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

Authors: Stephane Malek
Journal: Abstract and Applied Analysis
 (2020)

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## Abstract

A family of linear singularly perturbed difference differential equations is examined. These 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. These functions share a common asymptotic expansion in the perturbation parameter, which is shown to carry a double scale structure, which pairsq-Gevrey and Gevrey bounds.


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## Body

## 1. Introduction

In this work, we focus on singularly perturbed linear partialq-difference differential equations which couple two categories of operators acting both on the time variable, so-called q-difference 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(1)Q∂zut,z,ϵ=RD∂zϵδDkmDtδDkσq;tδDt∂tmDut,z,ϵ+Pt,z,ϵ,tkσq;t,t∂t,∂zut,z,ϵ+ft,z,ϵ,for vanishing initial data u0,z,ϵ≡0, where k,δD,mD≥1 are integers, σq;t represents the dilation map t⟶qt acting on time t for some real number q>1, and QX,RDX stand for polynomials in ℂX. The main block Pt,z,ϵ,V1,V2,V3 is polynomial in the arguments t,V1,V2,V3, holomorphic in the perturbation parameter ϵ on a disc D0,ϵ0⊂ℂ centered at 0 and in the space variable z on a horizontal strip of the form Hβ=z∈ℂ/Imz<β, for some β>0. The forcing term ft,z,ϵ is analytic relatively to z,ϵ∈Hβ×D0,ϵ0 and defines an entire function w.r.t t in ℂ with (at most) q-exponential growth (see (60), for precise bounds).This 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(2)Q∂zyt,z,ϵ=RD∂zϵkδDtk+1∂tδDt∂tmDyt,z,ϵ+Hz,ϵ,tk+1∂t,t∂t,∂zyt,z,ϵ+ht,z,ϵ,for vanishing initial data y0,z,ϵ≡0, where QX,RDX,Hz,ϵ,V1,V2,V3 stand for polynomials in their arguments X,V1,V2,V3 as above and ht,z,ϵ is like the forcing term ft,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 Lγp=ℝ+exp−1γp and Fourier inverse integral in space z:(3)ypt,z,ϵ=k2π1/2∫−∞+∞∫LγpVpτ,m,ϵexp−τϵtkeizmdττdm,where the Borel/Fourier map Vpτ,m,ϵ is itself set forth as a Laplace transform of order k′=kδD/mD:(4)Vpτ,m,ϵ=k′∫LγpWpu,m,ϵexp−uτk′duu,where Wpu,m,ϵ has (at most) exponential growth along Lγp and exponential decay in phase m on ℝ. The resulting maps ypt,z,ϵ are therefore expressed as iterated Laplace transforms following a so-called multisummability procedure introduced by Balser, see [4]. These functions define bounded holomorphic functions on domains T×Hβ×ℰp for a well-selected bounded sector T at 0 and where ℰ¯=ℰp0≤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⟼ypt,z,ϵ share on ℰp a common asymptotic expansion y^t,z,ϵ=∑n≥0ynt,zϵn with bounded holomorphic coefficients ynt,z on T×Hβ. This 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 Cp,Mp>0 such that(5)supt∈T,z∈Hβypt,z,ϵ−∑m=0n−1ymt,zϵm≤CpMpnΓ1+nκϵn,for all n≥1, all ϵ∈ℰp.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 tk+1∂t is replaced by the discrete operator tkσq;t. This terminology originates from the basic observation that the expression fqt−ft/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 t∂t is not discretized in the process. This 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 t∂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 Γ1+n/κ has to be replaced by qn2/2k in the control term of (3)). The 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/kδD, that we call Gevrey asymptotic expansion of mixed order 1/kδD;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(6)tσq;tr2z∂zr1∂zSXt,z=Bz,tσq;t,σq−1;z,∂zXt,z,for suitably chosen analytic Cauchy data(7)∂zjXt,0=φjt,0≤j≤S−1,and properly selected complex number q∈ℂ∗ with q>1, where r1≥0 and r2,S≥1 are integers and B stands for a polynomial. When r1≥1, the Fuchsian operator z∂zr1 is responsible of the classical Gevrey part of the asymptotic expansion X^t,z=∑n≥0Xnztn of the true solution Xt,z which is shown to be of mixed order r1/r2;q,1 (in the sense of Definition 8) outside some q-spiral λqℤ 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 Γr1/r2n 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 iteratedq-Laplace and Laplace transforms stems from a new work by Yamazawa. In [7], he examines linear q-difference differential equations of the form(8)Lt,σq;t,∂xut,x=ft,x,for the given holomorphic forcing term ft,x near the origin and where Lt,V1,V2 is a polynomial in V1,V2 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 ut,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≥1ukxtk of (4). Furthermore, he gets in particular that ut,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 ascomplete iterated q-Laplace and classical Laplace transforms that are shown to be exact solutions of our problem (1). This 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 inq-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 [8–10]. Notice that the fields of applications of q-difference equations have also encountered a rapid growth in the last years. Some forefront studies in this respect are given, for instance, by [11–13] and references therein.Now, we describe a little more precisely our main results obtained in Theorems1 and 3. Namely, under convenient restrictions on the shape of (1) detailed in the statement of Theorem 1, we can manufacture a family of bounded holomorphic solutions upt,z,ϵ on domains T×Hβ×ℰp for a suitable bounded sector T at 0, Hβ a strip of width β>0, and ℰp belonging to a good covering in ℂ∗, which can be displayed as a q-Laplace transform of order k along a halfline Lγp=ℝ+exp−1γp and Fourier integral:(9)upt,z,ϵ=1πq1/k2π1/2∫−∞+∞∫LγpWdpu,m,ϵ1Θq1/ku/ϵteizmduudm.Theq-Borel/Fourier map Wdpu,m,ϵ is itself shaped as a classical Laplace transform of order kδD along Lγp:(10)Wdpu,m,ϵ=kδD∫Lγpwdph,m,ϵexp−hukδDdhh,where wdph,m,ϵ has (at most) q-exponential growth of some order 0<k1<k along Lγp (see (180)) and exponential decay in phase m∈ℝ. In Theorem 3, we explain the reason for which all the partial functions ϵ⟼upt,z,ϵ share a common asymptotic expansion u^t,z,ϵ=∑m≥0hmt,zϵm on ℰp with bounded holomorphic coefficients hmt,z on T×Hβ, which turns out to be of mixed order 1/kδD;q,1/k. This last result leans on a new version of the classical Ramis—Sibuya theorem fitting the above asymptotics, which is fully expounded in Theorem 2.Our paper is arranged as follows.In Section2, 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 Section3, 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 kδD of a second Borel map w with again appropriate behaviour. The expression w is then contrived to solve a second principal integro q-difference equation (81).In Section4, bounds for linear convolution and q-difference operators acting on Banach spaces of functions with q-exponential growth are displayed. The second key equation (81) is then solved within these spaces at the hand of a fixed point argument.In Section5, 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 Section6, we achieve our goal in finding a set of true holomorphic solutions (176) to our initial problem (33).In Section7, the existence of a common asymptotic expansion of the Gevrey type with mixed order 1/kδD;q,1/k is established for the solutions set up in Section 6. The decisive technical tool for its construction is detailed in Theorem 2.


## 2. Laplace Transforms of Orderk′, q-Laplace Transforms of Order k, and Fourier Inverse Maps

Letk′≥1 be an integer. We remind the reader the definition of the Laplace transform of order k′ as introduced in [14].Definition1.
We setSd,δ=τ∈ℂ∗:d−argτ<δ as some unbounded sector with bisecting direction d∈ℝ and aperture 2δ>0 and D0,ρ as a disc centered at 0 with radius ρ>0. Consider a holomorphic function w:Sd,δ∪D0,ρ⟶ℂ that vanishes at 0 and withstands the bounds. There exist C>0 and K>0 such that(11)wτ≤CτexpKτk′,for all τ∈Sd,δ. We define the Laplace transform of w of order k′ in the direction d as the integral transform(12)ℒk′dwT=k′∫Lγwuexp−uTk′duu,along a halfline Lγ=ℝ+e−1γ⊂Sd,δ∪0, where γ depends on T and is chosen in such a way that cosk′γ−argT≥δ1, for some fixed real number δ1>0. The function ℒk′dwT is well defined, holomorphic, and bounded on any sector:(13)Sd,θ,R1/k′=T∈ℂ∗:T<R1/k′,d−argT<θ2,where 0<θ<π/k′+2δ and 0<R<δ1/K.
If one setswτ=∑n≥1wnτn, the Taylor expansion of w, which converges on the disc D0,ρ/2, the Laplace transform ℒk′dwT has the formal series(14)X^T=∑n≥1wnΓnk′Tn,as Gevrey asymptotic expansion of order 1/k′. This means that for all 0<θ1<θ, two constants C,M>0 can be selected with the bounds:(15)ℒk′dwT−∑p=1n−1wpΓpk′Tp≤CMnΓ1+nk′Tn,for all n≥2, all T∈Sd,θ1,R1/k′.
In particular, ifwτ represents an entire function w.r.t τ∈ℂ with bound (11), its Laplace transform ℒk′dwT does not depend on the direction d in ℝ and represents a bounded holomorphic function on D0,R1/k′, whose Taylor expansion is represented by the convergent series XT=∑n≥1wnΓn/k′Tn on D0,R1/k′.
Letk≥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 Theta function of orderk defined as the Laurent series:(16)Θq1/kx=∑n∈ℤq−nn−1/2kxn,for all x∈ℂ∗. This analytic function can be factorized as a product known as Jacobi’s triple product formula:(17)Θq1/kx=∏n≥01−q−n−1/k1+xq−n/k1+q−n−1/kx,for all x∈ℂ∗, from which we deduce that its zeros are the set of real numbers −qm/k/m∈ℤ. We recall the next lower bound estimates on a domain, bypassing the set of zeroes of Θq1/kx, from [5] Lemma 3, which are crucial in the sequel.Lemma1.
LetΔ>0. There exists a constant Cq,k>0 depending on q,k and independent of Δ such that(18)Θq1/kx≥Cq,kΔexpk2log2xlogqx1/2,for all x∈ℂ∗ satisfying 1+xqm/k>Δ, for all m∈ℤ.Definition2.
Letρ>0 be a real number and Sd be an unbounded sector centered at 0 with bisecting direction d∈ℝ. Let f:D0,ρ∪Sd⟶ℂ be a holomorphic function, continuous on the adherence D0,ρ¯, such that there exist constants K,α>0 and δ>1 with(19)fx≤Kxexpk2log2x+δlogq+αlogx+δ,for all x∈Sd∪D0,ρ. Let γ∈ℝ with e−1γ∈Sd. We put πq1/k=logq/k. We define the q-Laplace transform of order k of f in direction γ as(20)ℒq;1/kγfxT=1πq1/k∫LγfuΘq1/ku/Tduu,where Lγ=ℝ+e−1γ is a halfline in the direction γ.
The following lemma is a slightly modified version of Lemma4 from [5].Lemma2.
LetΔ>0 chosen as in Lemma 1 above. The integral transform ℒq;1/kγfxT defines a bounded holomorphic function on the domain ℛγ,Δ∩D0,r1 for any radius 0<r1≤q−1/kα+1/2, where(21)ℛγ,Δ=T∈ℂ∗1+e−1γ/Tr>Δ,for allr≥0.
Notice that the valueℒq;1/kγfxT does not depend on γ∈ℝ such that e−1γ∈Sd due to the Cauchy formula.
The next lemma describes conditions under which theq-Laplace transform defines a convergent series near the origin.Lemma3.
Letf:ℂ⟶ℂ be an entire function with Taylor expansion fx=∑n≥1fnxn fulfilling bound (19) for all x∈ℂ. Then, its q-Laplace transform of order k, ℒq;1/kdfT, does not depend on the direction d∈ℝ and represents a bounded holomorphic function on D0,r1 with the restriction 0<r1≤q−1/kα+1/2 whose Taylor expansion is given by the convergent series YT=∑n≥1fnqnn−1/2kTn.Proof.
The proof is a direct consequence of the next formulas:(22)1πq1/k∫Lγun−1Θq1/ku/Tdu=Tn1πq1/k∫Lγ′vn−1Θq1/kvdv=Tnqnn−1/2k,whenever T∈ℛγ,δ and γ′=γ−argT, 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].Definition3.
Letβ,μ∈ℝ. We set Eβ,μ as the vector space of continuous functions h:ℝ⟶ℂ such that(23)hmβ,μ=supm∈ℝ1+mμexpβmhm,is finite. The space Eβ,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].Definition4.
Letf∈Eβ,μ with β>0 and μ>1. The inverse Fourier transform f is given by(24)ℱ−1fx=12π1/2∫−∞+∞fmexpixmdm,for all x∈ℝ. The function ℱ−1f extends to an analytic bounded function on the strips(25)Hβ′=z∈ℂImz<β′,for all given 0<β′<β.(a)
Define the functionm⟼ϕm=imfm which belongs to the space Eβ,μ−1. Then, the next identity(26)∂zℱ−1fz=ℱ−1ϕz,occurs.(b)
Takeg∈Eβ,μ and set(27)ψm=12π1/2∫−∞+∞fm−m1gm1dm1,as the convolution product off and g. Then, ψ belongs to Eβ,μ, and moreover,(28)ℱ−1fzℱ−1gz=ℱ−1ψz,for allz∈Hβ.


## 3. Layout of the Principal Initial Value Problem and Associated Auxiliary Problems

We setk≥1 as an integer. Let mD,δD≥1 be integers. We set(29)k′=kδD.We consider a finite setI of ℕ3 that fulfills the next feature,(30)l0≥1+l2k′,whenever l0,l1,l2∈I and we set nonnegative integers Δl≥0 with(31)Δl−l0≥0,for all l=l0,l1,l2∈I.LetQX,RDX,RlX∈ℂX, and l∈I be polynomials such that(32)degQ≥degRD≥degRl,Qim≠0,RDim≠0,for all m∈ℝ, all l∈I.We consider a family of linear singularly perturbed initial value problems(33)Q∂zut,z,ϵ=RD∂zϵδDkmDtδDkσq;tδDt∂tmDut,z,ϵ+∑l=l0,l1,l2∈IϵΔlclz,ϵRl∂ztl0σq;tl1t∂tl2ut,z,ϵ+ft,z,ϵ,for vanishing initial data u0,z,ϵ≡0. Here, q>1 stands for a real number and the operator σq;t is defined as the dilation by q acting on the variable t through σq;tut,z,ϵ=uqt,z,ϵ.The coefficientsclz,ϵ are built in the following manner. For each l∈I, we consider a function m⟼Clm,ϵ that belongs to the Banach space Eβ,μ for some β,μ>0, which depends holomorphically on the parameter ϵ on some disc D0,ϵ0 with radius ϵ0>0 and for which one can find a constant Cl>0 with(34)supϵ∈D0,ϵ0Clm,ϵβ,μ≤Cl.We construct(35)clz,ϵ=12π1/2∫−∞+∞Clm,ϵeizmdm,as the inverse Fourier transform of the map Clm,ϵ for all l∈I. As a result, clz,ϵ is bounded holomorphically w.r.t ϵ on D0,ϵ0 and w.r.t z on any strip Hβ′ for 0<β′<β in view of Definition 4.The presentation of the forcing term requires some preliminary groundwork. We consider a sequence of functionsm⟼ψnm,ϵ, for n≥1, that belongs to the Banach space Eβ,μ with the parameters β,μ>0 given above and which relies analytically and is bounded w.r.t ϵ on the disc D0,ϵ0. We assume that the next bounds,(36)supϵ∈D0,ϵ0ψnm,ϵβ,μ≤K01T0n,hold for all n≥1 and given constants K0,T0>0. We define the formal series(37)ψτ,m,ϵ=∑n≥1ψnm,ϵτnq1/k1nn−1/2,for some real number 0<k1<k. We introduce the next Banach space.Definition5.
Letk1,β,μ,r,α>0 and q,δ>1 be real numbers. Let Ud be an open unbounded sector with bisecting direction d∈ℝ centered at 0 in ℂ. We denote Expk1,β,μ,α,rq the vector space of complex valued continuous functions u,m⟼hu,m on the adherence Ud¯∪D0,r¯×ℝ, which are holomorphic w.r.t u on Ud∪D0,r and such that the norm(38)hu,mk1,β,μ,α,r=supu∈Ud¯∪D0,r¯,m∈ℝ1+mμeβm1u×exp−k12log2u+δlogq−αlogu+δhu,m,is finite. One can check that the normed space Expk1,β,μ,α,rq,.k1,β,μ,α,r represents a Banach space.Remark 1.
The spaces above are faint modifications of the Banach spaces already introduced in the works of Dreyfus and Lastra [16–18].
The next lemma is a proper adjustment of Lemma5 out of [5] to the new Banach spaces from Definition 5.Lemma4.
LetT0 be fixed as in (36). We take a number α>0 such that(39)T0>q1/2k1qα+1/k1.
Letk1,β,μ be chosen as above. Then, the function u,m⟼ψu,m,ϵ belongs to the Banach space Expk1,β,μ,α,rq for any unbounded sector Ud, any disc D0,r. Moreover, one can find a constant C1>0 (depending on q,k1,α,T0) with(40)supϵ∈D0,ϵ0ψu,m,ϵk1,β,μ,α,r≤K0C1.Proof.
ound (36) implies that(41)ψu,m,ϵk1,β,μ,α,r≤∑n≥1ψnm,ϵuq1/k1nn−1/2k1,β,μ,α,r≤∑n≥1K01T0n1q1/k1nn−1/2supu∈Ud¯∪D0,r¯un−1exp−k12log2u+δlogq−αlogu+δ.
According to the elementary fact that the polynomialhx=xn−1−α−k1/2x2/logq admits its maximum value logq/2k1n−1−α2 at x=logq/k1n−1−α, we deduce by means of the change of variable x=logu+δ that(42)supu∈Ud¯∪D0,r¯un−1exp−k12log2u+δlogq−αlogu+δ≤supx∈ℝexpxn−1−α−k1x22logq≤qn−1−α2/2k1,for all n≥1. Therefore, we deduce that(43)ψu,m,ϵk1,β,μ,α,r≤K0q1+α2/2k1∑n≥1q1/2k1T0qα+1/k1n,which converges, provided that (39) holds, whenever ϵ∈D0,ϵ0.
We define(44)Ψdτ,m,ϵ=k′∫Ldψu,m,ϵexp−uτk′duu,as the Laplace transform of ψu,m,ϵ w.r.t u of order k′ in direction d∈ℝ. Notice that two constants K1,K2>0 (depending on k1,q,α,δ,k′) can be found such that(45)k12log2u+δlogq+αlogu+δ≤K1uk′+K2,for all u∈ℂ. As a result, owing to bound (40) and the last part of Definition 1, we deduce that Ψd does not depend on the direction d and can be written as a convergent series:(46)Ψdτ,m,ϵ=∑n≥1ψnm,ϵΓn/k′q1/k1nn−1/2τn,w.r.t τ near the origin. Now, we fix some real number k2 such that 0<k1<k2<k. Then, one can sort a constant C2 (depending on q,k1,k2,k′) such that(47)Γn/k′q1/k1nn−1/2≤C21q1/k2nn−1/2,for all n≥1. This inequality is a consequence of the Stirling formula, which states that(48)Γx∼2πe−xx−1/2xx,as x tends to +∞ and from the existence of a constant K3>0 (depending on q,k1,k2,k′) with(49)−nk′+nk′lognk′+nn−1logq2k2≤nn−1logq2k1+K3,for all n≥1. Consequently, it turns out that Ψdτ,m,ϵ represents an entire function w.r.t τ such that(50)Ψdτ,m,ϵ≤K0C21+m−μe−βm∑n≥1τ/T0nq1/k2nn−1/2,for all τ∈ℂ. Furthermore, owing to bound (42), we know that(51)τn≤qn−α2/2k2expk22log2τ+δ+αlogτ+δ,for all n≥1, all τ∈ℂ. Henceforth, we get the next global bounds(52)Ψdτ,m,ϵ≤K0C2qα2/2k2∑n≥1q1−2α/2k2T0n1+m−μe−βm×expk22log2τ+δ+αlogτ+δ,provided that(53)T0>q1−2α/2k2,for all τ∈ℂ, m∈ℝ, and ϵ∈D0,ϵ0.
Next, we set(54)FdT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫LdΨdu,m,ϵ1Θq1/ku/Teizmduudm,as the q-Laplace transform of Ψdu,m,ϵ w.r.t u of order k in direction d and Fourier inverse integral w.r.t m. We put(55)Fnz,ϵ=12π1/2∫−∞+∞ψnm,ϵeizmdm,for all n≥1. We first provide bounds for this sequence of functions. Namely, we can get a constant Cμ,β,β′>0 (relying on μ,β,β′) with(56)Fnz,ϵ≤K02π1/21T0n∫−∞+∞1+m−μe−βme−Imzmdm≤K02π1/21T0n∫−∞+∞1+m−μeβ′−βmdm≤Cμ,β,β′K02π1/21T0n,for all n≥1, whenever ϵ∈D0,ϵ0 and z belongs to the horizontal strip Hβ′ for some 0<β′<β (see Definition 4). Owing to Lemma 3, we deduce that the function FdT,z,ϵ converges near the origin w.r.t T, where it carries the next Taylor expansion:(57)FdT,z,ϵ=∑n≥1Fnz,ϵΓnk′q1/k−1/k1nn−1/2Tn,for all ϵ∈D0,ϵ0 and z∈Hβ′. In particular, the function Fd is independent of the direction d chosen.
We now show thatFdT,z,ϵ represents an entire function w.r.t T and supply explicit upper bounds. Namely, in accordance with (47), we obtain(58)FdT,z,ϵ≤∑n≥1K0Cμ,β,β′C22π1/21q1/k2−1/knn−1/2TT0n.
Again, estimate (42) yields(59)Tn≤q1/2κ2n−α2expκ22log2T+δlogq+αlogT+δ,for all T∈ℂ, all n≥1, where κ2>0 is defined by 1/κ2=1/k2−1/k. By gathering the two last above inequalities, the next global estimates can be figured out:(60)FdT,z,ϵ≤K0Cμ,β,β′C22π1/2qα2/2κ2∑n≥1q1−2α/2κ2T0nexpκ22log2T+δlogq+αlogT+δ,for all T∈ℂ, all z∈Hβ′ and ϵ∈D0,ϵ0, provided that(61)T0>q1−2α/2κ2.
Lastly, we define the forcing termf as a time rescaled version of Fd,(62)ft,z,ϵ=Fdϵt,z,ϵ,that represents a bounded holomorphic function w.r.t z∈Hβ′ and ϵ∈D0,ϵ0 and an entire function w.r.t t with q-exponential growth of order κ2.
Throughout this paper, we are looking for time rescaled solutions of (33) of the form(63)ut,z,ϵ=Uϵt,z,ϵ.
As a consequence, the expressionUT,z,ϵ, through the change of variable T=ϵt, is asked to solve the next singular problem:(64)Q∂zUT,z,ϵ=RD∂zTδDkσq;TδDT∂TmDUT,z,ϵ+∑l=l0,l1,l2∈IϵΔl−l0clz,ϵRl∂zTl0σq;Tl1T∂Tl2UT,z,ϵ+FdT,z,ϵ.
At the onset, we seek for a solutionUT,z,ϵ that can be expressed as an integral representation via a q-Laplace transform of order k and Fourier inverse integral:(65)UγT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫LγWu,m,ϵ1Θq1/ku/Teizmduudm,where the inner integration is performed along a halfline Lγ=ℝ+e−1γ in direction γ∈ℝ. Overall this section, we assume that the partial functions u⟼Wu,m,ϵ have at most q-exponential growth of order k on some unbounded sector Sd centered 0 with bisecting direction d and m⟼Wu,m,ϵ belong to the Banach space Eβ,μ mentioned in Definition 3, whenever ϵ∈D0,ϵ0. Precise bounds will be given later in Section 5. Here, we assume that Lγ⊂Sd∪0.
Our aim is now the presentation of a related problem fulfilled by the expressionWu,m,ϵ. We first need to state two identities which concern the action of q-difference and Fuchsian operators on q-Laplace tranforms.Lemma5.
The actions of theq-difference operators Tl0σq;Tl1 for integers l0,l1≥0 and the Fuchsian differential operator T∂T are given by(66)Tl0σq;Tl1UγT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫Lγul0q1/kl0l0−1/2×σq;ul1−l0/kWu,m,ϵ1Θq1/ku/Teizmduudm,(67)T∂TUγT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫Lγu∂uWu,m,ϵ1Θq1/ku/Teizmduudm.Proof.
The 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:(68)T∂TUγT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫Lγ−1TWu,m,ϵ1Θq1/k′u/Teizmdudm=−1πq1/k2π1/2∫−∞+∞∫LγWu,m,ϵ∂u1Θq1/ku/Teizmdudm=−1πq1/k2π1/2∫−∞+∞Wu,m,ϵ1Θq1/ku/Tu=0u=∞eizmdm+1πq1/k2π1/2∫−∞+∞∫Lγ∂uWu,m,ϵ1Θq1/ku/Teizmdudm,from which the forecast formula follows since the map u⟼Wu,m,ϵ is assumed to possess a growth of q-exponential order k and vanishes at u=0.
The application of the above identities (66) and (67) in a row with (26) and (28) leads to the first integrodifferential q-difference equation fulfilled by the expression Wu,m,ϵ as long as UγT,z,ϵ solves (64):(69)QimWτ,m,ϵ=RDim1q1/kmDδDkδDk−1/2τδDk+1∂τmDWτ,m,ϵ+∑l=l0,l1,l2∈IϵΔl−l012π1/2∫−∞+∞Clm−m1,ϵ×τl0q1/kl0l0−1/2σq;τl1−l0/kτ∂τl2Wτ,m1,ϵRlim1dm1+Ψdτ,m,ϵ.
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 orderk′ since by construction Ψdτ,m,ϵ owns this structure after (44). Namely, we take for granted that(70)Wτ,m,ϵ=k′∫Lγwu,m,ϵexp−uτk′duu,where Lγ=ℝ+e−1γ stands for a halfline with direction γ∈ℝ, which belongs to Ud∪0 where Ud represents an unbounded sector centered at 0 with bisecting direction d. Within this step, we assume that the expression u,m⟼wu,m,ϵ belongs to the Banach space Expk1,β,μ,α,rq introduced in Definition 5, for all ϵ∈D0,ϵ0, where the constants k1,β,μ, and α are selected accordingly to the construction of the forcing term ft,z,ϵ.
The next lemma has already been stated in our previous work [3].Lemma6.
For all integersl≥1, positive integers aq,l≥1 and 1≤q≤l can be found such that(71)t∂tl=∑q=1laq,ltq∂tq.With the help of this last expansion, equation (69) can be recast in the form(72)QimWτ,m,ϵ=RDim1q1/kmDδDkδDk−1/2τδDk+1∂τmDWτ,m,ϵ+∑l=l0,l1,l2∈IϵΔl−l012π1/2∫−∞+∞Clm−m1,ϵRlim11q1/kl0l0−1/2×σq;τl1−l0/k∑h=1l2ah,l2ql1−l0/kl0τl0+h∂τhWτ,m1,ϵdm1+Ψdτ,m,ϵ.This 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 τk′+1∂τ.Lemma7.
Letk′,δ≥1 be integers. Then, there exist real numbers Aδ,p, 1≤p≤δ−1, such that(73)τδk′+1∂τδ=τk′+1∂τδ+∑1≤p≤δ−1Aδ,pτk′δ−pτk′+1∂τp.
By convention, we take for granted that the above sum∑1≤p≤δ−1.. vanishes when δ=1.Indeed, by construction of the finite setI, we can represent the next integers in a specific way:(74)l0+h=h1+k′+eh,l0,where eh,l0=l0−hk′≥1 for all l0,l1,l2∈I and 1≤h≤l2. As a consequence, we can further expand the next piece of (72) in its final convenient form:(75)τl0+h∂τhWτ,m1,ϵ=τeh,l0τh1+k′∂τhWτ,m1,ϵ=τeh,l0τk′+1∂τh+∑1≤p≤h−1Ah,pτk′h−pτk′+1∂τpWτ,m1,ϵ,and we can remodel equation (72) in such a way that it contains only primitive building blocs:(76)QimWτ,m,ϵ=RDim1q1/kmDk′k′−1/2τk′+1∂τmDWτ,m,ϵ+∑l=l0,l1,l2∈IϵΔl−l012π1/2∫−∞+∞Clm−m1,ϵRlim11q1/kl0l0−1/2×σq;τl1−l0/k∑h=1l2ah,l2ql1−l0/kl0τeh,l0τk′+1∂τh+∑1≤p≤h−1Ah,pτk′h−pτk′+1∂τpWτ,m1,ϵdm1+Ψdτ,m,ϵ.Similarly to our previous technical Lemma5, we disclose some useful commutation formulas dealing with the actions of the basic irregular operator τk′+1∂τ, multiplication by monomials τm′, and of the q-difference operator σq;τ,Lemma8.
(1)
The action of the differential operatorsτk′+1∂τ on Wτ,m,ϵ is given by(77)τk′+1∂τWτ,m,ϵ=k′∫Lγk′uk′wu,m,ϵexp−uτk′duu.(2)
Letm′≥1 be an integer. The action of the multiplication by τm′ on Wτ,m,ϵ is described through the next formula:(78)τm′Wτ,m,ϵ=k′∫Lγuk′Γm′/k′∫0uk′uk′−sm′/k′−1ws1/k′,m,ϵdssexp−uτk′duu.(3)
Letγ∈ℤ be an integer. The action of the operator σq;τγ is represented through the following integral transform:(79)σq;τγWτ,m,ϵ=k′∫Lγσq;uγwu,m,ϵexp−uτk′duu.Proof.
The first two formulas have already been given in our previous works [3, 20]. We focus on the third equality. By definition,(80)σq;τγWτ,m,ϵ=k′∫Lγwu,m,ϵexp−uqγτk′duu,and if one deforms the path of integration Lγ through u=qγv which keeps the path invariant since qγ∈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 wu,m,ϵ,(81)Qimwu,m,ϵ=RDimk′mDq1/kmDk′k′−1/2uk′mDwu,m,ϵ+∑l=l0,l1,l2∈IϵΔl−l012π1/2∫−∞+∞Clm−m1,ϵRlim11q1/kl0l0−1/2σq;ul1−l0/k∑h=1l2ah,l2ql1−l0/kl0uk′Γeh,l0/k′∫0uk′uk′−seh,l0/k′−1k′hshws1/k′,m1,ϵdss+∑1≤p≤h−1Ah,puk′Γeh,l0+k′h−p/k′∫0uk′uk′−seh,l0+k′h−p/k′−1k′pspws1/k′,m1,ϵdssdm1+ψu,m,ϵ.


## 4. An Integralq-Difference Equation with Complex Parameter

The objective of this section is the construction of a unique solution of equation (81) just established overhead. This 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 Expk1,β,μ,α,rq is discussed.Proposition1.
Letk′≥1 be an integer and γ1>0 and γ2,γ3 be real numbers such that k′γ2+γ3+2 is an integer that are submitted to the next constraint:(82)γ2+1>0,γ3+1k′+1>0,γ1≥k′γ2+γ3+2≥0.
Letaγ1u be a continuous function on Ud¯∪D0,r¯, holomorphic w.r.t u on Ud∪D0,r submitted to the bound(83)aγ1u≤Mγ11+uγ1,for all u∈Ud¯∪D0,r¯, for some constant Mγ1>0. Then, the linear function(84)f⟼Ck′,γ1,γ2,γ3fu,m=aγ1uuk′∫0uk′uk′−sγ2sγ3fs1/k′,mds,represents a continuous map from the Banach space Expk1,β,μ,α,rq into itself. In other words, some constant M1>0 (depending on k′,γ1,γ2,γ3) can be found with(85)Ck′,γ1,γ2,γ3fu,mk1,β,μ,α,r≤M1Mγ1fu,mk1,β,μ,α,r,for all f∈Expk1,β,μ,α,rq.Proof.
Letf belong to Expk1,β,μ,α,rq. The map Ck′,γ1,γ2,γ3fu,m can be rewritten using the parametrization s=uk′p for 0≤p≤1, namely,(86)Ck′,γ1,γ2,γ3fu,m=aγ1uuk′γ2+γ3+2∫011−pγ2pγ3fup1/k′,mdp,for all u∈Ud∪D0,r, whenever m∈ℝ. By definition, the next upper bounds(87)fu,m≤fu,mk1,β,μ,α,ruexpk12log2u+δlogq+αlogu+δ1+m−μe−βm,hold for all u∈Ud∪D0,r, m∈ℝ. It follows that(88)Ck′,γ1,γ2,γ3fu,m≤Mγ1uk′γ2+γ3+21+uγ1∫011−pγ2pγ3+1/k′dpfu,mk1,β,μ,α,ru×expk12log2u+δlogq+αlogu+δ1+m−μe−βm,for all u∈Ud∪D0,r, all m∈ℝ. Under condition (82), the expected bound (85) follows.Proposition2.
Leta∈ℂ be a complex number, γ1≥0 be an integer, and γ2≥0 be a real number withstanding the following condition:(89)k1γ2≥γ1.
Then, we can sort a constantM2>0 (depending on γ1,γ2,q,α,δ,r,a) with(90)u+aγ1σq;u−γ2fu,mk1,β,μ,α,r≤M2fu,mk1,β,μ,α,r,for all f∈Expk1,β,μ,α,rq.Proof.
The proof is proximate to the one of Proposition1 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.
Letfu,m belong to Expk1,β,μ,α,rq. By definition, we can perform the next factorization(91)u+aγ1σq;u−γ2fu,mk1,β,μ,α,r=supu∈Ud¯∪D0,r¯1+mμeβm1u×exp−k12log2u+δlogq−αlogu+δu+aγ1×fuqγ2,m1+mμeβm1u/qγ2exp−k12log2u/qγ2+δlogq−αloguqγ2+δ×1+m−μe−βmuqγ2expk12log2u/qγ2+δlogq+αloguqγ2+δ.
Since the contractive mapu⟼u/qγ2 keeps the domain Ud¯∪D0,r¯ invariant, we deduce(92)u+aγ1σq;u−γ2fu,mk1,β,μ,α,r≤M˜2fu,mk1,β,μ,α,r,with M˜2=1/qγ2supu∈Ud¯∪D0,r¯Au, where(93)Au=u+aγ1exp−k12log2u+δlogq−αlogu+δ×expk12log2u/qγ2+δlogq+αloguqγ2+δ.
We observe that(94)M˜2≤1qγ2maxM˜2.1,M˜2.2,where M˜2.1=supu∈D0,r¯Au is finite since Au is continuous on D0,r¯ and M˜2.2 is equal to supu∈Ud¯,u>rAu.
In the remaining part of the proof, we show thatM˜2.2 is also finite. We first need to rearrange the pieces of Au. Namely, we expand(95)log2u+δ=log2u+2logulog1+δu+log21+δu,log2uqγ2+δ=log2u−2γ2logqlogu+γ2logq2+2loguqγ2log1+qγ2uδ+log21+qγ2uδ,(96)logu+δ=logu+log1+δu,loguqγ2+δ=logu−γ2logq+log1+qγ2uδ.
Sincelog1+x∼x as x⟶0, we get two constants A1,A2∈ℝ (depending on r,δ,q,γ2) with(97)A1≤2logulog1+δu≤A2,A1≤2loguqγ2log1+qγ2uδ≤A2,for all u∈Ud¯, u>r. Gathering (95) and (96) with (97) gives rise to the bound(98)M˜2.2≤supu∈Ud¯,u>ru+aγ1uγ2k1expk1γ222logq+A2k12logq+k12logqlog21+qγ2rδ−A1k12logq−αγ2logq+αlog1+qγ2rδ,which is finite owing to (89).Proposition3.
We set polynomialsQX,RX∈ℂX such that(99)degR≥degQ,Rim≠0,μ>degQ+1.
Consideru,m⟼bu,m a continuous function on Ud¯∪D0,r¯×ℝ, holomorphic w.r.t u on Ud∪D0,r with the bound(100)supm∈ℝu∈Ud¯∪D0,r¯bu,m≤Mb,for some constant Mb>0. Then, there exists a constant M3>0 (depending on Q, R, and μ) such that(101)1Rim∫−∞+∞fm−m1Qim1bu,mgu,m1dm1k1,β,μ,α,r≤M3Mbfmβ,μgu,mk1,β,μ,α,r,whenever f belongs to Eβ,μ and g belongs to Expk1,β,μ,α,rq.Proof.
The proof shares the same ingredients as the one of Proposition2 of [3]. Again, we give a thorough explanation of the result. We take f inside Eβ,μ and select g belonging to Expk1,β,μ,α,rq. We first recast the norm of the convolution operator as follows:(102)N2=1Rim∫−∞+∞fm−m1Qim1bu,mgu,m1dm1k1,β,μ,α,r=supu∈Ud¯∪D0,r¯,m∈ℝ1+mμ1ueβmexp−k12log2u+δlogq−αlogu+δ×1Rim∫−∞+∞1+m−m1μexpβm−m1fm−m1bu,m×1+m1μeβm11uexp−k12log2u+δlogq−αlogu+δgu,m1Au,m,m1dm1,where(103)Au,m,m1=Qim1exp−βm1exp−βm−m11+m1μ1+m−m1μuexpk12log2u+δlogq+αlogu+δ.
By construction of the polynomialsQ and R, one can sort two constants Q,ℜ>0 with(104)Qim1≤Q1+m1degQ,Rim≥ℜ1+mdegR,for all m,m1∈ℝ. As a consequence of (102), (104), and (100), with the help of the triangular inequality m≤m1+m−m1, we are led to the bounds(105)N2≤M3Mbfmβ,μgu,mk1,β,μ,α,r,where(106)M3=Qℜsupm∈ℝ1+mμ−degR∫−∞+∞11+m−m1μ1+m1μ−degQdm1,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 polynomialsQX and RDX. These 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:(107)SQ,RD=z∈ℂ∗z>rQ,RD,argz−dQ,RD≤ηQ,RD,where direction dQ,RD∈ℝ and aperture ηQ,RD>0 for some given inner radius rQ,RD>0 with the feature:(108)Qim/RDimm∈ℝ⊂SQ,RD.
We consider the next polynomial:(109)Pmu=Qim−RDimk′mDq1/kmDk′k′−1/2uk′mD.
In the following, we need lower bounds of the expressionPmu with respect to both variables m and u. In order to achieve this goal, we can factorize the polynomial w.r.t u, namely,(110)Pmu=−RDimk′mDq1/kmDk′k′−1/2∏l=0k′mD−1u−qlm,where its roots qlm can be displayed explicitely as(111)qlm=QimRDimk′mD/q1/kmDk′k′−1/21/k′mDexp−1k′mDargQimRDimk′mD/q1/kmDk′k′−1/2+2πl,for all 0≤l≤k′mD−1, for all m∈ℝ.
We set an unbounded sectorUd centered at 0, a small disc D0,r, and we adjust the sector SQ,RD in a way that the next condition holds. A constant M>0 can be chosen with(112)u−qlm≥M1+u,for all 0≤l≤k′mD−1, all m∈ℝ, provided that u∈Ud∪D0,r. Indeed, inclusion (108) implies in particular that all the roots qlm, 0≤l≤k′mD−1 remain a part of some neighborhood of the origin, i.e., satisfy qlm≥2r for an appropriate choice of r>0. Furthermore, when the aperture ηQ,RD>0 is taken close enough to 0, all these roots qlm 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 Ud with(113)Ud∩U=Ø.
By construction, the quotientsqlm/u live outside some small disc centered at 1 in ℂ for all u∈Ud, m∈ℝ, 0≤l≤k′mD−1. Then, (112) follows.
We are now ready to supply lower bounds forPmu.Lemma9.
A constantCP>0 (depending on k,k′,mD,q,M) can be found with(114)Pmu≥CPRDim1+uk′mD,for u∈Ud∪D0,r, all m∈ℝ.Proof.
Departing from factorization (110), the lower bound (112) entails(115)Pmu≥k′mDq1/kmDk′k′−1/2Mk′mDRDim1+uk′mD,for all u∈Ud∪D0,r.
The next proposition discusses sufficient conditions under which a solutionwdu,m,ϵ of the main integral q-difference equation (81) can be built up in the space Expk1,β,μ,α,rq.Proposition4.
Let us assume the next extra requirements:(116)l0k−l1≥0,μ>degRl,for all l=l0,l1,l2∈I. Furthermore, for each l=l0,l1,l2∈I, we set an integer pl0,l1 such that(117)0≤pl0,l1≤k1l0k−l1,and we take for granted that(118)pl0,l1+k′mD≥l0,holds. Then, for an appropriate choice of the constants Cl>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⟼wdu,m,ϵ in the space Expk1,β,μ,α,rq with the condition:(119)wdu,m,ϵk1,β,μ,α,r≤ϖ,whenever ϵ∈D0,ϵ0, where Ud,r are chosen as above and k1,β,μ,α are specified in Section 3 on the way to the construction of the forcing term ft,z,ϵ.Proof.
The proof relies strongly on the next lemma which discusses contractive properties of a linear map.Lemma10.
For allϵ∈D0,ϵ0, we define the map ℋϵ as(120)ℋϵwu,m≔∑l=l0,l1,l2∈IϵΔl−l012π1/2∫−∞+∞Clm−m1,ϵRlim11q1/kl0l0−1/21Pmuσq;ul1−l0/k∑h=1l2ah,l2ql1−l0/kl0uk′Γeh,l0/k′∫0uk′uk′−seh,l0/k′−1k′hshws1/k′,m1dss+∑1≤p≤h−1Ah,puk′Γeh,l0+k′h−p/k′∫0uk′uk′−seh,l0+k′h−p/k′−1k′pspws1/k′,m1dssdm1+ψu,m,ϵPmu.
Under the additional requirement (116)–(118), one can select the constants Cl>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 Expk1,β,μ,α,rq in the following way:(i)
The inclusion(121)ℋϵB0,ϖ¯⊂B0,ϖ¯,holds, whereB0,ϖ¯ stands for the closed ball of radius ϖ centered at 0 in Expk1,β,μ,α,rq, for all ϵ∈D0,ϵ0.(ii)
The mapℋϵ is contractive, namely,(122)ℋϵw2u,m−ℋϵw1u,mk1,β,μ,α,r≤12w2u,m−w1u,mk1,β,μ,α,r,wheneverw1,w2∈B¯0,ϖ, for all ϵ∈D0,ϵ0.Proof.
We first control the forcing term. Owing to bound (76) in Lemma 4, together with (114), we can exhibit a constant C1>0 (relying on q,k1,α,T0) with(123)ψu,m,ϵPmuk1,β,μ,α,r≤K0C1CPminm∈ℝRDim,where K0>0 is a constant that is set in (36), whenever ϵ∈D0,ϵ0.
We deal with the first property (121). Let us take wτ,m in Expk1,β,μ,α,rq under the constraint wτ,mk1,β,μ,α,r≤ϖ.
We fix some complex numberωd such that ωd∉Ud¯∪D0,r¯, and we redraft the norm of the next integral expression as follows:(124)I1=∫−∞+∞Clm−m1,ϵRlim11Pmu×σq;ul1−l0/kuk′∫0uk′uk′−seh,l0/k′−1sh−1ws1/k′,m1dsdm1k1,β,μ,α,r=1RDim∫−∞+∞Clm−m1,ϵRlim1ℬu,mu−ωdpl0,l1σq;u−l0/k−l1uk′u−ωdl0×∫0uk′uk′−seh,l0/k′−1sh−1ws1/k′,m1dsdm1k1,β,μ,α,r,for all l∈I, 1≤h≤l2, where pl0,l1≥0 is an integer chosen as in (117) and(125)ℬu,m=q−l0/k−l1u−ωdl0RDimu−ωdpl0,l1Pmu.
We observe that a constantMℬ>0 (depending on q,l0,l1,pl0,l1,k,k′,mD,ωd,M) can be picked up with(126)ℬu,m≤Mℬ,for all u∈Ud∪D0,r, m∈ℝ. Indeed, from (114), we obtain(127)ℬu,m≤1CPsupu∈Ud¯∪D0,r¯q−l0/k−l1u−ωdl0u−ωdpl0,l11+uk′mD,for all u∈Ud∪D0,r, m∈ℝ where the right-hand side is finite owing to the suitable choices of ωd and pl0,l1 in (118).
Under requirements (32) and (116), an application of Proposition 3 yields a constant M3.1>0 (depending on RD,Rl and μ) such that(128)I1≤M3.1MℬClm,ϵβ,μI2,where(129)I2=u−ωdpl0,l1σq;u−l0/k−l1uk′u−ωdl0∫0uk′uk′−seh,l0/k′−1sh−1ws1/k′,mdsk1,β,μ,α,r.
Conditions (116) and (117) allow us to call back Proposition 2 in order to get a constant M2.1>0 (depending on l0,l1,pl0,l1,k,q,α,δ,r,ωd) with(130)I2≤M2.1I3,where(131)I3=uk′u−ωdl0∫0uk′uk′−seh,l0/k′−1sh−1ws1/k′,mdsk1,β,μ,α,r.
Lastly, Proposition1 gives rise to constants Mωd,l0>0 (depending on ωd,l0) and M1.1>0 (depending on k′,l0,l2) with(132)I3≤M1.1Mωd,l0wu,mk1,β,μ,α,r.
By compiling (128)–(132), we obtain(133)I1≤M1.1M2.1M3.1MℬMωd,l0Clm,ϵβ,μwu,mk1,β,μ,α,r,for all l∈I, 1≤h≤l2.
We now turn to the second principal pieces ofℋϵ. Following the same lines of arguments as above, we obtain that(134)J1=∫−∞+∞Clm−m1,ϵRlim11Pmu×σq;ul1−l0/kuk′∫0uk′uk′−seh,l0+k′h−p/k′−1sp−1ws1/k′,m1dsdm1k1,β,μ,α,r≤M2.1M3.1MℬClm,ϵβ,μJ3,where(135)J3=uk′u−ωdl0∫0uk′uk′−seh,l0+k′h−p/k′−1sp−1ws1/k′,mdsk1,β,μ,α,r,for all l∈I, 2≤h≤l2 and 1≤p≤h−1. In order to give bounds for J3, we make use of Proposition 1 which affords a constant M1.2>0 (depending on k′,l0,l2) with(136)J3≤M1.2Mωd,l0wu,mk1,β,μ,α,r.
By combining (134) and (136), we obtain(137)J1≤M1.2M2.1M3.1MℬMωd,l0Clm,ϵβ,μwu,mk1,β,μ,α,r,for all l∈I, 2≤h≤l2 and 1≤p≤h−1.
In the next step, we impose the constantsCl>0, l∈I, to stay close enough to 0 in order that a constant ϖ>0 can be singled out with(138)∑l=l0,l1,l2∈Iϵ0Δl−l012π1/21q1/kl0l0−1/2∑h=1l2ah,l2ql1−l0/kl0×k′hΓeh,l0/k′M1.1M2.1M3.1MℬMωd,l0Clϖ+∑1≤p≤h−1Ah,pk′pΓeh,l0+k′h−p/k′M1.2M2.1M3.1MℬMωd,l0Clϖ+K0C1CPminm∈ℝRDim≤ϖ.
The collection of (123), 133 and (137) submitted to condition (138) yields the inclusion (121).
The next part of the proof is devoted to the explanation of the contractive property (122). Indeed, consider two functions w1u,m and w2u,m inside the ball B0,ϖ¯⊂Expk1,β,μ,α,rq. Then, an application of the two inequalities (133) and (137) overhead leads to(139)∫−∞+∞Clm−m1,ϵRlim11Pmu×σq;ul1−l0/kuk′∫0uk′uk′−seh,l0/k′−1sh−1w2s1/k′,m1−w1s1/k′,m1dsdm1k1,β,μ,α,r≤M1.1M2.1M3.1MℬMωd,l0Clm,ϵβ,μw2u,m−w1u,mk1,β,μ,α,r,for all l∈I, 1≤h≤l2 and(140)∫−∞+∞Clm−m1,ϵRlim11Pmu×σq;ul1−l0/kuk′∫0uk′uk′−seh,l0+k′h−p/k′−1sp−1w2s1/k′,m1−w1s1/k′,m1dsdm1k1,β,μ,α,r≤M1.2M2.1M3.1MℬMωd,l0Clm,ϵβ,μw2u,m−w1u,mk1,β,μ,α,r,for all l∈I, 2≤h≤l2 and 1≤p≤h−1.
This time, we require the constantsCl>0, l∈I, to withstand the next inequality(141)∑l=l0,l1,l2∈Iϵ0Δl−l012π1/21q1/kl0l0−1/2∑h=1l2ah,l2ql1−l0/kl0×k′hΓeh,l0/k′M1.1M2.1M3.1MℬMωd,l0Cl+∑1≤p≤h−1Ah,pk′pΓeh,l0+k′h−p/k′M1.2M2.1M3.1MℬMωd,l0Cl≤12.
Owing to (139) and (140), under demand (141), we obtain (122).
In conclusion, we choose the constantsCl>0, l∈I in order that both (138) and (141) hold conjointly. This yield Lemma 10.
We go back to the core of Proposition4. For ϖ>0, chosen as in the lemma above, we consider the closed ball B0,ϖ¯⊂Expk1,β,μ,α,rd that stands for a complete metric space for the distance dx,y=x−yk1,β,μ,α,r. According to the same lemma, we observe that ℋϵ induces a contractive application from B0,ϖ¯,d into itself. Then, according to the classical contractive mapping theorem, the map ℋϵ carries a unique fixed point that we set as wdu,m,ϵ; meaning that(142)ℋϵwdu,m,ϵ=wdu,m,ϵ,that belongs to the ball B0,ϖ¯, for all ϵ∈D0,ϵ0. Furthermore, the function wdu,m,ϵ depends holomorphically on ϵ in D0,ϵ0. Let the term(143)RDimk′mDq1/kmDk′k′−1/2uk′mDwu,m,ϵ,be taken from the right to the left-hand side of (81) and then divide by the polynomial Pmu defined in (109). These operations allows (81) to be exactly recast into equation (142) above. Consequently, the unique fixed point wdu,m,ϵ of ℋϵ obtained overhead in B¯0,ϖ precisely solves equation (81).


## 5. An Integrodifferentialq-Difference Equation with a Complex Parameter

In this section, we build up a solutionWdτ,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.Proposition5.
We depart from the solutionwdu,m,ϵ of the integral equation (81) that has just been constructed in Proposition 4. We define(144)Wdτ,m,ϵ=k′∫Lγwdu,m,ϵexp−uτk′duu,as the Laplace transform of wdu,m,ϵ of order k′ in direction d, where the integration path Lγ=ℝ+e−1γ belongs to the sector Ud∪0. Then, for all ϵ∈D0,ϵ0, the map τ,m,ϵ⟼Wdτ,m,ϵ is continuous on a domain Sd×ℝ×D0,ϵ0 and depends holomorphically on τ,ϵ in Sd×D0,ϵ0, where Sd represents an unbounded sector with bisecting direction d and opening θk′ that fulfills(145)0<θk′<πk′+ApUd,for ApUd defined as the aperture of the sector Ud. Furthermore, the map τ,m,ϵ⟼Wdτ,m,ϵ withstands the next accurate bounds:(1)
One can single out three constantsE1>0 (depending on k′,k1,δ,q,α), E2>1 (depending on k1,k′,q), and 0<E3<1 (relying on k′,θk′,ApU) such that(146)Wdτ,m,ϵ≤E1ϖ1+m−μe−βmexpk12log2τ/E3logq+α+E2logτ/E3×expk1logτ/E3k′logq+12logk1logτ/E3k′logq+1,for allτ∈Sd with τ≥21/k′E3δ, m∈ℝ and ϵ∈D0,ϵ0.(2)
One can find a constantE0>0 (depending on k′,k1,δ,q,α) with(147)Wdτ,m,ϵ≤E0ϖ1+m−μe−βm,whenever τ∈Sd with τ≤21/k′E3δ, all m∈ℝ, all ϵ∈D0,ϵ0.
In particular, one can sort two constantsE4,E5>0 (depending on k′,k1,δ,q,α,θk′,ApUd) with(148)Wdτ,m,ϵ≤E4ϖ1+m−μe−βmexpk22log2τ+δlogq+α+E5logτ+δ,for all τ∈Sd, m∈ℝ and ϵ∈D0,ϵ0, where k1<k2<k was introduced in Section 3 just above (47).
Finally,Wdτ,m,ϵ satisfies the first auxiliary integrodifferential q-difference equation (69) on the domain Sd×ℝ×D0,ϵ0.Proof.
Bound (119) in Proposition 4 can be recast as(149)wdu,m,ϵ≤ϖ1+m−μe−βmuexpk12log2u+δlogq+αlogu+δ,which holds for all u,m,ϵ∈Ud∪D0,r×ℝ×D0,ϵ0. The integral representation (144) yields(150)Wdτ,m,ϵ≤k′∫0+∞ϖ1+m−μe−βmexpk12log2r+δlogq+αlogr+δ×exp−rk′τk′cosk′γ−argτdr≤k′∫0+∞ϖ1+m−μe−βm×expk12log2r+δlogq+αlogr+δexp−rk′τk′δ2dr,whenever τ∈Sd for a well-chosen direction γ (that may depend on τ) such that cosk′γ−argτ≥δ2 for some fixed constant 0<δ2<1 that exists under the requirement (145).
In the second part of the proof, we are scaled down to provide bounds for the next associated function:(151)Ex=∫0+∞expk12log2r+δlogq+αlogr+δexp−rk′xdr,when x>0 is chosen large enough. The next lemma holds.Lemma11.
One can select two constantsE1>0 (depending on k′,k1,δ,q,α) and E2>1 (relying in k1,k′,q) such that(152)Ex≤E1expk12log2x1/k′logq+α+E2logx1/k′×expk1logx1/k′k′logq+12logk1logx1/k′k′logq+1,for all x≥2δk′.Proof.
We first make the change of variabler˜=rk′/x in the integral above:(153)Ex=∫0+∞expk12log2r˜1/k′x1/k′+δlogq+αlogr˜1/k′x1/k′+δe−r˜r˜1/k′−1k′x1/k′dr˜.
On the other hand, we need the next expansions:(154)log2r˜1/k′x1/k′+δ=log2x1/k′+2logx1/k′logr˜1/k′+δx1/k′+log2r˜1/k′+δx1/k′,logr˜1/k′x1/k′+δ=logx1/k′+logr˜1/k′+δx1/k′.
We cut the integral expression in two pieces:(155)Ex=E1x+E2x,where(156)E1x=∫01−δ/x1/k′k′expk12log2r˜1/k′x1/k′+δlogq+αlogr˜1/k′x1/k′+δe−r˜r˜1/k′−1k′x1/k′dr˜,(157)E2x=∫1−δ/x1/k′k′+∞expk12log2r˜1/k′x1/k′+δlogq+αlogr˜1/k′x1/k′+δe−r˜r˜1/k′−1k′x1/k′dr˜,provided that x≥2δk′.
We control the first pieceE1x. We observe that logr˜1/k′+δ/x1/k′≤0 when r˜∈0,1−δ/x1/k′k′. From (154), we deduce the inequalities(158)log2r˜1/k′x1/k′+δ≤log2x1/k′+log2r˜1/k′+δx1/k′,logr˜1/k′x1/k′+δ≤logx1/k′,for all r˜∈0,1−δ/x1/k′k′ and x≥2δk′. Therefore,(159)E1x≤expk12log2x1/k′logq+αlogx1/k′x1/k′E1.2x,where(160)E1.2x=1k′∫01−δ/x1/k′k′expk12log2r˜1/k′+δ/x1/k′logqe−r˜r˜1/k′−1dr˜.
By construction, a constantE1.2>0 (depending on k′,k1,δ,q) can be found with(161)E1.2x≤E1.2,for all x≥2δk′.
In a second step, we evaluate the partE2x. Expansion (154) affords us to write(162)E2x≤expk12log2x1/k′logq+αlogx1/k′x1/k′E2.1x,where(163)E2.1x=∫1−δ/x1/k′k′+∞r˜1/k′+δx1/k′k1logx1/k′/logq×expk12log2r˜1/k′+δ/x1/k′logq+αlogr˜1/k′+δx1/k′e−r˜1k′r˜1/k′−1dr˜.
Besides, we can check that there exists a constantE2.1>0 (depending on k′,k1,δ,q,α) such that(164)expk12log2r˜1/k′+δ/x1/k′logq+αlogr˜1/k′+δx1/k′1k′r˜1/k′−1≤E2.1er˜/2,provided that r˜≥1−δ/x1/k′k′, when x≥2δk′. We deduce that(165)E2.1x≤E2.1∫1−δ/x1/k′k′+∞r˜1/k′+δx1/k′k1logx1/k′/logqe−r˜/2dr˜≤E2.1E2.2x,where(166)E2.2x=∫1−1/21/k′k′+∞r˜1/k′+121/k′k1logx1/k′/logqe−r˜/2dr˜,when x≥2δk′. Furthermore, one can sort a constant E2.2>1 (depending on k′) such that(167)r˜1/k′+121/k′≤E2.2r˜1/k′,for all r˜≥1−1/21/k′k′. Hence,(168)E2.2x≤E2.2k1logx1/k′/logq∫1−1/21/k′k′+∞r˜k1/k′logx1/k′/logqe−r˜/2dr˜≤E2.2k1logx1/k′/logqE2.3x,with(169)E2.3x=∫0+∞r˜k1/k′logx1/k′/logqe−r˜/2dr˜,when x≥2δk′. We perform the linear change of variable h=r˜/2 in this latter integral(170)E2.3x=2k1logx1/k′/k′logq+1∫0+∞hk1logx1/k′/k′logqe−hdh=2k1logx1/k′/k′logq+1Γk1logx1/k′k′logq+1,in order to express it in terms of the Gamma function Γx. Keeping in mind the Stirling formula (48), we get a constant E2.3>0 (which depends on k1,k′,q,δ) such that(171)E2.3x≤E2.32ek1logx1/k′/k′logq+1expk1logx1/k′k′logq+12logk1logx1/k′k′logq+1≤E2.3expk1logx1/k′k′logq+12logk1logx1/k′k′logq+1,for all x≥2δk′.
Finally, splitting (155) together with the collection of bounds (159)–(162) and (165)–(171) gives rise to the expected bound (152).
This last lemma combined with estimate (150) yield the announced upper bounds (146) and (147). In order to deduce the particular estimate (148) from (146) and (147), we observe that for any given χ>0 (even close to 0), we can sort a constant Aχ,k,k′,q,E3,δ>0 (which relies on χ,k,k′,q,E3,δ) such that(172)logk1k′logqlogzE3+1≤χlogz+δ+Aχ,k,k′,q,E3,δ,for all z≥21/k′E3δ.
In the final part of the proof, the functionWdτ,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 wdu,m,ϵ solves (81) and appertains to the space Expk1,β,μ,α,rq, for a well-chosen sector Ud, the three identities of Lemma 8 can be applied in order to check that Wdτ,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).


## 6. 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 Section3 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 inℂ∗, 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].Definition6.
Letς≥2 be an integer. We consider a set ℰ¯ of open sectors ℰp centered at 0, with radius ϵ0>0 for all 0≤p≤ς−1 for which the next three properties hold:(i)
The intersectionℰp∩ℰp+1 is not empty for all 0≤p≤ς−1 (with the convention that ℰς=ℰ0)(ii)
The intersection of any three elements ofℰ¯ is empty(iii)
The union∪p=0ς−1ℰp equals U∖0 for some neighborhood U of 0 in ℂ
Then, the set of sectorsℰ¯ is named a good covering of ℂ∗.Definition7.
We consider(i)
A good coveringℰ¯=ℰp0≤p≤ς−1 of ℂ∗ whose radius ϵ0 satisfies 0<ϵ0<1(ii)
A setU¯ of unbounded sectors Udp, 0≤p≤ς−1, centered at 0 with bisecting direction dp∈ℝ and small opening θUdp>0(iii)
A setS¯ of unbounded sectors Sdp, 0≤p≤ς−1, centered at 0 with bisecting direction dp∈ℝ and aperture 0<θSdp<π/k′+θUdp, for some integer k′≥1(iv)
A fixed bounded sectorT centered at 0 with radius rT>0 and a disc D0,r suitably selected in a way that the next features are conjointly satisfied:(a)
Bound (112) holds, provided that u∈Udp∪D0,r, for all 0≤p≤ς−1(b)
The setS¯ fulfills the next properties:(1)
The intersectionSdp∩Sdp+1 is not empty for all 0≤p≤ς−1 (with the convention that Sdς=Sd0)(2)
The union∪p=0ς−1Sdp covers ℂ∖0(c)
For all0≤p≤ς−1, all ϵ∈ℰp and all t∈T:(173)ϵt∈ℛdp,Δ,where(174)ℛdp,Δ=T∈ℂ∗1+e−1dp/Tr>Δ,for all r≥0,withΔ>0 any fixed real number close to 0.
When the above features are verified, we say that the set of dataℰ¯,U¯,S¯,T,D0,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 ℰp, 0≤p≤ς−1, of a good covering in ℂ∗. Besides, we are able to monitor the difference between consecutive solutions on the intersections ℰp∩ℰp+1.Theorem1.
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(175)A¯=ℰ¯=ℰp0≤p≤ς−1,U¯=Udp0≤p≤ς−1,S¯=Sdp0≤p≤ς−1,T,D0,r,as described in the definition above.
Then, for a suitable choice of the constantsCl>0 (c.f. (34)) close enough to 0 for all l∈I, a collection upt,z,ϵ0≤p≤ς−1 of true solutions of (33) can be singled out. More precisely, each function upt,z,ϵ stands for a bounded holomorphic map on the product T∩D0,σ×Hβ′×ℰp for any given 0<β′<β and appropriate small radius σ>0. Additionally, upt,z,ϵ is represented as a q-Laplace transform of order k and Fourier inverse integral:(176)upt,z,ϵ=1πq1/k2π1/2∫−∞+∞∫LγpWdpu,m,ϵ1Θq1/ku/ϵteizmduudm,where Lγp=ℝ+e−1γp⊂Sdp∪0. Furthermore, the map τ,m,ϵ⟶Wdpτ,m,ϵ is itself fashioned as a Laplace transform of order k′:(177)Wdpτ,m,ϵ=k′∫Lγp′wdpu,m,ϵexp−uτk′duu,whose integration halfline Lγp′ is enclosed in Udp∪0 and where u,m⟼wdpu,m,ϵ belongs to the Banach space Expk1,β,μ,α,rq for the unbounded sector Udp, provided that ϵ∈D0,ϵ0.
Finally, some constantsAp,Bp>0 can be found with(178)supt∈T∩D0,σ,z∈Hβ′up+1t,z,ϵ−upt,z,ϵ≤ApBpNΓN+1k′qN2/2kϵN,for all integers N≥1, all 0≤p≤ς−1, whenever ϵ∈ℰp+1∩ℰp, where by convention, we set uςt,z,ϵ=u0t,z,ϵ.Proof.
We first single out an admissible set of dataA¯. Under the requirements enounced in Theorem 1, Proposition 5 can be called in order to find a family of functions:(179)Wdpτ,m,ϵ=k′∫Lγp′wdpu,m,ϵexp−uτk′duu,expressed as a Laplace transform of order k′ in direction γp′ such that Lγp′=ℝ+e−1γp′⊂Udp∪0 of a Borel map wdpu,m,ϵ which is holomorphic w.r.t u on Udp∪D0,r, w.r.t ϵ on D0,ϵ0, and continuous relatively to m∈ℝ, coming along with a constant ϖdp>0 such that(180)wdpu,m,ϵ≤ϖdp1+m−μe−βmuexpk12log2u+δlogq+αlogu+δ,for all u∈Udp∪D0,r, m∈ℝ, and ϵ∈D0,ϵ0. Furthermore, the function Wdpτ,m,ϵ solves the first auxiliary integrodifferential q-difference equation (69) on Sdp×ℝ×D0,ϵ0 and suffers the bounds:(181)Wdpτ,m,ϵ≤ρ1dp1+m−μe−βmexpk22log2τ+δlogq+α+ρ2dplogτ+δ,for some constants ρjdp>0, for j=1,2, provided that τ∈Sdp, m∈ℝ, and ϵ∈D0,ϵ0.
We now revisit the first stage of the formal construction from Section3. Namely, we set the next q-Laplace transform of order k and Fourier inverse map(182)UγpT,z,ϵ=1πq1/k2π1/2∫−∞+∞∫LγpWdpτ,m,ϵ1Θq1/kτ/Teizmdττdm,along a halfline Lγp=ℝ+e−1γp⊂Sdp∪0. 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 UγpT,z,ϵ stands for(a)
A bounded holomorphic function w.r.tT on a domain ℛdp,Δ∩D0,r0 for some small radius r0>0, where ℛdp,Δ is described in (173)(b)
A bounded holomorphic application relatively to the couplez,ϵ on Hβ′×D0,ϵ0, for any given 0<β′<β
Additionally, sinceWdpτ,m,ϵ solves (69), Lemma 5 leads to the claim that UγpT,z,ϵ must fulfill the singular equation (64) on ℛdp,Δ∩D0,r0×Hβ′×D0,ϵ0.
In conclusion, the function defined as(183)upt,z,ϵ=Uγpϵt,z,ϵ,represents a bounded holomorphic function w.r.t t on T∩D0,σ for some σ>0 close enough to 0, ϵ∈ℰp, z∈Hβ′ for any given 0<β′<β, owing to assumption 3 of Definition 7. Moreover, upt,z,ϵ solves the main initial value problem (33) on the domain T∩D0,σ×Hβ′×ℰp, for any 0≤p≤ς−1.
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(184)τ⟼Wdpτ,m,ϵΘq1/kτ/ϵtτ,is holomorphic on the sector Sdl. By the Cauchy theorem, we can bend each straight halfline Lγl, l=p,p+1 into the union of three curves with appropriate orientation depicted as follows:(1)
A halflineLγl,r1=r1,+∞ for a given real number r1>0(2)
An arc of circle with radiusr1 denoted Cr1,γl,γp,p+1 joining the point r1exp−1γp,p+1 which is taken inside the intersection Sdp∩Sdp+1 (that is assumed to be nonempty, see Definition 7, 2.1) to the halfline Lγl,r1(3)
A segmentLγp,p+1,0,r1=0,r1exp−1γp,p+1
As a result, the differenceup+1−up can be decomposed into a sum of five integrals along these curves:(185)up+1t,z,ϵ−upt,z,ϵ=1πq1/k2π1/2∫−∞+∞∫Lγp+1,r1Wdp+1τ,m,ϵ1Θq1/kτ/ϵteizmdττdm−1πq1/k2π1/2∫−∞+∞∫Lγp,r1Wdpτ,m,ϵ1Θq1/kτ/ϵteizmdττdm+1πq1/k2π1/2∫−∞+∞∫Cr1,γp+1,γp,p+1Wdp+1τ,m,ϵ1Θq1/kτ/ϵteizmdττdm−1πq1/k2π1/2∫−∞+∞∫Cr1,γp,γp,p+1Wdpτ,m,ϵ1Θq1/kτ/ϵteizmdττdm+1πq1/k2π1/2∫−∞+∞∫Lγp,p+1,0,r1Wdp+1τ,m,ϵ−Wdpτ,m,ϵ1Θq1/kτ/ϵteizmdττdm.
Bounds for the first piece,(186)I1=1πq1/k2π1/2∫−∞+∞∫Lγp+1,r1Wdp+1τ,m,ϵ1Θq1/kτ/ϵteizmdττdm,are now considered. The arguments followed are proximate to the ones displayed in the proof of Theorem 1 from [5]. Owing to Lemma 1 and bound (181), we obtain(187)I1≤1πq1/k2π1/2∫−∞+∞∫r1+∞ρ1dp+11+m−μe−βm×expk22log2r+δlogq+α+ρ2dp+1logr+δ1Cq,kΔr1/2/ϵt1/2exp−k2log2r/ϵtlogq×exp−mImzdrr,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. We need the next two expansions:(188)log2r+δ=log2r+2logrlog1+δr+log21+δr,log2rϵt=log2r−2logrlogϵ−2logrlogt+log2ϵ+2logϵlogt+log2t.
Hence,(189)I1≤ρ1dp+1πq1/k2π1/2Cq,kΔϵt1/2exp−k2log2ϵlogq×∫−∞+∞exp−β−β′mdmexp−klogqlogϵlogtexp−k2logqlog2t×∫r1+∞exp−log2r2logqk−k21r3/2expk2logqlogrlog1+δr+k22logqlog21+δr+α+ρ2dp+1logr+δ+klogqlogrlogt+klogqlogrlogϵdr,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. We now specify estimates for some pieces of these last upper bounds. Namely, since log1+x∼x as x tends to 0, we get a constant A1.1>0 (depending on r1,δ) such that(190)logrlog1+δr≤A1.1,for all r≥r1. Since 0<ϵ0<1, we also notice that(191)exp−klogqlogϵlogt≤ϵ−k/logqlogrT,expklogqlogrlogϵ≤ϵk/logqlogr1,for all t∈T∩D0,σ, ϵ∈ℰp+1∩ℰp, and r≥r1. Furthermore,(192)expklogqlogrlogt≤tk/logqlogr1,whenever r1≤r≤1 and 0<σ<1 together with(193)expklogqlogrlogt≤rk/logqlogσ,provided that r≥1. Finally, there exists a constant Kk,r1,q>0 (depending on k,r1,q) with(194)supx>0xk/logqlogr1exp−k2logqlog2x≤Kk,r1,q.
Inequality (189) together with the collection of bounds (190)–(194) yield two constants I1.1>0 and I1.2∈ℝ (which rely on k,q,Δ,k2,δ,r1,ρ1dp+1,ρ2dp+1,β,β′) such that(195)I1≤I1.1ϵI1.2exp−k2log2ϵlogq,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. We want to express these last bounds in terms of sequences now. The discussion hinges on the next lemma.Lemma12.
The following inequality(196)ϵ−NϵI1.2exp−k2log2ϵlogq≤qI1.22/2kq−I1.2/kNqN2/2k,holds for all ϵ∈ℂ∗, all integers N≥1.Proof.
By performing the change of variablex=logϵ with the help of the computation already undertaken in Lemma 4, we obtain(197)expI1.2−Nlogϵ−k2log2ϵlogq≤supx∈ℝexpI1.2−Nx−k2x2logq≤qI1.2−N2/2k=qI1.22/2kq−I1.2/kNqN2/2k,for all given ϵ∈ℂ∗ and integer N≥1.
Consequently to (195) and (196), two constants I1.3,I1.4>0 (depending on I1.1,I1.2,q,k) can be picked up with(198)I1≤I1.3I1.4NqN2/2kϵN,whenever ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′, for all integers N≥1.
With a similar discussion, we can exhibit comparable bounds for the next term:(199)I2=1πq1/k2π1/2∫−∞+∞∫Lγp,r1Wdpτ,m,ϵ1Θq1/kτ/ϵteizmdττdm.
Namely, two constantsI2.1>0 and I2.2∈ℝ (depending on k,q,Δ,k2,δ,r1,ρ1dp,ρ2dp,β,β′) can be found with(200)I2≤I2.1ϵI2.2exp−k2log2ϵlogq,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. Furthermore, we can single out two constants I2.3,I2.4>0 (resting on I2.1,I2.2,q,k) such that(201)I2≤I2.3I2.4NqN2/2kϵN,provided that ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′, for all integers N≥1.
In the next step, we turn to the first integral along an arc of circle:(202)I3=1πq1/k2π1/2∫−∞+∞∫Cr1,γp+1,γp,p+1Wdp+1τ,m,ϵ1Θq1/kτ/ϵteizmdττdm.
Making use of Lemma1 and (181), gives rise to the inequality(203)I3≤1πq1/k2π1/2∫−∞+∞∫γp+1γp,p+1ρ1dp+11+m−μe−βm×expk22log2r1+δlogq+α+ρ2dp+1logr1+δ1Cq,kΔr11/2/ϵt1/2×exp−k2log2r1/ϵtlogqexp−mImzdθdm,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. We require once more the expansion:(204)log2r1ϵt=log2r1−2logr1logϵ−2logr1logt+log2ϵ+2logϵlogt+log2t.
We deduce that(205)I3≤2ρ1dp+1πq1/k2π1/2Cq,kΔr11/2β−β′γp+1−γp,p+1×expk22log2r1+δlogq+α+ρ2dp+1logr1+δexp−k2log2r1logqϵt1/2exp−k2log2ϵlogq×expklogqlogr1logϵ−klogqlogϵlogtexpklogqlogr1logt×exp−k2log2tlogq.
Owing to the hypothesis0<ϵ0<1, we check that (191) holds and bearing in mind (194), we arrive at the existence of two constants I3.1>0 and I3.2∈ℝ (depending on the constants k,q,Δ,k2,δ,r1,ρ1dp+1,ρ2dp+1,β,β′) with(206)I3≤I3.1ϵI3.2exp−k2log2ϵlogq,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. Calling back Lemma 12 gives rise to two additional constants I3.3,I3.4>0 (subjected to I3.1,I3.2,q,k) with(207)I3≤I3.3I3.4NqN2/2kϵN,for all ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′, for all given integers N≥1.
The second integral along an arc of circle(208)I4=1πq1/k2π1/2∫−∞+∞∫Cr1,γp,γp,p+1Wdpτ,m,ϵ1Θq1/kτ/ϵteizmdττdmcan be managed in the same way. Indeed, one can single out two constants I4.1>0 and I4.2∈ℝ (relying on k,q,Δ,k2,δ,r1,ρ1dp,ρ2dp,β,β′) such that(209)I4≤I4.1ϵI4.2exp−k2log2ϵlogq,provided that ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′. Moreover, we can find two constants I4.3,I4.4>0 (that hinge on I4.1,I4.2,q,k) such that(210)I4≤I4.3I4.4NqN2/2kϵN,whenever ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′, for all integers N≥1.
In the remaining part of the proof, we inspect the last integral along the segment:(211)I5=1πq1/k2π1/2∫−∞+∞∫Lγp,p+1,0,r1Wdp+1τ,m,ϵ−Wdpτ,m,ϵ×1Θq1/kτ/ϵteizmdττdm.
We require a lead-in lemma which supplies exponential flatness for the differenceWdp+1−Wdp.Lemma13.
For each0≤p≤ς−1, we can sort two constants KpW,MpW>0 such that(212)Wdp+1τ,m,ϵ−Wdpτ,m,ϵ≤KpWexp−MpWτk′1+m−μe−βm,for all ϵ∈D0,ϵ0, all m∈ℝ and all τ∈Sdp+1∩Sdp∩D0,r1, whenever it is assumed that(213)0<r1≤δ1−δ2Δ21/k′,for some fixed 0<δ2<δ1 close enough to 0 and any given positive real number Δ2>0, under the convention that Wdς=Wd0.Proof.
We first observe that all the mapsu⟼wdpu,m,ϵ, 0≤p≤ς−1, are analytic continuations on the sector Udp of a unique holomorphic function that we name u⟼wu,m,ϵ on the disc D0,r which suffers the same bound (180). Furthermore, the application u⟼wu,m,ϵexp−u/τk′/u is holomorphic on D0,r when τ∈Sdp+1∩Sdp, and its integral is therefore vanishing along an oriented path described as the union of(a)
A segment linking 0 tor/2exp−1γp+1′(b)
An arc of circle with radiusr/2 joining the points r/2exp−1γp+1′ and r/2exp−1γp′(c)
A segment attachingr/2exp−1γp′ and the origin
As a result, taking heed of the integral representation (179) of Wdp+1 and Wdp, we can convert the difference Wdp+1−Wdp into a sum of three integrals:(214)Wdp+1τ,m,ϵ−Wdpτ,m,ϵ=k′∫Lγp+1′,r/2wdp+1u,m,ϵexp−uτk′duu−k′∫Lγp′,r/2wdpu,m,ϵexp−uτk′duu+k′∫Cr/2,γp′,γp+1′wu,m,ϵexp−uτk′duu,where the integrations paths are two halflines and an arc of circle staying aside from the origin that are depicted as follows:(215)Lγp+1′,r/2=r2,+∞exp−1γp+1′,Lγp′,r/2=r2,+∞exp−1γp′,Cr/2,γp′,γp+1′=r2exp−1θ:θ∈γp′,γp+1′.
We consider the first integral along a halfline in the above splitting:(216)J1=k′∫Lγp+1′,r/2wdp+1u,m,ϵexp−uτk′duu.
The directionγp+1′ (which might depend on τ) is properly chosen in order that(217)cosk′γp+1′−argτ≥δ1,for all τ∈Sdp+1∩Sdp, for some fixed δ1>0. Besides, let Δ2>0 be any given positive real number (even close to 0). Then, we can find a constant B1.1>0 (depending on k1,k′,δ,q,α,r,Δ2) such that(218)k12log2s+δlogq+αlogs+δ≤Δ2sk′+B1.1,for all s≥r/2. According to estimate (180), we obtain that(219)J1≤k′∫r/2+∞ϖdp+11+m−μe−βmsexpk12log2s+δlogq+αlogs+δ×exp−cosk′γp+1′−argττk′sk′dss≤k′ϖdp+1eB1.11+m−μe−βm∫r/2+∞exp−sk′τk′δ1−τk′Δ2ds≤k′ϖdp+1eB1.11+m−μe−βm∫r/2+∞exp−sk′τk′δ2ds≤k′ϖdp+1eB1.11+m−μe−βmτk′δ21k′r/2k′−1∫r/2+∞δ2τk′k′sk′−1exp−sk′τk′δ2ds=k′ϖdp+1eB1.11+m−μe−βmτk′δ21k′r/2k′−1exp−r/2k′τk′δ2,for all ϵ∈D0,ϵ0, all m∈ℝ, provided that τ∈Sdp+1∩Sdp with(220)τ≤δ1−δ2Δ21/k′,for a given 0<δ2<δ1.
In a similar manner, we disclose bounds for the next integral over a halfline:(221)J2=k′∫Lγp′,r/2wdpu,m,ϵexp−uτk′duu.
Indeed, the directionγp′ (that relies on τ) is properly chosen in order that(222)cosk′γp′argτ≥δ1,for all τ∈Sdp+1∩Sdp, for some fixed δ1>0. The use of (180) together with a record of bounds ressembling (219) allows(223)J2≤k′ϖdpeB1.11+m−μe−βmτk′δ21k′r/2k′−1exp−r/2k′τk′δ2to hold whenever ϵ∈D0,ϵ0, m∈ℝ, and τ∈Sdp+1∩Sdp restricted to (220) for given 0<δ2<δ1 and Δ2>0.
In the remaining part of the lemma, we evaluate the third integral along an arc of circle:(224)J3=k′∫Cr/2,γp′,γp+1′wu,m,ϵexp−uτk′duu.
The circleCr/2,γp′,γp+1′ satisfies the lower bounds(225)cosk′θ−argτ≥δ1,for some fixed δ1>0, for all θ∈γp′,γp+1′ (if γp′<γp+1′) or θ∈γp+1′,γp′ (if γp+1′<γp′) granting that τ∈Sdp+1∩Sdp. Again, estimate (180) together with (218) for s=r/2 brings on(226)J3≤k′∫γp′γp+1′maxϖdp,ϖdp+11+m−μe−βmr2expk12log2r/2+δlogq+αlogr2+δ×exp−cosk′θ−argττk′r2k′dθ≤k′maxϖdp,ϖdp+1γp+1′−γp′eB1.1×1+m−μe−βmr2×exp−r/2k′τk′δ1−τk′Δ2≤k′maxϖdp,ϖdp+1γp+1′−γp′eB1.1×1+m−μe−βmr2×exp−r/2k′τk′δ2,for all ϵ∈D0,ϵ0, m∈ℝ and τ∈Sdp+1∩Sdp submitted to (220) for given 0<δ2<δ1 and Δ2>0.
By collecting the above inequalities (219), (223), and (226) applied to the splitting of (214), we achieve the announced bounds (212).
Onwards, we take for granted that the real numberr1>0 selected in the above deformation (1), (2), and (3) suffers restriction (213) and 0<r1≤1. Bound (212) in a row with Lemma 1 yields:(227)I5≤KpWπq1/k2π1/2Cq,kΔ∫−∞+∞exp−β−β′mdmϵt1/2I˜5ϵt,where(228)I˜5ϵt=∫0r1exp−MpWrk′exp−k2log2r/ϵtlogq1r3/2dr,for all ϵ∈ℰp+1∩ℰp, all t∈T∩D0,σ and all z∈Hβ′. Bound control given below in (235) are now provided for this parameter depending on last integral. The ongoing reasoning leans on the next elementary lemma.Lemma14.
(1)
The next inequality(229)1rNexp−MpWrk′≤CΓ,k′CNNk′1/2ΓNk′
holds for all integersN≥1 and all positive real numbers r>0, where C=1/MpW1/k′ and CΓ,k′>0 is a constant depending on k′.
(2)
For allr>0, all ϵ,t∈ℂ∗, the estimate(230)rϵtNexp−k2log2r/ϵtlogq≤qN2/2k
arises for all integersN≥1.Proof.
For the first item (1), using the change of variable x=1/rk′ we observe that(231)supr>01rNexp−1rk′MpW=supx>0xN/k′exp−xMpW=Nk′MpWN/k′exp−Nk′,for all integers N≥1. On the other hand, from the Stirling formula (48), we get a constant CΓ,k′>0 (depending on k′) such that(232)e−xxx≤CΓ,k′x1/2Γx,for all x≥1/k′. Gathering (231) and (232) yields (229).
The second item (2) can be treated in a similar way through the successive changes of variables y=r/ϵt and x=logy by using the computation already carried out in Lemma 4,(233)supr>0,ϵ,t∈ℂ∗rϵtNexp−k2log2r/ϵtlogq=supy>0yNexp−k2log2ylogq=supx∈ℝexpxN−k2logqx2=qN2/2k,whenever N≥1.
A consecutive application of (229) and (230) gives rise to the bounds(234)I˜5ϵt≤CΓ,k′CN+1N+1k′1/2ΓN+1k′∫0r1rN+1exp−k2log2r/ϵtlogq1r3/2dr=CΓ,k′CN+1N+1k′1/2ΓN+1k′ϵtN+1∫0r1rϵtN+1exp−k2log2r/ϵtlogq1r3/2dr≤CΓ,k′CN+1N+1k′1/2ΓN+1k′ϵtN+1∫0r1r−1/2dr1ϵtqN2/2k≤2r1CΓ,k′CN+1σNN+1k′1/2ΓN+1k′qN2/2kϵN,for all ϵ∈ℂ∗, all t∈D0,σ, all integers N≥1. In other words, we get two constants CI5,1,CI5,2>0 (which rely on r1,σ,k′,MpW) such that(235)I˜5ϵt≤CI5,1CI5,2NΓN+1k′qN2/2kϵN,whenever ϵ∈ℂ∗, t∈D0,σ, for all integers N≥1.
Finally, blending (227) and (235) yields two constants I5.1,I5.2>0 (which rely on ϵ0,σ,KpW,q,k,Cq,k,Δ,β,β′,CI5,1,CI5,2) with the estimate(236)I5≤I5.1I5.2NΓN+1k′qN2/2kϵN,provided that ϵ∈ℰp+1∩ℰp, t∈T∩D0,σ, and z∈Hβ′, for all integers N≥1.
Lastly, the collection of estimates (198), (201), (207), (210), and (236) applied to the of splitting (185) induces the next bounds(237)supt∈T∩D0,σ,z∈Hβ′up+1t,z,ϵ−upt,z,ϵ≤∑d=14Id.3Id.4NqN2/2k+I5.1I5.2NΓN+1k′qN2/2kϵN,for all ϵ∈ℰp+1∩ℰp, all integers N≥1. Since the sequence ΓN+1/k′N≥1 grows faster than any geometric sequence INN≥1 with ratio I>0, the last inequality (237) warrants the forecast bounds (178).
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.Proposition6.
Letupt,z,ϵ0≤p≤ς−1 be the set of actual solutions of (33) built up in Theorem 1. Then, we can find constants ϑ>0 and A˜p,B˜p>0 (which rely on Ap,Bp,k,k′,q) such that(238)supt∈T∩D0,σ,z∈Hβ′up+1t,z,ϵ−upt,z,ϵ≤A˜pexp−logq2kΨ2ϵ,where(239)Ψx=kk′logqlogB˜pxk′−loglogB˜pxk′−1,provided that ϵ∈ℰp+1∩ℰp with ϵ<ϑ, for all 0≤p≤ς−1, where by convention uς=u0.Proof.
LetAp,Bp>0 be real numbers and k,k′≥1 be integers. We define the function(240)fx=infN≥1ApBpNΓN+1k′qN2/2kxN,for all x>0. Keeping in mind the Stirling formula (48), we can find two constants C,D>0 (depending on k′) with(241)ΓN+1k′≤CDNexpNk′logNk′,for all integers N≥1. Hence,(242)fx≤hx=infN≥1Ap,1Bp,1NNk′N/k′qN2/2kxN,where Ap,1=ApC and Bp,1=BpD. In the next part of the proof, we exhibit explicit bounds for the function hx. We follow a similar strategy as in the recent work [7]. We select a real number ϑ1>0 small enough in order that for all given x∈0,ϑ1, there exists a positive real number N˜>1 such that(243)N˜k′qN˜k′/k=1Bp,1k′xk′.
We focus on the integerN=N˜, where x denotes the floor function. By construction, we have N≤N˜. Therefore,(244)Nk′qN˜k′/k≤N˜k′qN˜k′/k=1Bp,1k′xk′,which implies that(245)Bp,1NxN≤1N/k′N/k′qN2/k,and hence(246)Ap,1Bp,1NNk′N/k′qN2/2kxN≤Ap,1q−N2/2k.
On the other hand, we can expressN˜ in term of the variable x by means of the Lambert function. Namely, we set Wz as the principal branch of the Lambert function defined on −e−1,+∞ and which solves the functional equation(247)WzexpWz=z,for all z∈−e−1,+∞. Since relation (243) can be recast in the form(248)N˜k′klogqexpN˜k′klogq=k′2logqkBp,1k′xk′,we deduce that(249)N˜=kk′logqWk′2logqkBp,1k′xk′.
Furthermore, owing to the paper [23], the next sharp lower bounds(250)Wz≥logz−loglogz,hold for all z>e. Finally, since N˜<N+1, the above facts (246)–(250) give rise to the bounds(251)Ap,1Bp,1NNk′N/k′qN2/2kxN≤Ap,1exp−logq2kΨ2x,where(252)Ψx=kk′logqlogk′2logqkBp,1k′xk′−loglogk′2logqkBp,1k′xk′−1,provided that 0<x<ϑ=minϑ1,ϑ2, where ϑ2=k′2logq/kBp,1k′e1/k′. Finally, from (242) and (251), we deduce that(253)fx≤Ap,1exp−logq2kΨ2x,whenever 0<x<ϑ, which implies the forseen bounds (238) when looking back to estimate (178).


## 7. Asymptotic Expansions in the Perturbation Parameter

### 7.1. Asymptotic Expansions with Double Gevrey andq-Gevrey Scales: A Related Version of the Ramis–Sibuya Theorem

We first put forward the notion of asymptotic expansion with double Gevrey andq-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.Definition8.
LetF,.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. Then, f is said to possess the formal series(254)f^ϵ=∑n≥0anϵn∈Fϵ,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(255)fϵ−∑n=0nanϵnF≤CMN+1ΓN+2k′qN+12/2kϵN+1,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 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.Theorem2.
Consider a complex Banach spaceF,.F and set a good covering ℰp0≤p≤ς−1 in ℂ∗ (described in Definition 6). Let Gp0≤p≤ς−1 be a set of holomorphic maps Gp from ℰp into F. We define the cocycle Δpϵ=Gp+1ϵ−Gpϵ, 0≤p≤ς−1, that stands for a holomorphic function from Zp=ℰp+1∩ℰp into F, with the convention ℰς=ℰ0 and Gς=G0.
Assume that the ensuing two requirements hold.(1)
The functionsGpϵ are bounded on ℰp, for 0≤p≤ς−1,(2)
The functionsΔpϵ suffer the next sequential constraint on Zp; there exist two constants Ap,Bp>0 with(256)ΔpϵF≤ApBpNΓN+1k′qN2/2kϵN,provided thatϵ∈Zp, for all integers N≥1, all 0≤p≤ς−1. In other words, Δpϵ has the null formal series 0^ as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on Zp, for 0≤p≤ς−1.
Then, all the functionsGpϵ, 0≤p≤ς−1, share a common formal power series G^ϵ∈Fϵ as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on ℰp.Proof.
The entire discussion leans on the following central lemma.Lemma15.
For all0≤p≤ς−1, the cocycle Δpϵ splits, which means that bounded holomorphic functions Ψp:ℰp⟶F can be singled out with the next feature:(257)Δpϵ=Ψp+1ϵ−Ψpϵ,for all ϵ∈Zp, where by convention Ψς=Ψ0. Furthermore, a sequence φmm≥0 of elements in F can be built up such that for each 0≤p≤ς−1 and any closed proper subsector W⊂ℰp with apex at 0, one can find K^p,M^p>0 with(258)Ψpϵ−∑m=0MφmϵmF≤K^pM^pM+1ΓM+2k′qM+12/2kϵM+1,for all ϵ∈W, all integers M≥0.Proof.
The proof mimics the arguments of Lemma XI-2-6 from [25] with fitting adjustment in the asymptotic expansions of the functions Ψp constructed by means of the Cauchy—Heine transform.
For all0≤p≤ς−1, we choose a segment:(259)Cp=te−1θp,t∈0,r⊂ℰp∩ℰp+1.
Theseς segments divide the open punctured disc D0,r∖0 into ς open sectors ℰ˜0,…,ℰ˜ς−1, where(260)ℰ˜p=ϵ∈ℂ∗θp−1<argϵ<θp,ϵ<r,0≤p≤ς−1,where by convention θ−1=θς−1. Let(261)Ψpϵ=−12π−1∑h=0ς−1∫ChΔhξξ−ϵdξ,for all ϵ∈ℰ˜p, for 0≤p≤ς−1, be defined as a sum of Cauchy—Heine transforms of the functions Δhϵ. By deformation of the paths Cp−1 and Cp without moving their endpoints and letting the other paths Ch, h≠p−1,p untouched (with the convention that C−1=Cς−1), one can continue analytically the function Ψp onto ℰp. Therefore, Ψp defines a holomorphic function on ℰp, for all 0≤p≤ς−1.
Now, takeϵ∈ℰp∩ℰp+1. In order to compute Ψp+1ϵ−Ψpϵ, we write(262)Ψpϵ=−12π−1∫C^pΔpξξ−ϵdξ+−12π−1∑h=0,h≠pς−1∫ChΔhξξ−ϵdξ,Ψp+1ϵ=−12π−1∫CˇpΔlξξ−ϵdξ+−12π−1∑h=0,h≠pς−1∫ChΔhξξ−ϵdξ,where the paths C^p and Cˇp are obtained by deforming the same path Cp without moving its endpoints in such a way that(a)
C^p⊂ℰp∩ℰp+1 and Cˇp⊂ℰp∩ℰp+1(b)
Γp,p+1≔−Cˇp+C^p is a simple closed curve with positive orientation whose interior contains ϵ
Therefore, due to the residue formula, we can write(263)Ψp+1ϵ−Ψpϵ=12π−1∫Γp,p+1Δpξξ−ϵdξ=Δpϵ,for all ϵ∈ℰp∩ℰp+1, for all 0≤p≤ς−1 (with the convention that Ψς=Ψ0).
In a second step, we derive asymptotic properties ofΨp. We fix an 0≤p≤ς−1 and a proper closed sector W contained in ℰp. Let C˜p (resp. C˜p−1) be a path obtained by deforming Cp (resp. Cp−1) without moving the endpoints in order that W is contained in the interior of the simple closed curve C˜p−1+γp−C˜p (which is itself contained in ℰp), where γp is a circular arc joining the two points re−1θp−1 and re−1θp. We get the representation(264)Ψpϵ=−12π−1∫C˜pΔpξξ−ϵdξ+−12π−1∫C˜p−1Δp−1ξξ−ϵdξ+−12π−1∑h=0,h≠p,p−1ν−1∫ChΔhξξ−ϵdξ,for all ϵ∈W. One assumes that the path C˜p is given as the union of a segment Lp=te−1wp/t∈0,r1, where r1<r and wp>θp and a curve Γp=μpτ/τ∈0,1 such that μp0=r1e−1wp, μp1=re−1θp, and r1≤μpτ<r for all τ∈0,1. We also assume that there exists a positive number σ<1 with ϵ≤σr1 for all ϵ∈W. By construction of the path Γp, we get that the function ϵ⟼1/2π−1∫ΓpΔpξ/ξ−ϵdξ defines an analytic function on the open disc D0,r1.
It remains to give estimates for the integralϵ⟼1/2π−1∫ΓpΔpξ/ξ−ϵdξ. Let M≥0 be an integer. From the usual geometric series expansion, one can write(265)12π−1∫LpΔpξξ−ϵdξ=∑m=0Mαp,mϵm+ϵM+1Ep,M+1ϵ,where(266)αp,m=12π−1∫LpΔlξξm+1dξ,Ep,M+1ϵ=12π−1∫LpΔpξξM+1ξ−ϵdξ,for all ϵ∈W.
Keeping in mind (256) for the special value N=m+1 and (266), we get some constants Ap,Bp>0 such that(267)αp,mF≤12π∫0r1ApBpm+1Γm+2k′qm+12/2kτm+1τm+1dτ≤r1Ap2πBpm+1Γm+2k′qm+12/2k,for all 0≤m≤M. In particular, we deduce the existence of two constants A^p,B^p>0 (depending on Ap,Bp,r1,q,k,k′) with(268)αp,mF≤A^pB^pmΓm+1k′qm2/2k,for all 0≤m≤M. Indeed, recall from [24], Appendix B, that for any given real number a>0, Γxxa∼Γx+a as x tends to +∞. Hence, a constant Kk′>0 (depending on k′) can be sort with(269)Γm+2k′≤Kk′m+1k′1/k′Γm+1k′,for all m≥0. Consequently, (268) follows from (267) and (269).
Moreover, one can choose a positive numberη>0 (depending on W) such that ξ−ϵ≥ξsinη for all ξ∈Lp and all ϵ∈W. Bringing to mind (256) for the peculiar value N=M+2 and (266) give rise to two constants Ap,Bp>0 such that(270)Ep,M+1ϵF≤12πsinη∫0r1ApBpM+2ΓM+3k′qM+22/2kτM+2τM+2dτ≤r1Ap2πsinηBpM+2ΓM+3k′qM+22/2k.
For that reason, we can find constantsAˇp,Bˇp>0 (relying on Ap,Bp,r1,q,k,k′,η) such that(271)Ep,M+1ϵF≤AˇpBˇpM+1ΓM+2k′qM+12/2k,for all ϵ∈W. Namely, from (269) we notice that(272)ΓM+3k′≤Kk′M+2k′1/k′ΓM+2k′.
Using comparable arguments, one can give estimates of the form (265), (266), (268), and (271) for the other integrals(273)−12π−1∫C˜p−1Δp−12ξξ−ϵdξ,−12π−1∫ChΔh2ξξ−ϵdξ,for all h≠p,p−1.
As a consequence, for any0≤p≤ς−1, there exist coefficients φp,m∈F, m≥0, and two constants K^p,M^p>0 such that(274)Ψpϵ−∑m=0Mφp,mϵmF≤K^pM^pM+1ΓM+2k′qM+12/2kϵM+1,for all integers M≥0, all ϵ∈W.
Besides, identity (263) and sequential assumption (256) imply in particular that the difference Ψp+1ϵ−Ψpϵ has the null formal series 0^ as asymptotic expansion (in the Poincaré sense) on Zp. Owing to the unicity of the asymptotic expansions on sectors, we deduce that all the formal series ∑m≥0φp,mϵm, 0≤p≤ς−1, are equal to some formal series denoted G^ϵ=∑m≥0φmϵm∈Eϵ. Lemma 15 follows.
We introduce the bounded holomorphic functions(275)apϵ=Gpϵ−Ψpϵ,for all 0≤p≤ς−1, all ϵ∈ℰp. By definition, for any p∈0,…,ς−1, we observe that(276)ap+1ϵ−apϵ=Gp+1ϵ−Gpϵ−Δpϵ=0,for all ϵ∈Zp. Therefore, each apϵ stands for the restriction on ℰp of a global holomorphic function called aϵ on D0,r∖0. Since aϵ remains bounded on D0,r∖0, the origin turns out to be a removable singularity for aϵ which, as a result, defines a convergent power series on D0,r.
Finally, one can recast(277)Gpϵ=aϵ+Ψpϵ,for all ϵ∈ℰp, all 0≤p≤ς−1. Furthermore, aϵ represents a convergent power series and Ψpϵ has the formal series G^ϵ=∑m≥0φmϵm as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on ℰp, for all 0≤p≤ς−1. The conclusion of Theorem 2 follows.


### 7.2. Parametric Gevrey Asymptotic Expansions of the Mixed Order for the Actual Solutions of the Main Initial Value Problem

Within this section, we explain the second principal result of our work.Theorem3.
We setF as the Banach space of complex value-bounded holomorphic functions on the product T∩D0,σ×Hβ′ equipped with the supremum norm, where the sector T, radius σ, and width β′ are settled in Theorem 1.
Then, for all0≤p≤ς−1, the bounded holomorphic functions ϵ⟼upt,z,ϵ from ℰp into F constructed in Theorem 1 share a common formal power series(278)u^t,z,ϵ=∑m≥0hmt,zϵm∈Fϵ,as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k. Videlicet, for all 0≤p≤ς−1, two constants Cp,Mp>0 can be found with(279)supt∈T∩D0,σ,z∈Hβ′upt,z,ϵ−∑m=0nhmt,zϵm≤CpMpn+1Γn+2k′qn+12/2kϵn+1,for all integers n≥1, provided that ϵ∈ℰp.Proof.
We focus on the set of functionsupt,z,ϵ, 0≤p≤ς−1, constructed in Theorem 1. For all 0≤p≤ς−1, we set(280)Gpϵ≔t,z⟼upt,z,ϵ.
EachGp defines a bounded holomorphic map from ℰp into the Banach space F described in the statement of Theorem 3. Furthermore, bound (178) implies that the cocycle Δpϵ=Gp+1ϵ−Gpϵ fulfills the sequential bound (256) on Zp=ℰp+1∩ℰp for any 0≤p≤ς−1. Then, Theorem 2 can be applied in order to get a formal power series G^ϵ∈Fϵ which stands for the Gevrey asymptotic expansion of mixed order 1/k′;q,1/k of each Gpϵ on ℰp, for all 0≤p≤ς−1.


## 7.1. Asymptotic Expansions with Double Gevrey andq-Gevrey Scales: A Related Version of the Ramis–Sibuya Theorem

We first put forward the notion of asymptotic expansion with double Gevrey andq-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.Definition8.
LetF,.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. Then, f is said to possess the formal series(254)f^ϵ=∑n≥0anϵn∈Fϵ,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(255)fϵ−∑n=0nanϵnF≤CMN+1ΓN+2k′qN+12/2kϵN+1,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 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.Theorem2.
Consider a complex Banach spaceF,.F and set a good covering ℰp0≤p≤ς−1 in ℂ∗ (described in Definition 6). Let Gp0≤p≤ς−1 be a set of holomorphic maps Gp from ℰp into F. We define the cocycle Δpϵ=Gp+1ϵ−Gpϵ, 0≤p≤ς−1, that stands for a holomorphic function from Zp=ℰp+1∩ℰp into F, with the convention ℰς=ℰ0 and Gς=G0.
Assume that the ensuing two requirements hold.(1)
The functionsGpϵ are bounded on ℰp, for 0≤p≤ς−1,(2)
The functionsΔpϵ suffer the next sequential constraint on Zp; there exist two constants Ap,Bp>0 with(256)ΔpϵF≤ApBpNΓN+1k′qN2/2kϵN,provided thatϵ∈Zp, for all integers N≥1, all 0≤p≤ς−1. In other words, Δpϵ has the null formal series 0^ as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on Zp, for 0≤p≤ς−1.
Then, all the functionsGpϵ, 0≤p≤ς−1, share a common formal power series G^ϵ∈Fϵ as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on ℰp.Proof.
The entire discussion leans on the following central lemma.Lemma15.
For all0≤p≤ς−1, the cocycle Δpϵ splits, which means that bounded holomorphic functions Ψp:ℰp⟶F can be singled out with the next feature:(257)Δpϵ=Ψp+1ϵ−Ψpϵ,for all ϵ∈Zp, where by convention Ψς=Ψ0. Furthermore, a sequence φmm≥0 of elements in F can be built up such that for each 0≤p≤ς−1 and any closed proper subsector W⊂ℰp with apex at 0, one can find K^p,M^p>0 with(258)Ψpϵ−∑m=0MφmϵmF≤K^pM^pM+1ΓM+2k′qM+12/2kϵM+1,for all ϵ∈W, all integers M≥0.Proof.
The proof mimics the arguments of Lemma XI-2-6 from [25] with fitting adjustment in the asymptotic expansions of the functions Ψp constructed by means of the Cauchy—Heine transform.
For all0≤p≤ς−1, we choose a segment:(259)Cp=te−1θp,t∈0,r⊂ℰp∩ℰp+1.
Theseς segments divide the open punctured disc D0,r∖0 into ς open sectors ℰ˜0,…,ℰ˜ς−1, where(260)ℰ˜p=ϵ∈ℂ∗θp−1<argϵ<θp,ϵ<r,0≤p≤ς−1,where by convention θ−1=θς−1. Let(261)Ψpϵ=−12π−1∑h=0ς−1∫ChΔhξξ−ϵdξ,for all ϵ∈ℰ˜p, for 0≤p≤ς−1, be defined as a sum of Cauchy—Heine transforms of the functions Δhϵ. By deformation of the paths Cp−1 and Cp without moving their endpoints and letting the other paths Ch, h≠p−1,p untouched (with the convention that C−1=Cς−1), one can continue analytically the function Ψp onto ℰp. Therefore, Ψp defines a holomorphic function on ℰp, for all 0≤p≤ς−1.
Now, takeϵ∈ℰp∩ℰp+1. In order to compute Ψp+1ϵ−Ψpϵ, we write(262)Ψpϵ=−12π−1∫C^pΔpξξ−ϵdξ+−12π−1∑h=0,h≠pς−1∫ChΔhξξ−ϵdξ,Ψp+1ϵ=−12π−1∫CˇpΔlξξ−ϵdξ+−12π−1∑h=0,h≠pς−1∫ChΔhξξ−ϵdξ,where the paths C^p and Cˇp are obtained by deforming the same path Cp without moving its endpoints in such a way that(a)
C^p⊂ℰp∩ℰp+1 and Cˇp⊂ℰp∩ℰp+1(b)
Γp,p+1≔−Cˇp+C^p is a simple closed curve with positive orientation whose interior contains ϵ
Therefore, due to the residue formula, we can write(263)Ψp+1ϵ−Ψpϵ=12π−1∫Γp,p+1Δpξξ−ϵdξ=Δpϵ,for all ϵ∈ℰp∩ℰp+1, for all 0≤p≤ς−1 (with the convention that Ψς=Ψ0).
In a second step, we derive asymptotic properties ofΨp. We fix an 0≤p≤ς−1 and a proper closed sector W contained in ℰp. Let C˜p (resp. C˜p−1) be a path obtained by deforming Cp (resp. Cp−1) without moving the endpoints in order that W is contained in the interior of the simple closed curve C˜p−1+γp−C˜p (which is itself contained in ℰp), where γp is a circular arc joining the two points re−1θp−1 and re−1θp. We get the representation(264)Ψpϵ=−12π−1∫C˜pΔpξξ−ϵdξ+−12π−1∫C˜p−1Δp−1ξξ−ϵdξ+−12π−1∑h=0,h≠p,p−1ν−1∫ChΔhξξ−ϵdξ,for all ϵ∈W. One assumes that the path C˜p is given as the union of a segment Lp=te−1wp/t∈0,r1, where r1<r and wp>θp and a curve Γp=μpτ/τ∈0,1 such that μp0=r1e−1wp, μp1=re−1θp, and r1≤μpτ<r for all τ∈0,1. We also assume that there exists a positive number σ<1 with ϵ≤σr1 for all ϵ∈W. By construction of the path Γp, we get that the function ϵ⟼1/2π−1∫ΓpΔpξ/ξ−ϵdξ defines an analytic function on the open disc D0,r1.
It remains to give estimates for the integralϵ⟼1/2π−1∫ΓpΔpξ/ξ−ϵdξ. Let M≥0 be an integer. From the usual geometric series expansion, one can write(265)12π−1∫LpΔpξξ−ϵdξ=∑m=0Mαp,mϵm+ϵM+1Ep,M+1ϵ,where(266)αp,m=12π−1∫LpΔlξξm+1dξ,Ep,M+1ϵ=12π−1∫LpΔpξξM+1ξ−ϵdξ,for all ϵ∈W.
Keeping in mind (256) for the special value N=m+1 and (266), we get some constants Ap,Bp>0 such that(267)αp,mF≤12π∫0r1ApBpm+1Γm+2k′qm+12/2kτm+1τm+1dτ≤r1Ap2πBpm+1Γm+2k′qm+12/2k,for all 0≤m≤M. In particular, we deduce the existence of two constants A^p,B^p>0 (depending on Ap,Bp,r1,q,k,k′) with(268)αp,mF≤A^pB^pmΓm+1k′qm2/2k,for all 0≤m≤M. Indeed, recall from [24], Appendix B, that for any given real number a>0, Γxxa∼Γx+a as x tends to +∞. Hence, a constant Kk′>0 (depending on k′) can be sort with(269)Γm+2k′≤Kk′m+1k′1/k′Γm+1k′,for all m≥0. Consequently, (268) follows from (267) and (269).
Moreover, one can choose a positive numberη>0 (depending on W) such that ξ−ϵ≥ξsinη for all ξ∈Lp and all ϵ∈W. Bringing to mind (256) for the peculiar value N=M+2 and (266) give rise to two constants Ap,Bp>0 such that(270)Ep,M+1ϵF≤12πsinη∫0r1ApBpM+2ΓM+3k′qM+22/2kτM+2τM+2dτ≤r1Ap2πsinηBpM+2ΓM+3k′qM+22/2k.
For that reason, we can find constantsAˇp,Bˇp>0 (relying on Ap,Bp,r1,q,k,k′,η) such that(271)Ep,M+1ϵF≤AˇpBˇpM+1ΓM+2k′qM+12/2k,for all ϵ∈W. Namely, from (269) we notice that(272)ΓM+3k′≤Kk′M+2k′1/k′ΓM+2k′.
Using comparable arguments, one can give estimates of the form (265), (266), (268), and (271) for the other integrals(273)−12π−1∫C˜p−1Δp−12ξξ−ϵdξ,−12π−1∫ChΔh2ξξ−ϵdξ,for all h≠p,p−1.
As a consequence, for any0≤p≤ς−1, there exist coefficients φp,m∈F, m≥0, and two constants K^p,M^p>0 such that(274)Ψpϵ−∑m=0Mφp,mϵmF≤K^pM^pM+1ΓM+2k′qM+12/2kϵM+1,for all integers M≥0, all ϵ∈W.
Besides, identity (263) and sequential assumption (256) imply in particular that the difference Ψp+1ϵ−Ψpϵ has the null formal series 0^ as asymptotic expansion (in the Poincaré sense) on Zp. Owing to the unicity of the asymptotic expansions on sectors, we deduce that all the formal series ∑m≥0φp,mϵm, 0≤p≤ς−1, are equal to some formal series denoted G^ϵ=∑m≥0φmϵm∈Eϵ. Lemma 15 follows.
We introduce the bounded holomorphic functions(275)apϵ=Gpϵ−Ψpϵ,for all 0≤p≤ς−1, all ϵ∈ℰp. By definition, for any p∈0,…,ς−1, we observe that(276)ap+1ϵ−apϵ=Gp+1ϵ−Gpϵ−Δpϵ=0,for all ϵ∈Zp. Therefore, each apϵ stands for the restriction on ℰp of a global holomorphic function called aϵ on D0,r∖0. Since aϵ remains bounded on D0,r∖0, the origin turns out to be a removable singularity for aϵ which, as a result, defines a convergent power series on D0,r.
Finally, one can recast(277)Gpϵ=aϵ+Ψpϵ,for all ϵ∈ℰp, all 0≤p≤ς−1. Furthermore, aϵ represents a convergent power series and Ψpϵ has the formal series G^ϵ=∑m≥0φmϵm as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k on ℰp, for all 0≤p≤ς−1. The conclusion of Theorem 2 follows.


## 7.2. Parametric Gevrey Asymptotic Expansions of the Mixed Order for the Actual Solutions of the Main Initial Value Problem

Within this section, we explain the second principal result of our work.Theorem3.
We setF as the Banach space of complex value-bounded holomorphic functions on the product T∩D0,σ×Hβ′ equipped with the supremum norm, where the sector T, radius σ, and width β′ are settled in Theorem 1.
Then, for all0≤p≤ς−1, the bounded holomorphic functions ϵ⟼upt,z,ϵ from ℰp into F constructed in Theorem 1 share a common formal power series(278)u^t,z,ϵ=∑m≥0hmt,zϵm∈Fϵ,as Gevrey asymptotic expansion of mixed order 1/k′;q,1/k. Videlicet, for all 0≤p≤ς−1, two constants Cp,Mp>0 can be found with(279)supt∈T∩D0,σ,z∈Hβ′upt,z,ϵ−∑m=0nhmt,zϵm≤CpMpn+1Γn+2k′qn+12/2kϵn+1,for all integers n≥1, provided that ϵ∈ℰp.Proof.
We focus on the set of functionsupt,z,ϵ, 0≤p≤ς−1, constructed in Theorem 1. For all 0≤p≤ς−1, we set(280)Gpϵ≔t,z⟼upt,z,ϵ.
EachGp defines a bounded holomorphic map from ℰp into the Banach space F described in the statement of Theorem 3. Furthermore, bound (178) implies that the cocycle Δpϵ=Gp+1ϵ−Gpϵ fulfills the sequential bound (256) on Zp=ℰp+1∩ℰp for any 0≤p≤ς−1. Then, Theorem 2 can be applied in order to get a formal power series G^ϵ∈Fϵ which stands for the Gevrey asymptotic expansion of mixed order 1/k′;q,1/k of each Gpϵ on ℰp, for all 0≤p≤ς−1.



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