) just established overhead.is solution will be built among the Banach space displayed in Definition 5. Within the next three propositions, continuity of linear convolutions and q-difference operators acting on Exp q (k 1 ,β,μ,α,r) is discussed.Proposition 1.Let k ′ ≥ 1 be an integer and c 1 > 0 and c 2 , c 3 be real numbers such that k ′ (c 2 + c 3 + 2) is an integer that are submitted to the next constraint: for all u ∈ U d ∪ D(0, r), for some constant M c 1 > 0. en, the linear function represents a continuous map from the Banach space Exp q (k 1 ,β,μ,α,r) into itself.In other words, some constant M 1 > 0 (depending on k ′ , c 1 , c 2 , c 3 ) can be found with for all u ∈ U d ∪ D(0, r), whenever m ∈ R. By definition, the next upper bounds Abstract and Applied Analysis for all u ∈ U d ∪ D(0, r), all m ∈ R.Under condition (82), the expected bound (85) follows.

□
Proposition 2. Let a ∈ C be a complex number, c 1 ≥ 0 be an integer, and c 2 ≥ 0 be a real number withstanding the following condition: en, we can sort a constant M 2 > 0 (depending on c 1 , c 2 , q, α, δ, r, a) with Proof. e proof is proximate to the one of Proposition 1 in [5] and similar to the one of Proposition 1 from [18].We provide, however, a complete proof for the sake of a better readability.
Let f(u, m) belong to Exp r) .By definition, we can perform the next factorization Since the contractive map u ⟼ u/q c 2 keeps the domain U d ∪ D(0, r) invariant, we deduce with  M 2 � (1/q c 2 )sup u∈U d ∪D(0,r) A(u), where We observe that where In the remaining part of the proof, we show that  M 2.2 is also finite.We first need to rearrange the pieces of A(u).Namely, we expand Abstract and Applied Analysis Since log(1 + x) ∼ x as x ⟶ 0, we get two constants A 1 , A 2 ∈ R (depending on r, δ, q, c 2 ) with for all u ∈ U d , |u| > r.Gathering (95) and ( 96) with (97) gives rise to the bound which is finite owing to (89). (99) for some constant M b > 0. en, there exists a constant M 3 > 0 (depending on Q, R, and μ) such that whenever f belongs to E (β,μ) and g belongs to Exp Proof. e proof shares the same ingredients as the one of Proposition 2 of [3].Again, we give a thorough explanation of the result.We take f inside E (β,μ) and select g belonging to Exp q (k 1 ,β,μ,α,r) .We first recast the norm of the convolution operator as follows: where 12 Abstract and Applied Analysis By construction of the polynomials Q and R, one can sort two constants Q, R > 0 with for all m, m 1 ∈ R. As a consequence of (102), (104), and (100), with the help of the triangular inequality |m| ≤ |m 1 | + |m − m 1 |, we are led to the bounds where is a finite constant under the first and last restriction of (99) according to the estimates of Lemma 2.2 from [21] or Lemma 4 of [22].
We disclose now additional assumptions on the leading polynomials Q(X) and R D (X).ese requirements will be essential in the transformation of our main problem (81) into a fixed point equation, as explained later in Proposition 4.
With this respect, the guideline is close to our previous study [3].Namely, we assume the existence of an unbounded sectorial annulus: where direction d Q,R D ∈ R and aperture η Q,R D > 0 for some given inner radius r Q,R D > 0 with the feature: We consider the next polynomial: In the following, we need lower bounds of the expression P m (u) with respect to both variables m and u.In order to achieve this goal, we can factorize the polynomial w.r.t u, namely, where its roots q l (m) can be displayed explicitely as for all 0 ≤ l ≤ k′m D − 1, for all m ∈ R.
We set an unbounded sector U d centered at 0, a small disc D(0, r), and we adjust the sector S Q,R D in a way that the next condition holds.A constant M > 0 can be chosen with Indeed, inclusion (108) implies in particular that all the roots q l (m), 0 ≤ l ≤ k ′ m D − 1 remain a part of some neighborhood of the origin, i.e., satisfy |q l (m)| ≥ 2r for an appropriate choice of r > 0. Furthermore, when the aperture η Q,R D > 0 is taken close enough to 0, all these roots q l (m) stay inside a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in C * .We assign a sector U d with By construction, the quotients q l (m)/u live outside some small disc centered at 1 in C for all u ∈ U d , m ∈ R, 0 ≤ l ≤ k ′ m D − 1. en, (112) follows.
We are now ready to supply lower bounds for P m (u).□ Lemma 9. A constant C P > 0 (depending on k, k ′ , m D , q, M) can be found with Proof.Departing from factorization (110), the lower bound (112) entails for all u ∈ U d ∪ D(0, r). e next proposition discusses sufficient conditions under which a solution w d (u, m, ϵ) of the main integral qdifference equation (81) can be built up in the space Exp q (k 1 ,β,μ,α,r) .

□ Proposition 4. Let us assume the next extra requirements:
Abstract and Applied Analysis for all l � (l 0 , l 1 , l 2 ) ∈ I. Furthermore, for each l � (l 0 , l 1 , l 2 ) ∈ I, we set an integer p l 0 ,l 1 such that and we take for granted that holds.en, for an appropriate choice of the constants C l > 0 (see (34)) that need to be taken close enough to 0 for all l ∈ I, a constant ϖ > 0 can be singled out in a manner that equation (81) gets a unique solution (u, m) ⟼ w d (u, m, ϵ) in the space Exp q (k 1 ,β,μ,α,r) with the condition: whenever ϵ ∈ D(0, ϵ 0 ), where U d ,r are chosen as above and k 1 , β, μ, α are specified in Section 3 on the way to the construction of the forcing term f(t, z, ϵ).
Proof. e proof relies strongly on the next lemma which discusses contractive properties of a linear map.

□
Lemma 10.For all ϵ ∈ D(0, ϵ 0 ), we define the map H ϵ as Under the additional requirement ( 116)-( 118), one can select the constants C l > 0, for l ∈ I, and a real number ϖ > 0 in a way that this map acts on some neighborhood of the origin of the space Exp q (k 1 ,β,μ,α,r) in the following way: (i) e inclusion holds, where B(0, ϖ) stands for the closed ball of radius ϖ centered at 0 in Exp r) , for all ϵ ∈ D(0, ϵ 0 ).(ii) e map H ϵ is contractive, namely, whenever w 1 , w 2 ∈ B(0, ϖ), for all ϵ ∈ D(0, ϵ 0 ).
Proof.We first control the forcing term.Owing to bound (76) in Lemma 4, together with (114), we can exhibit a constant C 1 > 0 (relying on q, k 1 , α, T 0 ) with where K 0 > 0 is a constant that is set in (36), whenever ϵ ∈ D(0, ϵ 0 ).We deal with the first property (121).Let us take w(τ, m) in Exp q (k 1 ,β,μ,α,r) under the constraint ‖w(τ, m)‖ (k 1 ,β,μ,α,r) ≤ ϖ.We fix some complex number ω d such that ω d ∉ U d ∪ D(0, r), and we redraft the norm of the next integral expression as follows: 14 Abstract and Applied Analysis for all l ∈ I, 1 ≤ h ≤ l 2 , where p l 0 ,l 1 ≥ 0 is an integer chosen as in (117) and We observe that a constant M B > 0 (depending on q, l 0 , l 1 , p l 0 ,l 1 , k, k ′ , m D , ω d , M) can be picked up with , we obtain for all u ∈ U d ∪ D(0, r), m ∈ R where the right-hand side is finite owing to the suitable choices of ω d and p l 0 ,l 1 in (118).Under requirements (32) and ( 116), an application of Proposition 3 yields a constant M 3.1 > 0 (depending on R D , R l and μ) such that where Conditions ( 116) and (117) allow us to call back Proposition 2 in order to get a constant M 2.1 > 0 (depending on l 0 , l 1 , p l 0 ,l 1 , k, q, α, δ, r, ω d ) with where Lastly, Proposition 1 gives rise to constants M ω d ,l 0 > 0 (depending on ω d , l 0 ) and M 1.1 > 0 (depending on k ′ , l 0 , l 2 ) with By compiling (128)-(132), we obtain We now turn to the second principal pieces of H ϵ .Following the same lines of arguments as above, we obtain that Abstract and Applied Analysis where for all l ∈ I, 2 ≤ h ≤ l 2 and 1 ≤ p ≤ h − 1.In order to give bounds for J 3 , we make use of Proposition 1 which affords a constant M 1.2 > 0 (depending on k ′ , l 0 , l 2 ) with By combining ( 134) and (136), we obtain for all l ∈ I, 2 ≤ h ≤ l 2 and 1 ≤ p ≤ h − 1.
In the next step, we impose the constants C l > 0, l ∈ I, to stay close enough to 0 in order that a constant ϖ > 0 can be singled out with e collection of (123), 133 and (137) submitted to condition (138) yields the inclusion (121).
e next part of the proof is devoted to the explanation of the contractive property (122).Indeed, consider two functions w 1 (u, m) and w 2 (u, m) inside the ball B(0, ϖ) ⊂ Exp q (k 1 ,β,μ,α,r) .en, an application of the two inequalities (133) and (137) overhead leads to is time, we require the constants C l > 0, l ∈ I, to withstand the next inequality 16 Abstract and Applied Analysis (141) Owing to (139) and (140), under demand (141), we obtain (122).
In conclusion, we choose the constants C l > 0, l ∈ I in order that both (138) and (141) hold conjointly. is yield Lemma 10.
We go back to the core of Proposition 4. For ϖ > 0, chosen as in the lemma above, we consider the closed ball B(0, ϖ) ⊂ Exp d (k 1 ,β,μ,α,r) that stands for a complete metric space for the distance d(x, y) � ‖x − y‖ (k 1 ,β,μ,α,r) .According to the same lemma, we observe that H ϵ induces a contractive application from (B(0, ϖ), d) into itself.en, according to the classical contractive mapping theorem, the map H ϵ carries a unique fixed point that we set as w d (u, m, ϵ); meaning that that belongs to the ball B(0, ϖ), for all ϵ ∈ D(0, ϵ 0 ).Furthermore, the function w d (u, m, ϵ) depends holomorphically on ϵ in D(0, ϵ 0 ).Let the term be taken from the right to the left-hand side of (81) and then divide by the polynomial P m (u) defined in (109).ese operations allows (81) to be exactly recast into equation (142) above.Consequently, the unique fixed point w d (u, m, ϵ) of H ϵ obtained overhead in B(0, ϖ) precisely solves equation (81).□

An Integrodifferential q-Difference Equation with a Complex Parameter
In this section, we build up a solution W d (τ, m, ϵ) to the integrodifferential q-difference equation (69) with the shape of a Laplace transform of order k ′ in direction d.Furthermore, we provide sharp bounds of this solution for large values of its q-Borel and Fourier variables τ and m.
In the second part of the proof, we are scaled down to provide bounds for the next associated function: when x > 0 is chosen large enough.e next lemma holds.

□ Lemma 11. One can select two constants
for all x ≥ 2δ k′ .
Proof.We first make the change of variable  r � r k′ /x in the integral above: On the other hand, we need the next expansions: We cut the integral expression in two pieces: Abstract and Applied Analysis where provided that x ≥ 2δ k′ .We control the first piece E 1 (x).We observe that log(( r) 1/k′ + (δ/x 1/k′ )) ≤ 0 when  r ∈ [0, (1 − (δ/x 1/k′ )) k′ ].From (154), we deduce the inequalities log 2 ( r) for all  r ∈ [0, (1 − (δ/x 1/k′ )) k′ ] and x ≥ 2δ k′ .erefore, By construction, a constant E 1.2 > 0 (depending on k ′ , k 1 , δ, q) can be found with for all x ≥ 2δ k′ .In a second step, we evaluate the part E 2 (x).Expansion (154) affords us to write where Besides, we can check that there exists a constant provided that  r ≥ (1 − (δ/x 1/k′ )) k′ , when x ≥ 2δ k′ .We deduce that when x ≥ 2δ k′ .Furthermore, one can sort a constant E 2.2 > 1 (depending on k ′ ) such that with when x ≥ 2δ k′ .We perform the linear change of variable h �  r/2 in this latter integral Abstract and Applied Analysis in order to express it in terms of the Gamma function Γ(x).
In the final part of the proof, the function W d (τ, m, ϵ) is shown to fulfill the second main equation (69).In this respect, we tread rearwards the construction discussed in Section 3. Indeed, according to the fact that w d (u, m, ϵ) solves (81) and appertains to the space Exp q (k 1 ,β,μ,α,r) , for a well-chosen sector U d , the three identities of Lemma 8 can be applied in order to check that W d (τ, m, ϵ) is a genuine solution of the integrodifferential-q-difference equation in prepared form (76). Ultimately, a successive play of Lemma 7 followed by Lemma 6 transforms equation ( 76) into the expected one (69).□

Construction of a Finite Set of True Sectorial Solutions to the Main Initial Value Problem
We return to the first part of the formal constructions undertaken in Section 3 in view of the gain made in solving the two auxiliary problems (81) and (69) throughout Sections 4 and 5.
We need to state the definition of a good covering in C * , and we introduce a fitted version of a so-called associated sets of sectors to a good covering which is analog to the one proposed in our previous work [3].Definition 6.Let ς ≥ 2 be an integer.We consider a set E of open sectors E p centered at 0, with radius ϵ 0 > 0 for all 0 ≤ p ≤ ς − 1 for which the next three properties hold: (i) e intersection E p ∩ E p+1 is not empty for all 0 ≤ p ≤ ς − 1 (with the convention that E ς � E 0 ) (ii) e intersection of any three elements of E is empty (iii) e union ∪ ς− 1 p�0 E p equals U\ 0 { } for some neighborhood U of 0 in C en, the set of sectors E is named a good covering of C * .Definition 7. We consider centered at 0 with bisecting direction d p ∈ R and small opening θ , for some integer k ′ ≥ 1 (iv) A fixed bounded sector T centered at 0 with radius r T > 0 and a disc D(0, r) suitably selected in a way that the next features are conjointly satisfied: (a) Bound (112) holds, provided that u ∈ U d p ∪ D(0, r), for all 0 ≤ p ≤ ς − 1 (b) e set S fulfills the next properties: (1) e intersection S d p ∩ S d p+1 is not empty for all 0 ≤ p ≤ ς − 1 (with the convention that ) For all 0 ≤ p ≤ ς − 1, all ϵ ∈ E p and all t ∈ T: where with Δ > 0 any fixed real number close to 0.
When the above features are verified, we say that the set of data E, U, S, T, D(0, r)   is admissible.We settle now the first principal result of the work.We construct a set of actual holomorphic solutions to the main initial value problem (33) defined on sectors E p , 0 ≤ p ≤ ς − 1, of a good covering in C * .Besides, we are able to monitor the difference between consecutive solutions on the intersections E p ∩ E p+1 .Theorem 1.We ask the record of requirements ( 29)-( 32), ( 34), ( 36), ( 39), ( 53), ( 61

), (108), and (116)-(118) to hold. Let us distinguish an admissible set of data
Abstract and Applied Analysis as described in the definition above.en, for a suitable choice of the constants C l > 0 (c.f. ( 34)) close enough to 0 for all l ∈ I, a collection u p (t, z, ϵ)   0 ≤ p ≤ ς− 1 of true solutions of (33) can be singled out.More precisely, each function u p (t, z, ϵ) stands for a bounded holomorphic map on the product (T ∩ D(0, σ)) × H β′ × E p for any given 0 < β ′ < β and appropriate small radius σ > 0. Additionally, u p (t, z, ϵ) is represented as a q-Laplace transform of order k and Fourier inverse integral: where where (u, m) ⟼ w d p (u, m, ϵ) belongs to the Banach space Exp q (k 1 ,β,μ,α,r) for the unbounded sector U d p , provided that ϵ ∈ D(0, ϵ 0 ).
Finally, some constants A p , B p > 0 can be found with where by convention, we set u ς (t, z, ϵ) � u 0 (t, z, ϵ).
Proof.We first single out an admissible set of data A. Under the requirements enounced in eorem 1, Proposition 5 can be called in order to find a family of functions: r), w.r.t ϵ on D(0, ϵ 0 ), and continuous relatively to m ∈ R, coming along with a constant ϖ d p > 0 such that for all u ∈ U d p ∪ D(0, r), m ∈ R, and ϵ ∈ D(0, ϵ 0 ).Furthermore, the function W d p (τ, m, ϵ) solves the first auxiliary integrodifferential q-difference equation (69) on S d p × R × D(0, ϵ 0 ) and suffers the bounds: for some constants ρ We now revisit the first stage of the formal construction from Section 3. Namely, we set the next q-Laplace transform of order k and Fourier inverse map Paying heed to the upper bound (181) and to Lemma 2 together with basic features about Fourier transforms discussed in Definition 4, we notice that U c p (T, z, ϵ) stands for (a) A bounded holomorphic function w.r.t T on a domain R d p ,Δ ∩ D(0, r 0 ) for some small radius r 0 > 0, where R d p ,Δ is described in (173) (b) A bounded holomorphic application relatively to the couple (z, ϵ) on H β′ × D(0, ϵ 0 ), for any given 0 < β ′ < β Additionally, since W d p (τ, m, ϵ) solves (69), Lemma 5 leads to the claim that U c p (T, z, ϵ) must fulfill the singular equation ( 64 In conclusion, the function defined as represents a bounded holomorphic function w.r.t t on T ∩ D(0, σ) for some σ > 0 close enough to 0, ϵ ∈ E p , z ∈ H β′ for any given 0 < β ′ < β, owing to assumption 3 of Definition 7.Moreover, u p (t, z, ϵ) solves the main initial value problem (33) on the domain In the second half of the proof, we explain bound (178).Here, we follow a similar roadmap based on path deformation arguments as in our previous work [3].Indeed, for l � p, p + 1, the partial function is holomorphic on the sector S d l .By the Cauchy theorem, we can bend each straight halfline L c l , l � p, p + 1 into the union of three curves with appropriate orientation depicted as follows: Abstract and Applied Analysis 21 (1) A halfline L c l ,r 1 � [r 1 , +∞) for a given real number r 1 > 0 (2) An arc of circle with radius r 1 denoted C r 1 ,c l ,c p,p+1 joining the point r 1 exp( ) which is taken inside the intersection S d p ∩ S d p+1 (that is assumed to be nonempty, see Definition 7, 2.1) to the halfline As a result, the difference u p+1 − u p can be decomposed into a sum of five integrals along these curves: e izm dτ τ dm e izm dτ τ dm Bounds for the first piece, are now considered.e arguments followed are proximate to the ones displayed in the proof of eorem 1 from [5].
Owing to Lemma 1 and bound (181), we obtain for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .We need the next two expansions: Hence, for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .We now specify estimates for some pieces of these last upper bounds.Namely, since log(1 + x) ∼ x as x tends to 0, we get a constant A 1.1 > 0 (depending on r 1 , δ) such that log(r)log 1 for all r ≥ r 1 .Since 0 < ϵ 0 < 1, we also notice that 22 Abstract and Applied Analysis exp whenever r 1 ≤ r ≤ 1 and 0 < σ < 1 together with provided that r ≥ 1.Finally, there exists a constant K k,r 1 ,q > 0 (depending on k, r 1 , q) with sup Inequality (189) together with the collection of bounds (190)-(194) yield two constants I 1.1 > 0 and for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .We want to express these last bounds in terms of sequences now.e discussion hinges on the next lemma.
Proof.By performing the change of variable x � log(|ϵ|) with the help of the computation already undertaken in Lemma 4, we obtain exp for all given ϵ ∈ C * and integer N ≥ 1. Consequently to (195) and (196), two constants I 1.3 , I 1.4 > 0 (depending on I 1.1 , I 1.2 , q, k) can be picked up with With a similar discussion, we can exhibit comparable bounds for the next term: e izm dτ τ dm                     .
(199) Namely, two constants I 2.1 > 0 and for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .Furthermore, we can single out two constants I 2.3 , I 2.4 > 0 (resting on I 2.1 , I 2.2 , q, k) such that In the next step, we turn to the first integral along an arc of circle: (202) Making use of Lemma 1 and (181), gives rise to the inequality for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .We require once more the expansion:

Abstract and Applied Analysis
We deduce that Owing to the hypothesis 0 < ϵ 0 < 1, we check that (191) holds and bearing in mind (194), we arrive at the existence of two constants I 3.1 > 0 and I 3.2 ∈ R (depending on the constants k, q, Δ, k 2 , δ, r 1 , ρ , β, β ′ ) with for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ .Calling back Lemma 12 gives rise to two additional constants I 3.3 , I 3.4 > 0 (subjected to I 3.1 , I 3.2 , q, k) with for all ϵ ∈ E p+1 ∩ E p , t ∈ T ∩ D(0, σ), and z ∈ H β′ , for all given integers N ≥ 1. e second integral along an arc of circle In the remaining part of the proof, we inspect the last integral along the segment: for some fixed 0 < δ 2 < δ 1 close enough to 0 and any given positive real number Δ 2 > 0, under the convention that Proof.We first observe that all the maps u ⟼ w d p (u, m, ϵ), 0 ≤ p ≤ ς − 1, are analytic continuations on the sector U d p of a unique holomorphic function that we name u ⟼ w(u, m, ϵ) on the disc D(0, r) which suffers the same bound (180).Furthermore, the application u ⟼ w(u, m, ϵ)exp(− (u/τ) k′ )/u is holomorphic on D(0, r) when τ ∈ S d p+1 ∩ S d p , and its integral is therefore vanishing along an oriented path described as the union of (a) A segment linking 0 to (r/2)exp( where the integrations paths are two halflines and an arc of circle staying aside from the origin that are depicted as follows: We consider the first integral along a halfline in the above splitting: e direction c p+1 ′ (which might depend on τ) is properly chosen in order that for all τ ∈ S d p+1 ∩ S d p , for some fixed δ 1 > 0. Besides, let Δ 2 > 0 be any given positive real number (even close to 0). en, we can find a constant B 1.1 > 0 (depending on k 1 , k ′ , δ, q, α, r, Δ 2 ) such that for all s ≥ r/2.According to estimate (180), we obtain that for a given 0 < δ 2 < δ 1 .
Onwards, we take for granted that the real number r 1 > 0 selected in the above deformation (1), (2), and (3) suffers restriction (213) and 0 < r 1 ≤ 1. Bound (212) in a row with Lemma 1 yields: where for all ϵ ∈ E p+1 ∩ E p , all t ∈ T ∩ D(0, σ) and all z ∈ H β′ .Bound control given below in (235) are now provided for this parameter depending on last integral.
e ongoing reasoning leans on the next elementary lemma.

□ Lemma 14
(1) e next inequality holds for all integers N ≥ 1 and all positive real numbers r > 0, where C � (1/M W p ) 1/k′ and C Γ,k′ > 0 is a constant depending on k ′ .

Abstract and Applied Analysis
Proof.For the first item (1), using the change of variable x � (1/r) k′ we observe that for all integers N ≥ 1.On the other hand, from the Stirling formula (48), we get a constant C Γ,k′ > 0 (depending on k ′ ) such that for all x ≥ 1/k ′ .Gathering ( 231) and ( 232) yields ( 229).
e second item (2) can be treated in a similar way through the successive changes of variables y � r/|ϵt| and x � log(y) by using the computation already carried out in Lemma 4, whenever N ≥ 1.
In the next proposition, we show that difference (178) of neighboring solutions of (33) turn out to be flat functions for which accurate bounds are displayed.□ Proposition 6.Let u p (t, z, ϵ)   0≤p≤ς− 1 be the set of actual solutions of (33) built up in eorem 1. en, we can find constants ϑ > 0 and  A p ,  B p > 0 (which rely on A p , B p , k, k ′ , q) such that where Abstract and Applied Analysis where by convention u ς � u 0 .
Proof.Let A p , B p > 0 be real numbers and k, k ′ ≥ 1 be integers.We define the function for all x > 0. Keeping in mind the Stirling formula (48), we can find two constants C, D > 0 (depending on k ′ ) with for all integers N ≥ 1.Hence, where A p,1 � A p C and B p,1 � B p D. In the next part of the proof, we exhibit explicit bounds for the function h(x).We follow a similar strategy as in the recent work [7].We select a real number ϑ 1 > 0 small enough in order that for all given x ∈ (0, ϑ 1 ), there exists a positive real number We focus on the integer N �  N, where x denotes the floor function.By construction, we have N ≤  N. erefore, which implies that and hence On the other hand, we can express  N in term of the variable x by means of the Lambert function.Namely, we set W(z) as the principal branch of the Lambert function defined on (− e − 1 , +∞) and which solves the functional equation for all z ∈ (− e − 1 , +∞).Since relation (243) can be recast in the form we deduce that Furthermore, owing to the paper [23], the next sharp lower bounds hold for all z > e.Finally, since  N < N + 1, the above facts (246)-(250) give rise to the bounds where , where ϑ 2 � ((k ′ ) 2 log(q)/kB k′ p,1 e) 1/k′ .Finally, from ( 242) and (251), we deduce that whenever 0 < x < ϑ, which implies the forseen bounds (238) when looking back to estimate (178).

Asymptotic Expansions with Double Gevrey and q-Gevrey
Scales: A Related Version of the Ramis-Sibuya eorem.We first put forward the notion of asymptotic expansion with double Gevrey and q-Gevrey scales for formal power series introduced by Lastraet al. [6].Here, we need a version that involves Banach valued functions which represents a straightforward adaptation of the original setting.Definition 8. Let (F, ‖.‖ F ) be a complex Banach space.We set k, k ′ ≥ 1 as two integers and q > 1 as a real number.Let E be a bounded sector in C * centered at 0 and f: E ⟶ F be a holomorphic function.en, f is said to possess the formal series as Gevrey asymptotic expansion of mixed order (1/k ′ ; (q, 1/k)) on E if for each closed proper subsector W of E centered at 0, one can choose two constants C, M > 0 with for all integers N ≥ 0 and any ϵ ∈ W.
In the literature, the Ramis-Sibuya theorem is known as a cohomological criterion which ensures the existence of a 28 Abstract and Applied Analysis common Gevrey asymptotic expansion of a given order for families of sectorial holomorphic functions (see [24], p.121 or [25], Lemma XI-2-6).Here, we propose a variant of this result which is adapted to the Gevrey asymptotic expansions of the mixed order disclosed in the above definition.
Assume that the ensuing two requirements hold.
(1) e functions G p (ϵ) are bounded on E p , for 0 ≤ p ≤ ς − 1, (2) e functions Δ p (ϵ) suffer the next sequential constraint on Z p ; there exist two constants A p , B p > 0 with In other words, Δ p (ϵ) has the null formal series  0 as Gevrey asymptotic expansion of mixed order en, all the functions G p (ϵ), 0 ≤ p ≤ ς − 1, share a common formal power series  G(ϵ) ∈ F[[ϵ]] as Gevrey asymptotic expansion of mixed order (1/k ′ ; (q, 1/k)) on E p . Proof.
e entire discussion leans on the following central lemma.

□
Lemma 15.For all 0 ≤ p ≤ ς − 1, the cocycle Δ p (ϵ) splits, which means that bounded holomorphic functions Ψ p : E p ⟶ F can be singled out with the next feature: for all ϵ ∈ Z p , where by convention Ψ ς � Ψ 0 .Furthermore, a sequence φ m   m ≥ 0 of elements in F can be built up such that for each 0 ≤ p ≤ ς − 1 and any closed proper subsector W ⊂ E p with apex at 0, one can find  K p ,  M p > 0 with for all ϵ ∈ W, all integers M ≥ 0.
Proof. e proof mimics the arguments of Lemma XI-2-6 from [25] with fitting adjustment in the asymptotic expansions of the functions Ψ p constructed by means of the Cauchy-Heine transform.
For all 0 ≤ p ≤ ς − 1, we choose a segment: where by convention θ − 1 � θ ς− 1 .Let for all ϵ ∈  E p , for 0 ≤ p ≤ ς − 1, be defined as a sum of Cauchy-Heine transforms of the functions Δ h (ϵ).By deformation of the paths C p− 1 and C p without moving their endpoints and letting the other paths C h , h ≠ p − 1, p untouched (with the convention that C − 1 � C ς− 1 ), one can continue analytically the function Ψ p onto E p .erefore, Ψ p defines a holomorphic function on E p , for all 0 ≤ p ≤ ς − 1.
Now, take ϵ ∈ E p ∩ E p+1 .In order to compute Ψ p+1 (ϵ) − Ψ p (ϵ), we write where the paths  C p and � C p are obtained by deforming the same path C p without moving its endpoints in such a way that where for all ϵ ∈ W. Keeping in mind (256) for the special value N � m + 1 and (266), we get some constants A p , B p > 0 such that for all 0 ≤ m ≤ M. In particular, we deduce the existence of two constants  A p ,  B p > 0 (depending on A p , B p , r 1 , q, k, k ′ ) with for all 0 ≤ m ≤ M. Indeed, recall from [24], Appendix B, that for any given real number a > 0, Γ(x)x a ∼ Γ(x + a) as x tends to +∞.Hence, a constant K k′ > 0 (depending on k ′ ) can be sort with for all m ≥ 0. Consequently, (268) follows from (267) and (269).Moreover, one can choose a positive number η > 0 (depending on W) such that |ξ − ϵ| ≥ |ξ|sin(η) for all ξ ∈ L p and all ϵ ∈ W. Bringing to mind (256) for the peculiar value N � M + 2 and (266) give rise to two constants A p , B p > 0 such that (270) For that reason, we can find constants � A p , � B p > 0 (relying on A p , B p , r 1 , q, k, k ′ , η) such that for all ϵ ∈ W. Namely, from (269) we notice that Using comparable arguments, one can give estimates of the form (265), (266), (268), and (271) for the other integrals for all h ≠ p, p − 1.
As a consequence, for any 0 ≤ p ≤ ς − 1, there exist coefficients φ p,m ∈ F, m ≥ 0, and two constants  for all ϵ ∈ Z p .erefore, each a p (ϵ) stands for the restriction on E p of a global holomorphic function called a(ϵ) on D(0, r)∖ 0 { }.Since a(ϵ) remains bounded on D(0, r)∖ 0 { }, the origin turns out to be a removable singularity for a(ϵ) which, as a result, defines a convergent power series on D(0, r).
Finally for all integers n ≥ 1, provided that ϵ ∈ E p .
Each G p defines a bounded holomorphic map from E p into the Banach space F described in the statement of eorem 3. Furthermore, bound (178) implies that the cocycle Δ p (ϵ) � G p+1 (ϵ) − G p (ϵ) fulfills the sequential bound (256) on Z p � E p+1 ∩ E p for any 0 ≤ p ≤ ς − 1. en, eorem 2 can be applied in order to get a formal power series  G(ϵ) ∈ F[[ϵ]] which stands for the Gevrey asymptotic expansion of mixed order (1/k ′ ; (q, 1/k)) of each G p (ϵ) on E p , for all 0 ≤ p ≤ ς − 1.

Theorem 2 .
Consider a complex Banach space (F, | |.| | F ) and set a good covering E (a)  C p ⊂ E p ∩ E p+1 and � C p ⊂ E p ∩ E p+1 (b) Γ p,p+1 ≔ − � C p +  C p is a simple closed curve with positive orientation whose interior contains ϵ erefore, due to the residue formula, we can writeΨ p+1 (ϵ) − Ψ p (ϵ) � 1 2π �� � − 1 √  Γ p,p+1 Δ p (ξ) ξ − ϵ dξ � Δ p (ϵ),(263)for all ϵ ∈ E p ∩ E p+1 , for all 0 ≤ p ≤ ς − 1 (with the convention that Ψ ς � Ψ 0 ).In a second step, we derive asymptotic properties of Ψ p .We fix an 0 ≤ p ≤ ς − 1 and a proper closed sector W contained in E p .Let  C p (resp. C p− 1 ) be a path obtained by deforming C p (resp.C p− 1 ) without moving the endpoints in order that W is contained in the interior of the simple closed curve  C p− 1 + c p −  C p (which is itself contained in E p ), where c p is a circular arc joining the two points re ��