# Asymptotics and Confluence for a Singular Nonlinearq-Difference-Differential Cauchy Problem

Authors: S. Malek
Journal: Abstract and Applied Analysis
 (2022)

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## Abstract

We examine a family of nonlinearq−difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation q−difference operators, recently investigated by the author, and quasilinear Kowalevski type problems that involve contraction q−difference operators. We build up local holomorphic solutions to these problems. Two aspects of these solutions are explored. One facet deals with asymptotic expansions in the complex time variable for which a mixed type Gevrey and q−Gevrey structure are exhibited. The other feature concerns the problem of confluence of these solutions as q>1 tends to 1.


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## Body

## 1. Introduction

In this paper, we study a particular family of nonlinearq−difference-differential Cauchy problems displayed as follows
(1)Ptk+1∂t∂zS~vt,z=Gt,z,∂tr1∂zr2vqr3t,zr¯=r1,r2,r3∈H,for prescribed Cauchy data
(2)∂zjvt,0=ϑjt,0≤j≤S~−1,where S~,k≥1 are integers, q>1 is a real number and where the symbol PT from the leading term of the equation (1) stands for a nonconstant element of ℂT, and H represents some well-chosen finite subset of ℕ2×ℤ. The right part Gt,z,Vr¯r¯∈H of (1) is a polynomial of degree at most 2 in the variables Vr¯ for r¯∈H, holomorphic relatively to z on some disc in ℂ centered at the origin and which depends polynomially on t along with the data (2). The detailed shape of (1), (2) is stated in Corollary 17 in Subsection 3.3.The present work is the sequel of the investigation initiated in [1] that focused on some linear q−difference-differential Cauchy problems outlined as
(3)Ptk+1∂t∂zSut,z=Pt,z,σq;t,tk+1∂t,∂zut,z,for given Cauchy data
(4)∂zjut,0=φjt,0≤j≤S−1,where S≥1 is a suitable natural number, the piece PT from the leading term of (3) is the polynomial appearing in (1), k≥1 is the integer stemming from (1), σq;t stands for the dilation operator acting on functions through σq;tft,z=fqt,z for q>1 arising in (1) and the right handside Pt,z,V1,V2,V3 together with the data (4) represent fittingly selected polynomials. As summed up in Theorem 3 of this work, under strong restrictions on the shape of (3) (not discussed in this paper but listed in [1]), a finite set upt,z0≤p≤ς−1, for some integer ς≥2, of holomorphic solutions to (3) and (4) could be modeled on products Tp×D, where D stands for some small disc centered at 0 in ℂ and where T¯=Tp0≤p≤ς−1 is a suitable set of bounded sectors whose union covers some neighborhood of 0 in ℂ∗, see Definition 1. These solutions were expressed through Laplace transforms of order k,
(5)upt,z=k∫Lγpwpu,zexp−utkduu,along half lines Lγp=0,+∞e−1γp in convenient directions γp∈ℝ where the Borel map wpu,z is holomorphic relatively to z on D and is compelled to bear q−exponential growth rate (17) w.r.t u on some unbounded sector Up. In [1], we addressed two important features of these solutions
(i)
Asymptotic behaviour w.r.t the time variablet as t⟶0(ii)
Confluence property asq>1 tends to 1Regarding the first point, a fine structure of mixed Gevrey andq−Gevrey type was disclosed. Namely, as expounded in Theorem 3 and 19, all the partial maps t↦upt,z share a common formal power series u^t,z=∑n≥0unztn, where unz is bounded holomorphic on D, as so-called Gevrey asymptotic expansion of mixed order 1/k;q,1 on Tp, meaning that two constants C,M>0 can be pinpointed with
(6)supz∈Dupt,z−∑n=0Nunztn≤CMN+1ΓN+1kqN+12/2tN+1,whenever t∈Tp, for all integers N≥0.Concerning the second item, discussed in Subsections5.1 and 5.5, we have shown that for any prescribed sector Tfrom the covering T¯, the corresponding solution u;qt,z (whose reliance on the parameter q is flagged by an index ;q) to (3), (4) merges uniformly on T×D, as q∈1,q0 tends to 1 for some fixed q0>0, to a holomorphic map u;1t,z on T×D which itself solves some linear PDE Cauchy problem given by (177), (178). More precisely, some constant K>0 (unrelated to q) could be singled out with
(7)supt∈T,z∈Du;qt,z−u;1t,z≤Kq−1,provided that q∈1,q0.The problem (1), (2) examined in this work is actually obtained by means of a procedure (described in Subsection2.2) which consists in coupling the singular linear Cauchy problem (3), (4) with a quasilinear Kowalevski type problem which involves the contraction operator σq−1;t acting on functions by means of σq−1;tft,z=fq−1t,z, framed as
(8)∂zκvt,z=P1t,z,σq−1;t,tk+1∂t,∂zvt,z+at,zv2t,z+ut,z,for assigned Cauchy data
(9)∂zjvt,0=φ~jt,0≤j≤κ−1,where κ≥1 is an appropriate integer, at,z is some polynomial in t with holomorphic coefficients on D, the linear piece P1t,z,V1,V2,V3 as well as the data (9) stand for properly chosen polynomials and where the forcing term ut,z is required to solve the linear problem (3), (4). Notice that the appearance of the q−difference operator σq−1;t is mandatory when some time derivative ∂t occurs in equation (8), according to the constraints (21).Our objectives remain similar to those in [1] and concern
(i)
The construction of local holomorphic solutions to (1), (2)(ii)
Asymptotic expansions of these solutions as timet borders the origin(iii)
Confluence aspects asq⟶1The first item is completed in Subsection3.3 (Theorem 16 and its Corollary 17) where a finite set vpt,z0≤p≤ς−1 of holomorphic solutions to (1), (2) is built up on domains Tp×D, provided that the radii of Tp and D are taken small enough. Furthermore, the solutions can be represented as Laplace transforms of order k,
(10)vpt,z=k∫LγpΘpu,zexp−utkduualong the same halflines Lγp as in (5) where the Borel map Θpu,z remains holomorphic w.r.t z on D but suffers now (at most) exponential growth rate (91) of order k relatively to u on Up (and not in general q−exponential increase as it was the case for wp).The second item is achieved in Subsection4.5 (Theorem 30) where the existence of a formal power series v^t,z=∑l≥0hlztl, with holomorphic coefficients hlz on D, is established which stands for the common Gevrey asymptotic expansion of mixed order 1/k;q,1 on Tp of the partial maps t↦vpt,z, 0≤p≤ς−1, satisfying therefore similar estimates to (6).The last item is explained in Subsection5.7 (Theorem 44). For any given sector T from the covering T¯, the related solution v;qt,z (whose dependence on q is marked by the index ;q) to (1), (2) converges uniformly on T×D, as q⟶1, to a holomorphic map v;1t,z on T×D which is the solution of some nonlinear PDE Cauchy problem, stated in (182), (183). Factually, comparable bounds to (7) hold, see (289).We draw attention to the fact that the proofs of our three main statements Theorems16, 30, and 44 lean on statements established in [1]. In essence, the features of the solutions vpt,z of (1), (2) reached in this paper are identical to those of the solutions upt,z of (3), (4) achieved in [1]. However, their proofs differ fundamentally. Indeed, the construction of the sectorial local holomorphic solutions vpt,z to (1), (2) is performed by means of a fixed point argument in suitably selected Banach spaces (Subsections 3.1 and 3.2) when a mere induction principle and elementary estimates were only required to reach the local solutions upt,z of (3), (4). The asymptotic features relatively to time t of the solutions upt,z were obtained by dint of a version the so-called Ramis-Sibuya theorem which banks on sharp estimates of the difference up+1−up. This approach fails to be applied in our nonlinear context. We use instead a majorant series method that reduces the problem to the construction of formal power series solutions to a related Cauchy problem in appropriate Banach spaces (Section 4). Regarding the confluence properties, both works hinge on an auxiliary result which studies the action of q−difference operators on the Borel maps of the limit maps u;1t,z and v;1t,z (see Propositions 36 and 37) but the proofs of the present work are again based on functional analytic arguments and the use of accurate bounds in Banach spaces, while induction principle was favored in [1].Observe that, by construction, the nonlinearq−difference differential equation (1) involves both dilations and contractions w.r.t the time variable t by means of the presence of operators σq;tl for both l>0 and l<0. The same property arises in the framework of nonlinear q−difference equation in the study of the so-called q− Painlevé equations. Indeed, the q−discrete versions of the first and second Painlevé equations are expressed through the next two equations
(11)wqxwxq=1wx−1xw2x,gqxgxq=αx2gx+x2gxgx−1,for x=x0qn with n∈ℤ for some x0∈ℂ, q∈ℂ\0,1, and α some parameter, see for instance [2, 3]. For an excellent comprehensive and introductive book to q-Painlevé equations and more generally to integrable discrete dynamical systems, we mention [4].Regarding the existence of local holomorphic solutions to nonlinearq−difference equations, we may refer to some recent works. Indeed, for meromorphic or holomorphic solutions around the origin for special type of nonlinear q−difference equations such as the q−Painlevé equations, we can mention [5, 6]. Some category of nonlinear q−difference equations of the form
(12)xyqx=yx+byx,x,where b is some polynomial has been investigated by Menous in the paper [7] who gave assumptions under which such equations can be analytically conjugated to well-studied models of linear q−difference equations with so-called irregular singularity at the origin, xzqx=zx or xzqx=zx+x. In the recent work [8], Gontsov et al. provide sufficient conditions for the convergence of so-called generalized power series
(13)φz=∑j≥0cjzλj,with complex coefficients cj∈ℂ and complex exponents λj∈ℂ that are solutions of algebraic q− difference equations
(14)Fz,φz,φqz,⋯,φqnz=0,where F stands for some polynomial, providing in particular local sectorial holomorphic solutions to these equations.In the context of nonlinearq−difference-differential equations, the literature concerning local existence of solutions is less profuse. Nevertheless, the important result by Yamazawa [9] ought to be quoted in that trend. The author constructs holomorphic and singular solutions of logarithmic type near the origin to equations of the form
(15)uqt,x=ut,x+Ft,x,∂xαuα≤m,for t∈ℂ, x∈ℂn, n≥1, q>1, and where F is some well-prepared analytic function in its arguments.On the subject of confluence for linearq−difference equations, some recent references have been pointed out on our latest contribution [1]. The confluence in the framework of nonlinear q−difference equations has been much less examined and represents a propitious direction for upcoming research. For instance, some aspects of confluence for the so-called q−Painlevé VI equation have been recently undertaken by Dreyfus and Heu in [10]. They construct some well-prepared q−analog Hamiltonian system
(16)∂q,ty=∂q,ZHy,Z,t+Oq−1,∂q,tZ=−∂q,yHy,Z,t+Oq−1,where ∂q,x=σq;x−1/qx−x and show that its discreet solution given in the form of two sequences yl=yqlt0, Zl=Zqlt0 for given t0∈ℂ∗ and y0,Z0∈ℂ\0,1,t0×ℂ encodes the Taylor series coefficients of the holomorphic solution yt,Zt to the (formal limit as q>1 tends to 1) nonautonomous Hamiltonian system
(17)y′t=∂H∂Zy,Z,t,Z′t=−∂H∂yy,Z,t,with initial condition yt0=y0 and Zt0=Z0, which defines the sixth Painlevé equation for some prescribed rational map H∈ℂy,Z,t.


## 2. The Main Problem Outlined

### 2.1. A Finite Set of Holomorphic Solutions to a Singular Linear Cauchy Problem

In this subsection, we remind the reader parts of the results obtained in our previous work [1] that will be used within the present Section 2. We first describe the linear Cauchy problem we have considered in that study.Letk,S≥1 be integers and q>1 be a real number. We set Pτ∈ℂτ a polynomial with complex coefficients such that
(18)P0≠0.LetA be a finite subset of ℕ4. For all l¯∈A and all 0≤j≤S−1, we fix polynomials cl¯z and φjt with complex coefficients.We focus on the next singular linear Cauchy problem with polynomial coefficients in time,(19)Ptk+1∂t∂zSut,z=∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2uql3t,z,for given Cauchy data
(20)∂zjut,0=φjt,0≤j≤S−1.In order to describe a set of solutions to (11), (12), we need to recall the definitions of good coverings and admissible sets of sectors introduced in Section 6 of [1].Definition 1.
Letς≥2 be an integer. For all 0≤p≤ς−1, we select open sectors Tp centered at 0 (and do not contain 0) with given radius rT that fulfill the next three features:
(i)
The intersection of any two consecutive sectors of the familyT¯=Tp0≤p≤ς−1 is nonempty, namely
(21)Tp∩Tp+1≠0,for all 0≤p≤ς−1, with the convention that Tς=T0.(ii)
The intersection of any three elements inT¯ is empty(iii)
The union of the sectorsTp covers some punctured neighborhood U̇ of the origin in ℂ∗(22)∪p=0ς−1Tp=U̇=U\0.
The familyT¯ is then named a good covering in ℂ∗.
The notion of admissible set of sectors is depicted in the next.Definition 2.
We setς≥2 as an integer and set T¯=Tp0≤p≤ς−1 as a good covering in ℂ∗. We consider a set U¯=Up0≤p≤ς−1 of unbounded sectors Up centered at 0 that endorse the next two properties:
(1)
Each sectorUp does not contain any of the roots of the polynomial u↦Pkuk, for 0≤p≤ς−1(2)
For all0≤p≤ς−1, there exists a constant Δp>0 such that for all t∈Tp, one can single out a direction γp∈ℝ (that may depend on t) such that both conditions
(23)Lγp=0,+∞exp−1γp⊂Up∪0,coskγp−argt>Δp,hold.
We say that the set of sectorsD¯=T¯,U¯ represents an admissible set of sectors.
In Section 6 of the paper [1], we have obtained the following result.Theorem 3.
Let us assume that all the requirements asked in Section2.1 of [1] hold true. Fix a good covering T¯=Tp0≤p≤ς−1 in ℂ∗ and a set U¯=Up0≤p≤ς−1 of unbounded sectors chosen in a way that the data D¯=T¯,U¯ forms an admissible set of sectors.
Then, for all0≤p≤ς−1, one can construct a solution upt,z to the Cauchy problem (11), (12) that is bounded and holomorphic on Tp×D1/2C4 and that can be expressed through a Laplace transform of order k,
(24)upt,z=k∫Lγpwpu,zexp−utkduu,for t,z∈Tp×D1/2C4 where D1/2C4 stands for the disc centered at 0 with radius 1/2C4 for some well-chosen constant C4>0. The Borel map wpu,z represents a holomorphic function on the domain Up×D1/2C4 whose Taylor expansion
(25)wpu,z=∑β≥0wp,βuzββ!,is subjected to the next bounds
(26)wp,βu≤C3C4ββ!uexpk1log2u+u0+αlogu+u0,for all β≥0, all u∈Up, for suitably chosen constants C3,k1>0 and u0>1, α≥0.Example 1.
A concrete example of a simple equation (19) that fulfills all the conditions required in Section 2.1 of [1] is given by
(27)t5∂t−1∂zut,z=zt3uqt,z.


### 2.2. Setup of the Main Nonlinear Cauchy Problem

Letκ≥1 be an integer and k,q as defined in the previous subsection. Let C be a finite subset of ℕ4. For all h¯∈C, we fix a bounded and analytic function dh¯z on a disc DR centered at 0 with some radius R>0. Furthermore, we define a polynomial
(28)at,z=∑h=0Aahzth,for some integer A≥0, where the coefficients ahz are bounded and holomorphic on DR and for all 0≤j≤κ−1, we denote φ⌣jt polynomials with complex coefficients written in the form
(29)φ⌣jt=∑h∈J⌣jp⌣j,hΓhkth,where J⌣j stands for a finite subset of ℕ\0.The finite setC is compelled to fulfill to the next list of requirements:(A1) There exists a real numberb>1 for which
(30)κ≥bh1+h2,κ>h2,for all h¯=h0,h1,h2,h3∈C.(A2) The next inequalities(31)κ≥1k+1h1+h2,κ>h2,h3≥h1,hold provided that h¯=h0,h1,h2,h3∈C.We consider the next nonlinear nonhomogeneous Cauchy problem(32)∂zκvpt,z=∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2vpq−h3t,z+at,zvp2t,z+upt,z,for given Cauchy data
(33)∂zjvpt,0=φ⌣jt,0≤j≤κ−1,where the forcing term upt,z is the holomorphic solution of the linear Cauchy problem (11), (12) disclosed in Theorem 3 of the former subsection.Example 2.
An explicit illustration of a plain equation (32) that is subjected to the requirements (20) and (21) is provided by
(34)∂z2vpt,z=t5∂tvpq−1t,z+vp2t,z+upt,z,where the forcing term upt,z is one genuine solution to the equation (27) built up in Theorem 3.We now unveil our main roadmap that will lead later on to the construction of suitable sets of solutions to our problem. We search for solutions to (22) and (23) in the form of a Laplace transform of orderk, namely,
(35)vpt,z=k∫LγpΘpu,zexp−utkduu,along the halfline Lγp=0,+∞e−1γp appearing in the representation (15). So far, the so-called Borel map Θpu,z is supposed to be holomorphic with respect to u on the unbounded sector Up and analytic w.r.t z on some small disc Dr centered at 0 with radius r>0. For the Laplace transform to be well defined, we make the further assumption that Θpu,z has at most exponential growth of order k w.r.t u on Up, uniformly in z on Dr, meaning the existence of two constants C,K>0 with
(36)supz∈DrΘpu,z≤CuexpKuk,for all u∈Up. Once we assume that such solutions exists, we will derive some functional equations that the Borel map Θpu,z will be asked to solve at a formal level only. Such equations will be described in the next subsection. Later on, in Subsection 3.2, these convolution equations will be solved in some Banach space of holomorphic functions, see Proposition 15, producing a genuine holomorphic map Θpu,z satisfying the above requirements.


### 2.3. An Auxiliary Cauchy Problem Satisfied by the Borel MapΘp

We first need to remind the reader the next proposition which is a slightly modified version of Proposition4 of [1].Proposition 4.
We setE,.E as a complex Banach space. Let k≥1 be an integer and let w:Sd,δ⟶E be a holomorphic function on the open unbounded sector Sd,δ=u∈ℂ∗:d−argu<δ, continuous on Sd,δ∪0. The existence of two constants C>0 and K>0 such that
(37)wuE≤CueKuk,is assumed for all u∈Sd,δ. Then, the Laplace transform of order k of w in the direction d is defined by
(38)Lkdwut=k∫Lγwue−u/tkduu,along a half-line Lγ=ℝ+eiγ⊂Sd,δ∪0, where γ depends on t and is chosen in such a way that coskγ−argt≥δ1>0, for some fixed δ1. The function Lkdwut is well defined, holomorphic, and bounded in any sector
(39)Sd,θ,R1/k=t∈ℂ∗:∣t∣<R1/k,d−argt<θ/2,where π/k<θ<π/k+2δ and 0<R<δ1/K.
(a)
The action of the Laplace transform on entire functions is described as follows: ifw is an entire function on ℂ, with growth estimates (26) and with Taylor expansion wu=∑n≥1bnun, then Lkdwut defines an analytic function near the origin w.r.t t with convergent Taylor expansion ∑n≥1Γn/kbntn(b)
The actions of the irregular operatortk+1∂t and the monomial tm on the Laplace transform are expressed through the next formulas
(40)Lkdkukwut=tk+1∂tLkdwut,mLkdwut=Lkdu↦um⋆kwut,for every integers m≥1, and for all t∈Sd,θ,R1/k with 0<R<δ1/K. Here, um⋆kwu stands for the convolution product
(41)um⋆kwu≔ukΓm/k∫0ukuk−sm/k−1ws1/kdss.(c)
Letw1,w2:Sd,δ⟶E be holomorphic maps with the same feature (26) as w above. Assume moreover, that E is equipped with a product ∗ in a way that E,∗ becomes a Banach algebra. Then, the next multiplicative formula
(42)Lkdw1ut×Lkdw2ut=Lkdw1u⋆kw2ut,holds for all t∈Sd,θ,R1/k with 0<R<δ1/K, where w1u⋆kw2u represents the convolution product
(43)w1u⋆kw2u≔uk∫0ukw1uk−s1/k∗w2s1/k1uk−ssds.(d)
The action of the dilationqδ commutes with the Laplace transform, for any integer δ≥1, namely
(44)Lkdwuqδt=Lkdwqδut,holds for all t∈Sd,θ,R11/k for 0<R1<δ1/Kqkδ.
The point (A) allows the Cauchy data (23) to be expressed through Laplace transforms of orderk,
(45)φ⌣jt=k∫LγP⌣juexp−utkduu,of polynomials given by P⌣ju=∑h∈J⌣jp⌣j,huh, for 0≤j≤κ−1.
Owing to the above identities (28), (29), and (30), we observe that the Laplace representation (25) solves the Cauchy problem (22), (23) if the Borel mapΘpu,z is subjected to the next nonlinear and nonhomogeneous convolution Cauchy problem
(46)∂zκΘpu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3ukh1∂zh2Θpq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1kq−h3s1/kkh1∂zh2Θpq−h3s1/k,zdss+a0zuk∫0ukΘpuk−s1/k,zΘps1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1s∫0sΘps−s11/k,zΘps11/k,z1s−s1s1ds1dss+wpu,z,for prescribed Cauchy data
(47)∂zjΘpu,0=P⌣ju,0≤j≤κ−1.


## 2.1. A Finite Set of Holomorphic Solutions to a Singular Linear Cauchy Problem

In this subsection, we remind the reader parts of the results obtained in our previous work [1] that will be used within the present Section 2. We first describe the linear Cauchy problem we have considered in that study.Letk,S≥1 be integers and q>1 be a real number. We set Pτ∈ℂτ a polynomial with complex coefficients such that
(18)P0≠0.LetA be a finite subset of ℕ4. For all l¯∈A and all 0≤j≤S−1, we fix polynomials cl¯z and φjt with complex coefficients.We focus on the next singular linear Cauchy problem with polynomial coefficients in time,(19)Ptk+1∂t∂zSut,z=∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2uql3t,z,for given Cauchy data
(20)∂zjut,0=φjt,0≤j≤S−1.In order to describe a set of solutions to (11), (12), we need to recall the definitions of good coverings and admissible sets of sectors introduced in Section 6 of [1].Definition 1.
Letς≥2 be an integer. For all 0≤p≤ς−1, we select open sectors Tp centered at 0 (and do not contain 0) with given radius rT that fulfill the next three features:
(i)
The intersection of any two consecutive sectors of the familyT¯=Tp0≤p≤ς−1 is nonempty, namely
(21)Tp∩Tp+1≠0,for all 0≤p≤ς−1, with the convention that Tς=T0.(ii)
The intersection of any three elements inT¯ is empty(iii)
The union of the sectorsTp covers some punctured neighborhood U̇ of the origin in ℂ∗(22)∪p=0ς−1Tp=U̇=U\0.
The familyT¯ is then named a good covering in ℂ∗.
The notion of admissible set of sectors is depicted in the next.Definition 2.
We setς≥2 as an integer and set T¯=Tp0≤p≤ς−1 as a good covering in ℂ∗. We consider a set U¯=Up0≤p≤ς−1 of unbounded sectors Up centered at 0 that endorse the next two properties:
(1)
Each sectorUp does not contain any of the roots of the polynomial u↦Pkuk, for 0≤p≤ς−1(2)
For all0≤p≤ς−1, there exists a constant Δp>0 such that for all t∈Tp, one can single out a direction γp∈ℝ (that may depend on t) such that both conditions
(23)Lγp=0,+∞exp−1γp⊂Up∪0,coskγp−argt>Δp,hold.
We say that the set of sectorsD¯=T¯,U¯ represents an admissible set of sectors.
In Section 6 of the paper [1], we have obtained the following result.Theorem 3.
Let us assume that all the requirements asked in Section2.1 of [1] hold true. Fix a good covering T¯=Tp0≤p≤ς−1 in ℂ∗ and a set U¯=Up0≤p≤ς−1 of unbounded sectors chosen in a way that the data D¯=T¯,U¯ forms an admissible set of sectors.
Then, for all0≤p≤ς−1, one can construct a solution upt,z to the Cauchy problem (11), (12) that is bounded and holomorphic on Tp×D1/2C4 and that can be expressed through a Laplace transform of order k,
(24)upt,z=k∫Lγpwpu,zexp−utkduu,for t,z∈Tp×D1/2C4 where D1/2C4 stands for the disc centered at 0 with radius 1/2C4 for some well-chosen constant C4>0. The Borel map wpu,z represents a holomorphic function on the domain Up×D1/2C4 whose Taylor expansion
(25)wpu,z=∑β≥0wp,βuzββ!,is subjected to the next bounds
(26)wp,βu≤C3C4ββ!uexpk1log2u+u0+αlogu+u0,for all β≥0, all u∈Up, for suitably chosen constants C3,k1>0 and u0>1, α≥0.Example 1.
A concrete example of a simple equation (19) that fulfills all the conditions required in Section 2.1 of [1] is given by
(27)t5∂t−1∂zut,z=zt3uqt,z.


## 2.2. Setup of the Main Nonlinear Cauchy Problem

Letκ≥1 be an integer and k,q as defined in the previous subsection. Let C be a finite subset of ℕ4. For all h¯∈C, we fix a bounded and analytic function dh¯z on a disc DR centered at 0 with some radius R>0. Furthermore, we define a polynomial
(28)at,z=∑h=0Aahzth,for some integer A≥0, where the coefficients ahz are bounded and holomorphic on DR and for all 0≤j≤κ−1, we denote φ⌣jt polynomials with complex coefficients written in the form
(29)φ⌣jt=∑h∈J⌣jp⌣j,hΓhkth,where J⌣j stands for a finite subset of ℕ\0.The finite setC is compelled to fulfill to the next list of requirements:(A1) There exists a real numberb>1 for which
(30)κ≥bh1+h2,κ>h2,for all h¯=h0,h1,h2,h3∈C.(A2) The next inequalities(31)κ≥1k+1h1+h2,κ>h2,h3≥h1,hold provided that h¯=h0,h1,h2,h3∈C.We consider the next nonlinear nonhomogeneous Cauchy problem(32)∂zκvpt,z=∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2vpq−h3t,z+at,zvp2t,z+upt,z,for given Cauchy data
(33)∂zjvpt,0=φ⌣jt,0≤j≤κ−1,where the forcing term upt,z is the holomorphic solution of the linear Cauchy problem (11), (12) disclosed in Theorem 3 of the former subsection.Example 2.
An explicit illustration of a plain equation (32) that is subjected to the requirements (20) and (21) is provided by
(34)∂z2vpt,z=t5∂tvpq−1t,z+vp2t,z+upt,z,where the forcing term upt,z is one genuine solution to the equation (27) built up in Theorem 3.We now unveil our main roadmap that will lead later on to the construction of suitable sets of solutions to our problem. We search for solutions to (22) and (23) in the form of a Laplace transform of orderk, namely,
(35)vpt,z=k∫LγpΘpu,zexp−utkduu,along the halfline Lγp=0,+∞e−1γp appearing in the representation (15). So far, the so-called Borel map Θpu,z is supposed to be holomorphic with respect to u on the unbounded sector Up and analytic w.r.t z on some small disc Dr centered at 0 with radius r>0. For the Laplace transform to be well defined, we make the further assumption that Θpu,z has at most exponential growth of order k w.r.t u on Up, uniformly in z on Dr, meaning the existence of two constants C,K>0 with
(36)supz∈DrΘpu,z≤CuexpKuk,for all u∈Up. Once we assume that such solutions exists, we will derive some functional equations that the Borel map Θpu,z will be asked to solve at a formal level only. Such equations will be described in the next subsection. Later on, in Subsection 3.2, these convolution equations will be solved in some Banach space of holomorphic functions, see Proposition 15, producing a genuine holomorphic map Θpu,z satisfying the above requirements.


## 2.3. An Auxiliary Cauchy Problem Satisfied by the Borel MapΘp

We first need to remind the reader the next proposition which is a slightly modified version of Proposition4 of [1].Proposition 4.
We setE,.E as a complex Banach space. Let k≥1 be an integer and let w:Sd,δ⟶E be a holomorphic function on the open unbounded sector Sd,δ=u∈ℂ∗:d−argu<δ, continuous on Sd,δ∪0. The existence of two constants C>0 and K>0 such that
(37)wuE≤CueKuk,is assumed for all u∈Sd,δ. Then, the Laplace transform of order k of w in the direction d is defined by
(38)Lkdwut=k∫Lγwue−u/tkduu,along a half-line Lγ=ℝ+eiγ⊂Sd,δ∪0, where γ depends on t and is chosen in such a way that coskγ−argt≥δ1>0, for some fixed δ1. The function Lkdwut is well defined, holomorphic, and bounded in any sector
(39)Sd,θ,R1/k=t∈ℂ∗:∣t∣<R1/k,d−argt<θ/2,where π/k<θ<π/k+2δ and 0<R<δ1/K.
(a)
The action of the Laplace transform on entire functions is described as follows: ifw is an entire function on ℂ, with growth estimates (26) and with Taylor expansion wu=∑n≥1bnun, then Lkdwut defines an analytic function near the origin w.r.t t with convergent Taylor expansion ∑n≥1Γn/kbntn(b)
The actions of the irregular operatortk+1∂t and the monomial tm on the Laplace transform are expressed through the next formulas
(40)Lkdkukwut=tk+1∂tLkdwut,mLkdwut=Lkdu↦um⋆kwut,for every integers m≥1, and for all t∈Sd,θ,R1/k with 0<R<δ1/K. Here, um⋆kwu stands for the convolution product
(41)um⋆kwu≔ukΓm/k∫0ukuk−sm/k−1ws1/kdss.(c)
Letw1,w2:Sd,δ⟶E be holomorphic maps with the same feature (26) as w above. Assume moreover, that E is equipped with a product ∗ in a way that E,∗ becomes a Banach algebra. Then, the next multiplicative formula
(42)Lkdw1ut×Lkdw2ut=Lkdw1u⋆kw2ut,holds for all t∈Sd,θ,R1/k with 0<R<δ1/K, where w1u⋆kw2u represents the convolution product
(43)w1u⋆kw2u≔uk∫0ukw1uk−s1/k∗w2s1/k1uk−ssds.(d)
The action of the dilationqδ commutes with the Laplace transform, for any integer δ≥1, namely
(44)Lkdwuqδt=Lkdwqδut,holds for all t∈Sd,θ,R11/k for 0<R1<δ1/Kqkδ.
The point (A) allows the Cauchy data (23) to be expressed through Laplace transforms of orderk,
(45)φ⌣jt=k∫LγP⌣juexp−utkduu,of polynomials given by P⌣ju=∑h∈J⌣jp⌣j,huh, for 0≤j≤κ−1.
Owing to the above identities (28), (29), and (30), we observe that the Laplace representation (25) solves the Cauchy problem (22), (23) if the Borel mapΘpu,z is subjected to the next nonlinear and nonhomogeneous convolution Cauchy problem
(46)∂zκΘpu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3ukh1∂zh2Θpq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1kq−h3s1/kkh1∂zh2Θpq−h3s1/k,zdss+a0zuk∫0ukΘpuk−s1/k,zΘps1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1s∫0sΘps−s11/k,zΘps11/k,z1s−s1s1ds1dss+wpu,z,for prescribed Cauchy data
(47)∂zjΘpu,0=P⌣ju,0≤j≤κ−1.


## 3. Solving the Main Nonlinear Cauchy Problem

Within this section, we construct actual holomorphic solutions to our main problem (22), (23) as Laplace transforms which boils down to build up actual holomorphic solutions to the Cauchy problem (32), (33) in suitable Banach spaces.


### 3.1. Some Banach Spaces of Analytic Functions

In this subsection, we disclose the definition and properties of the Banach spaces in which we search for solutions to the Cauchy problem (32), (33).Definition 5.
(a)
Letb>1 be a real number. We set
(48)rbβ=∑n=0β1n+1b,for all integers β≥0. We fix σ>0 a real number. We denote Eβ,σ,Upk the vector space of all functions Θτ that are holomorphic on the unbounded sector Up (determined in Theorem 3) for which the norm
(49)Θτβ,σ,Up≔supτ∈Up1+τ2kτexp−σrbβτkΘτ,is finite.(b)
LetZ0>0 be a real number. We set Gσ,Z0,Upk as the vector space of all holomorphic functions
(50)Θτ,z=∑β≥0Θβτzββ!,near z=0 with holomorphic coefficients on Up such that the norm
(51)Θτ,zσ,Z0,Up≔∑β≥0Θβτβ,σ,UpZ0ββ!,is finite. The normed space Gσ,Z0,Upk,.σ,Z0,Up turns out to be a Banach space.It is worth noticing that these Banach spaces are slight modifications of the Banach spaces introduced by Malek and Stenger in the work [11] and by Costin and Tanveer in [12].In the next list of propositions, we analyze the continuity of linear and nonlinear maps acting on these Banach spaces that will show to be useful in the next subsection.Proposition 6.
Leth1,h2,h3≥0 be integers such that
(52)h2≥bh1.
Then, one can find a constantM1>0 (relying on k,q,h1,h2,h3,σ,b) such that
(53)τkh1∂z−h2fq−h3τ,zσ,Z0,Up≤M1Z0h2fτ,zσ,Z0,Up,for all fτ,z∈Gσ,Z0,Upk, where ∂z−h2 stands for the h2-times iteration of the integration map ∂z−1fz=∫0zfsds.Proof.
Letfτ,z=∑β≥0fβτzβ/β! with fβτ∈Eβ,σ,Upk for all β≥0. We check that
(54)τkh1∂z−h2fq−h3τ,z=∑β≥h2τkh1fβ−h2q−h3τzββ!.
In the next lemma, we provide bounds for the coefficients of this last series.Lemma 7.
The next bounds(55)τkh1fβ−h2q−h3τβ,σ,Up≤q2k−1h3h1σh2h1e−h1β+1bh1fβ−h2τβ−h2,σ,Up,hold for all β≥h2.Proof.
We observe from (35) that(56)τkh1fβ−h2q−h3τβ,σ,Up=supτ∈Up1+τ2k∣τ∣exp−σrbβτkτkh1fβ−h2q−h3τ∣supτ∈Up1+q−h3τ2kq−h3τexp−σrbβ−h2q−h3τkfβ−h2q−h3τ×Aτ,β,where
(57)Aτ,β=q−h3ττ1+τ2k1+q−h3τ2kτkh1exp−σrbβτk+σrbβ−h2q−h3τk,for all β≥h2. Besides, we notice that
(58)1+τ2k1+q−h3τ2k=1+q2kh3q−h3τ2k1+q−h3τ2k≤supx≥01+q2h3kx1+x=q2h3k,for all τ∈Up, since the function hx=1+q2kh3x/1+x is increasing on 0,+∞ provided that q>1. We deduce that
(59)Aτ,β≤ψτ,β,where
(60)ψτ,β=q2k−1h3τkh1exp−σrbβ−rbβ−h2τk,for all τ∈Up, all β≥h2. In the next step, we supply bounds for the map ψ. We check that
(61)rbβ−rbβ−h2=∑n=β−h2+1β1n+1b≥h2β+1b,for all β≥h2 and in a row with the classical estimates
(62)supx≥0xm1e−m2x=m1m2m1e−m1,for any given integers m1≥0, m2≥1, we deduce that
(63)ψτ,β≤q2k−1h3τkh1exp−σh2β+1bτk≤q2k−1h3supx≥0xh1exp−σh2β+1bx=q2k−1h3h1σh2h1e−h1β+1bh1,for all τ∈Up, all β≥h2.
At last, collecting (40), (41), and (42) gives rise to the forecast bounds (39).
Owing to the above lemma, we deduce the next bounds(64)τkh1∂z−h2fq−h3τ,zσ,Z0,Up≤∑β≥h2τkh1fβ−h2q−h3τβ,σ,UpZ0ββ!≤q2k−1h3h1σh2h1e−h1∑β≥h2fβ−h2τβ−h2,σ,Upβ+1bh1Z0ββ!=q2k−1h3h1σh2h1e−h1∑β≥h2β+1bh1β−h2!β!Z0h2fβ−h2τβ−h2,σ,UpZ0β−h2β−h2!.
Furthermore, we see that(65)β+1bh1β−h2!β!Z0h2=β+1bh1ββ−1⋯β−h2+1Z0h2≤β+1bh1β−h2+1h2Z0h2≤M⌣1Z0h2,for some constant M⌣1 (relying on b,h1,h2), for all β≥h2 under the constraint (37). Finally, gathering (43) and (44) yields the expected bounds (38).Proposition 8.
There exists a constantM2>0 (depending on k) such that
(66)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssdsσ,Z0,Up≤M2fτ,zσ,Z0,Upgτ,zσ,Z0,Up,for all f,g∈Gσ,Z0,Upk.Proof.
Let(67)fτ,z=∑β≥0fβτzββ!,τ,z=∑β≥0gβτzββ!,where fβτ,gβτ∈Eβ,σ,Upk, for all β≥0. By construction, one can check that
(68)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssds=∑β≥0∑β1+β2=βτk∫0τkβ!fβ1τk−s1/kβ1!gβ2s1/kβ2!1τk−ssdszββ!.In the next lemma, estimates for the coefficients of the latter series are disclosed.Lemma 9.
There exists a constantBk>0 depending on k such that
(69)τk∫0τkfβ1τk−s1/kgβ2s1/k1τk−ssdsβ,σ,Up≤Bkfβ1τβ1,σ,Upgβ2τβ2,σ,Up,for all β≥0, all β1,β2≥0 such that β1+β2=β.Proof.
Departing from the very definition of the norms, we can factorize the upper bounds as follows(70)τk∫0τkfβ1τk−s1/kgβ2s1/k1τk−ssdsβ,σ,Up≤supτ∈Up1+τ2k∣τ∣exp−σrbβτk×τk∫0τk1+τk−s1/k2kτk−s1/kexp−σrbβ1τk−s1/kkfβ1τk−s1/k×1+s1/k2ks1/kexp−σrbβ2s1/kkgβ2s1/k×τk−s1/k1+τk−s1/k2ks1/k1+s1/k2k×expσrbβ1τk−s1/kkexpσrbβ2s1/kk1τk−ssds,≤Bβ1,β2fβ1τβ1,σ,Upgβ2τβ2,σ,Upwhere
(71)Bβ1,β2≔supτ∈Up1+τ2kτexp−σrbβτk×τk∫0τkτk−u1/k1+τk−u2u1/k1+u2expσrbβ1τk−u×expσrbβ2u1τk−uudu,for all β1,β2≥0 with β1+β2=β. Since β↦rbβ is an increasing sequence, we observe that rbβ1≤rbβ together with rbβ2≤rbβ, and we get
(72)σrbβ1τk−u+σrbβ2u≤σrbβτk,for all 0≤u≤τk. We deduce that
(73)Bβ1,β2≤Bk,with
(74)Bk=supτ∈Up1+τ2k∣τ∣τk∫0τkτk−u1/k−11+τk−u2u1/k−11+u2du.
From now on, we perform computations based on the ones already done in our previous work [13]. We first assume that k≥2.
We make the change of variableu=τkx, 0≤x≤1 in the above integral part
(75)∫0τkτk−u1/k−11+τk−u2u1/k−1=du1+u2=∫0111+τ2k1−x211+τ2kx211−x1−1/k1x1−1/kdx×τ−k+2.
Besides, using a partial fraction decomposition, we can split the next integral along0,1 as
(76)∫0111+τ2k1−x211+τ2kx211−x1−1/k1x1−1/kdx=14+τ2k∫013−2x1+τ2k1−x211−x1−1/k1x1−1/kdx+∫012x+11+τ2kx211−x1−1/k1x1−1/kdx,for all τ∈Up. Furthermore, the change of variable x1=τkx, for 0≤x≤1 enables us to reach the bounds
(77)∫012x+11+τ2kx211−x1−1/k1x1−1/kdx≤3∫0111+τ2kx211−x1−1/k1x1−1/kdx=3τ∫0τk11+x1211−x1/τk1−1/k1x11−1/kdx1≤Bk,1τ,for some constant Bk,1>0 provided that τ∈Up with ∣τ∣≥1. In a similar way, one can find a constant Bk,2>0 for which
(78)∫013−2x1+τ2k1−x211−x1−1/k1x1−1/kdx≤Bk,2∣τ∣,as long as τ∈Up with τ≥1.
Gathering (51), (52), (53), and (54) yields that(79)Bkisafinitequantity,provided that k≥2.
It remains to check the casek=1. In that situation, the quantity B1 can be computed explicitly
(80)B1=supτ∈Up1+τ2∫0∣τ∣11+∣τ∣−u21+u2du=supx≥01+x22log1+x2+xarctanxxx2+4,and turns out to be a finite positive real number.
Finally, collecting the intermediate upper estimates (48), (50), (55), and (56) gives rise to Lemma9.
From the expansion (46) along with the lemma9, we obtain the next bounds
(81)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssdsσ,Z0,Up≤∑β≥0∑β1+β2=ββ!Bkfβ1τβ1,σ,Upβ1!gβ2τβ2,σ,Upβ2!Z0ββ!,
from which Proposition8 follows.Proposition 10.
Leth0≥1 and h1,h2,h3≥0 natural numbers such that
(82)h2≥bh1.
Then, there exists a constantM3>0 (depending upon k,q,h0,h1,h2,h3,σ,b) such that
(83)τk∫0τkτk−sh0/k−1sh1−1∂z−h2fq−h3s1/k,zdsσ,Z0,Up≤M3Z0h2fτ,zσ,Z0,Up,for all f∈Gσ,Z0,Upk.Proof.
We check that Proposition10 is a direct consequence of Proposition 6 and Proposition 8. Indeed, we set
(84)g~τ,z=τkh1∂z−h2fq−h3τ,z,for a given f∈Gσ,Z0,Upk. According to Proposition 6, we observe that g~τ,z belongs to Gσ,Z0,Upk and that
(85)g~τ,zσ,Z0,Up≤M1Z0h2fτ,zσ,Z0,Up,for some constant M1>0 provided that (58) holds. Furthermore, we set f~τ,z=τh0. By construction, we notice that f~τ,z∈Gσ,Z0,Upk. Then, we get that
(86)τk∫0τkτk−sh0/k−1sh1−1∂z−h2fq−h3s1/k,zds=τk∫0τkf~τk−s1/k,zg~s1/k,z1τk−ssds.
Owing to Proposition8, we get the next bounds
(87)τk∫0τkf~τk−s1/k,zg~s1/k,z1τk−ssdsσ,Z0,Up≤M2f~τ,zσ,Z0,Upg~τ,zσ,Z0,Up≤M2M1Z0h2τh0σ,Z0,Upfτ,zσ,Z0,Up,for some constant M2 (relying on k). At last, we set M3=M2M1τh0σ,Z0,Up which yields the result.Proposition 11.
Letaz=∑β≥0aβzβ/β! be a holomorphic function on a disc DR with radius R>Z0. Then, the next bounds
(88)azfτ,zσ,Z0,Up≤aZ0fτ,zσ,Z0,Up,hold for all f∈Gσ,Z0,Upk, where
(89)aZ0=∑β≥0aβZ0ββ!.Proof.
Let(90)fτ,z=∑β≥0fβτzββ!,with fβτ∈Eβ,σ,Upk for all β≥0. We first expand the product
(91)azfτ,z=∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ2!zββ!,which allows to control its norm
(92)azfτ,zσ,Z0,Up≤∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ,σ,Upβ2!Z0ββ!.
Besides, sincerbβ2≤rbβ for β2≤β, we notice that
(93)fβ2τβ,σ,Up≤fβ2τβ2,σ,Up,for 0≤β2≤β. Hence,
(94)azfτ,zσ,Z0,Up≤∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ2,σ,Upβ2!Z0ββ!=∑β≥0aββ!Z0β∑β≥0fβτβ,σ,Upβ!Z0β,which confirms the statement of Proposition 11.In the next proposition, we show that the Cauchy data (33) belong to the Banach spaces considered in Definition5.Proposition 12.
We set(95)Ψ⌣τ,z=∑j=0κ−1P⌣jτzjj!,where the polynomials P⌣jτ are given by the Cauchy data (33). Then,
(a)
For all integersh1,h3≥0, 0≤h2≤κ−1, the maps
(96)τ,z↦τkh1∂zh2Ψ⌣q−h3τ,z,belong to Gσ,Z0,Upk, for all σ,Z0>0.(b)
For all integersh0≥1, h1,h3≥0, 0≤h2≤κ−1, the maps
(97)τ,z↦τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zds,appertain to Gσ,Z0,Upk, for all σ,Z0>0.Proof.
We first make the Taylor expansion(98)∂zh2Ψ⌣q−h3τ,z=∑j=0κ−1−h2P⌣j+h2q−h3τzjj!,explicit provided that 0≤h2≤κ−1.
We focus on (a). By the very definition of the norm, we get(99)τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up=∑j=0κ−1−h2τkh1P⌣j+h2q−h3τj,σ,UpZ0jj!,and we can find a constant Δ⌣j (relying on j,h1,h2,h3,k) for which
(100)τkh1P⌣j+h2q−h3τj,σ,Up=supτ∈Up1+τ2kτexp−σrbjτkP⌣j+h2q−h3τ≤supx≥01+x2kxxkh1P⌣j+h2q−h3xexp−σxk≔Δ⌣j,since rbj≥1, for all j≥1, where
(101)P⌣j+h2x=∑h∈J⌣j+h2p⌣j+h2,hxh,and J⌣j+h2⊂ℕ\0. As a consequence,
(102)τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up≤∑j=0κ−1−h2Δ⌣jZ0jj!,which is a finite quantity.
We turn our attention to (b). The definition of the norm yields(103)τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up=∑j=0κ−1−h2τk∫0τkτk−sh0/k−1sh1−1P⌣j+h2q−h3s1/kdsj,σ,UpZ0jj!.
We need bounds for the coefficients of this latter polynomial inZ0. Indeed,
(104)Cj≔τk∫0τkτk−sh0/k−1sh1−1P⌣j+h2q−h3s1/kdsj,σ,Up≤supτ∈Up1+τ2kτexp−σrbjτkτk∫0τkτk−hh0/k−1hh1−1P⌣j+h2q−h3h1/kdh,where ∣P⌣j+h2∣x is defined in (65). We make the change of variable h=τkx, for 0≤x≤1 in the above integral and observe that rbj≥1, j≥1. This gives rise to a constant ∇⌣j (depending on j,h0,h1,h2,h3,k) such that
(105)Cj≤supy≥01+y2kyexp−σykyh0+kh1P⌣j+h2q−h3y∫011−xh0/k−1xh1−1dx≔∇⌣j.
Finally,(106)τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up≤∑j=0κ−1−h2∇⌣jZ0jj!,which represents a positive real number.In the following proposition, we claim that the forcing termwp of the Cauchy problem (22), (23) belong to the Banach space described in Definition 5.Proposition 13.
For all0≤p≤ς−1, the map τ,z↦wpτ,z belongs to the space Gσ,Z0,Upk, for all σ>0, provided that Z0<1/2C4.Proof.
According to the Taylor expansion (16), the very definition of the norm yields(107)wpτ,zσ,Z0,Up=∑β≥0wp,βτβ,σ,UpZ0ββ!.
Then, we need to control the coefficients of this latter series. Namely, one can find a constantC~3>0 (relying on C3,k1,α,u0 given in (17) and on k,σ) with
(108)wp,βτβ,σ,Up≔supτ∈Up1+τ2kτexp−σrbβτkwp,βτ≤C3C4ββ!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0=C~3C4ββ!,since rbβ≥1 for all β≥0. Consequently,
(109)wpτ,zσ,Z0,Up≤C~3∑β≥0C4Z0β≤2C~3,whenever Z0<1/2C4 and Proposition 13 ensues.


### 3.2. Solving the Main Nonlinear Convolution Cauchy Problem

In the following, we search for a solution to the nonlinear convolution Cauchy problem (32), (33) expressed by means of the next shape(110)Θpu,z=∂z−κΞpu,z+Ψ⌣u,z,where the map Ψ⌣u,z is defined in (63), for some expression Ξpu,z.We check thatΘpu,z solves the problem (32), (33) if the quantity Ξpu,z fulfills the next fixed point equation
(111)Ξpu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3ukh1×∂z−κ−h2Ξpq−h3u,z+∂zh2Ψ⌣q−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Ξpq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdss+a0zuk∫0uk∂z−κΞpuk−s1/k,z+Ψ⌣uk−s1/k,z×∂z−κΞps1/k,z+Ψ⌣s1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1×s∫0s∂z−κΞps−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞps11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss+wpu,z.Our next task will be to seek for a solution of this last equation (111) in the Banach space we have discussed in the previous subsection 3.1. We introduce the nonlinear map
(112)BΞτ,z≔∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3τkh1×∂z−κ−h2Ξq−h3τ,z+∂zh2Ψ⌣q−h3τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zτkΓh0/k∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Ξq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdss+a0zτk∫0τk∂z−κΞτk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞs1/k,z+Ψ⌣s1/k,z1τk−ssds+∑h=1AahzτkΓh/k∫0τkτk−sh/k−1×s∫0s∂z−κΞs−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss+wpτ,z.In the next proposition, we give sufficient conditions under whichB represents a shrinking map on some small ball centered at 0 in the space Gσ,Z0,Upk.Proposition 14.
Under the assumption (20), there exists some small real numberχ>0 such that if 0<Z0<χ, one can select a radius υ>0 such that B satisfies the next two properties.
LetBυ be the ball centered at 0 in Gσ,Z0,Upk with radius υ.
(1)
B maps Bυ into Bυ, meaning that
(113)BBυ⊂Bυ.(2)
For allΞ1,Ξ2∈Bυ, we have
(114)BΞ1−BΞ2σ,Z0,Up≤12Ξ1−Ξ2σ,Z0,Up.Proof.
We discuss the first point 1. LetΞτ,z belong to Gσ,Z0,Upk with Ξτ,zσ,Z0,Up≤υ.
Under the constraint (20) and owing to Propositions6, 11, and 12, we get a constant M1>0 such that
(115)dh¯zτkh1∂z−κ−h2Ξq−h3τ,z+∂zh2Ψ⌣q−h3τ,zσ,Z0,Up≤dh¯Z0M1Z0κ−h2Ξτ,zσ,Z0,Up+τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up.
Condition (30) and Propositions 10, 11, and 12 allow us to reach a constant M3>0 for which
(116)dh¯zτk∫0τkτk−sh0/k−1sh1−1×∂z−κ−h2Ξq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up≤dh¯Z0M3Z0κ−h2Ξτ,zσ,Z0,Up+τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up.
According to Propositions6, 8, 11, and 12, we obtain constants M1,M2>0 with
(117)a0zτk∫0τk∂z−κΞτk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞs1/k,z+Ψ⌣s1/k,z1τk−ssdsσ,Z0,Up≤a0Z0M2∂z−κΞτ,z+Ψ⌣τ,zσ,Z0,Up2≤a0Z0M2∂z−κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2≤a0Z0M2M1Z0κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2.
Propositions6, 8, 10, 11, and 12 grant the existence of constants M1,M2,M3>0 such that
(118)ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞs−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dssσ,Z0,Up≤ahZ0M3τk∫0τk∂z−κΞτk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1τk−s1s1ds1σ,Z0,Up≤ahZ0M3M2∂z−κΞτ,z+Ψ⌣τ,zσ,Z0,Up2≤ahZ0M3M2∂z−κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2≤ahZ0M3M2M1Z0κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2.
We first choose the radiusυ>0 in a way that
(119)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0×τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up+a0Z0M2Ψ⌣τ,zσ,Z0,Up2+∑h=1AahZ01Γh/kM3M2Ψ⌣τ,zσ,Z0,Up2+wpτ,zσ,Z0,Up<υ,for some fixed Z0>0. Then, we select χ>0 sufficiently small with 0<Z0<χ in a way that the next inequality holds,
(120)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0M1Z0κ−h2υ+τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0×M3Z0κ−h2υ+τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up+a0Z0M2M1Z0κυ+Ψ⌣τ,zσ,Z0,Up2+∑h=1AahZ01Γh/kM3M2M1Z0κυ+Ψ⌣τ,zσ,Z0,Up2+wpτ,zσ,Z0,Up≤υ.
This last inequality can be achieved since the quantitiesZ0κ−h2 and Z0κ tend to zero as Z0>0 becomes close to the origin, accordingly to the assumption (20).
Collecting all the above estimates (74), (75), (76), and (77), under the latter constraint (78), sires the awaited property (72).
We now turn to the second feature 2. LetΞjτ,z∈Gσ,Z0,Upk with Ξjτ,zσ,Z0,Up≤υ, for j=1,2.
Under the condition (30), Propositions 6 and 11 yield a constant M1>0 for which
(121)dh¯zτkh1∂z−κ−h2Ξ1−Ξ2q−h3τ,zσ,Z0,Up≤dh¯Z0M1Z0κ−h2Ξ1−Ξ2τ,zσ,Z0,Up,
and Propositions10 and 11 breed a constant M3>0 such that
(122)dh¯zτk∫0τkτk−sh0/k−1sh1−1∂z−κ−h2Ξ1−Ξ2q−h3s1/k,zdsσ,Z0,Up≤dh¯Z0M3Z0κ−h2Ξ1τ,z−Ξ2τ,zσ,Z0,Up.
In order to deal with the nonlinear terms, we use the next identityab−cd=a−cb+cb−d which helps to factorize the next difference
(123)∂z−κΞ1τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1s1/k,z+Ψ⌣s1/k,z−∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ2s1/k,z+Ψ⌣s1/k,z=∂z−κΞ1−Ξ2τk−s1/k,z∂z−κΞ1s1/k,z+Ψ⌣s1/k,z+∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1−Ξ2s1/k,z,which leads, by means of Proposition 6, 8, 11, and 12, to constants M1,M2>0 for which
(124)a0zτk∫0τk∂z−κΞ1τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1s1/k,z+Ψ⌣s1/k,z1τk−ssds−a0zτk∫0τk∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ2s1/k,z+Ψ⌣s1/k,z1τk−ssdsσ,Z0,Up≤a0Z0M2∂z−κΞ1−Ξ2τ,zσ,Z0,Up×∂z−κΞ1τ,z+Ψ⌣τ,zσ,Z0,Up+∂z−κΞ2τ,z+Ψ⌣τ,zσ,Z0,Up×∂z−κΞ1−Ξ2τ,zσ,Z0,Up≤a0Z0M2M1Z0κΞ1τ,zσ,Z0,Up+M1Z0κΞ2τ,zσ,Z0,Up+2Ψ⌣τ,zσ,Z0,Up×M1Z0κΞ1τ,z−Ξ2τ,zσ,Z0,Up.
Furthermore, according to Proposition10, we get a constant M3>0 with
(125)A≔ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞ1s−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞ1s11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss−ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞ2s−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞ2s11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dssσ,Z0,Up≤ahZ0M3τk∫0τk∂z−κΞ1τk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞ1s11/k,z+Ψ⌣s11/k,z−∂z−κΞ2τk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞ2s11/k,z+Ψ⌣s11/k,z×1τk−s1s1ds1σ,Z0,Up.
Using the factorization (81), Propositions6, 8, 11, and 12 grant the existence of constants M1,M2>0 such that
(126)A≤ahZ0M3M2∂z−κΞ1−Ξ2τ,zσ,Z0,Up×∂z−κΞ1τ,z+Ψ⌣τ,zσ,Z0,Up+∂z−κΞ2τ,z+Ψ⌣τ,zσ,Z0,Up×∂z−κΞ1−Ξ2τ,zσ,Z0,Up≤ahZ0M3M2M1Z0κΞ1τ,zσ,Z0,Up+M1Z0κΞ2τ,zσ,Z0,Up+2Ψ⌣τ,zσ,Z0,Up×M1Z0κΞ1τ,z−Ξ2τ,zσ,Z0,Up.
From now on, we select the radiusυ>0 and χ>0 small enough with 0<Z0<χ such that 0<Z0<χ in order that the next condition holds,
(127)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0M1Z0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0M3Z0κ−h2+a0Z0M22M1Z0κυ+2Ψ⌣τ,zσ,Z0,UpM1Z0κ+∑h=1AahZ01Γh/kM3M22M1Z0κυ+2Ψ⌣τ,zσ,Z0,UpM1Z0κ≤12.
Gathering the list of bounds (79), (80), (82), (83), (84), subjected to (85), implies the shrinking condition (114) we are looking for.
Finally, in order to certify both properties (72) and (73), we impose on the constantsχ>0 and υ>0 the conjoint constraints (78) and (85). Proposition 14 follows.In the next proposition, we solve the nonlinear Cauchy problem (32), (33) in the Banach spaces described in Subsection3.1.Proposition 15.
We take for granted that assumption (20) holds. We fix the constantsχ>0 and υ>0 as in proposition 14. Then, the convolution Cauchy problem (32), (33) possesses a solution Θpu,z which belongs to the space Gσ,Z0,Upk, for any given σ>0 provided that 0<Z0<χ. Furthermore, one can single out a constant M1>0 (relying on k,q,κ,σ,b) such that
(128)Θpτ,zσ,Z0,Up≤M1Z0κυ+Ψ⌣τ,zσ,Z0,Up.Proof.
According to Proposition14, we can apply the classical fixed point theorem for shrinking maps in complete metric spaces to the map B:Bυ⟶Bυ. By construction, Bυ,d is a complete metric space for the distance dx,y=x−yσ,Z0,Up since the vector space Gσ,Z0,Upk equipped with the norm .σ,Z0,Up is a Banach space. Furthermore, under the constraints imposed in Proposition 14, the map B:Bυ⟶Bυ is shrinking of Lipschitz type 1/2. As a result, B:Bυ⟶Bυ has a unique fixed point, denoted Ξpτ,z∈Bυ, meaning that
(129)BΞp=Ξp.
This means in particular that we obtain a solution (unique in the ballBυ) Ξpτ,z for the equation (111). Moreover, owing to Proposition 6, we check that
(130)∂z−κΞpτ,zσ,Z0,Up≤M1Z0κΞpτ,zσ,Z0,Up≤M1Z0κυ.
The decomposition (69) then confirms that the mapΘpτ,z=∂z−κΞpτ,z+Ψ⌣τ,z belongs to Gσ,Z0,Upk, solves the convolution Cauchy problem (32), (33), and suffers the bounds (86).


### 3.3. Construction of Genuine Solutions to the Main Cauchy Problem (22), (23)

In this subsection, we state the first main result of this work.Theorem 16.
Assume that condition (30) is granted. Assume that the radius rT of each bounded sector Tp described in Theorem 3 fulfills
(131)rT<Δpσζb1/k,for 0≤p≤ς−1, where Δp is introduced in (14) and where ζb=∑n≥01/n+1b.
Then, the Cauchy problem (22), (23) possesses a holomorphic solutionvpt,z on the product Tp×DZ0/2, for some radius Z0>0 small enough. Furthermore, vpt,z can be expressed by means of a Laplace transform of order k,
(132)vpt,z=k∫LγpΘpu,zexp−utkduu,for t,z∈Tp×DZ0/2 along a halfline Lγp=0,+∞e−1γp which appears in the representation (15). The Borel map Θpu,z represents a holomorphic function on the domain Up×DZ0/2 whose Taylor expansion
(133)Θpu,z=∑β≥0Θp,βuzββ!,is submitted to the following estimates
(134)Θp,βu≤K1β!1Z0βu1+u2kexpσrbβuk,for all β≥0, all u∈Up, for well selected constants K1>0, any given σ>0, where rbβ is the sequence defined by (34). In particular, the Borel map Θpu,z suffers the next bounds estimates
(135)Θpu,z≤2K1uexpσζbuk,for u,z∈Up×DZ0/2.Proof.
The proof is a direct consequence of Proposition15 and of the construction made in Subsection 2.3.In the next corollary, we show thatvpt,z turns out to solve a nonlinear Cauchy problem with analytic coefficients in space z near the origin and polynomial in time t, which involves both differential operators and dilatations/contractions q−difference operators acting on time.Corollary 17.
The holomorphic mapvpt,z solves on the product Tp×DZ0/2 a particular nonlinear Cauchy problem which is polynomial in time t of the form
(136)Ptk+1∂t∂zS+κvpt,z=∑m¯=m1,m2,m3∈Mfm¯t,z∂tm1∂zm2vpqm3t,z+Gt,z,∂tr1∂zr2vpqr3t,zr¯=r1,r2,r3∈R,for given Cauchy data of the form
(137)∂zjvpt,0=φ⌣jt,0≤j≤κ−1,together with
(138)∂zjvpt,0=ϕ⌣jt,κ≤j≤κ+S−1,for well-chosen polynomials ϕ⌣jt with complex coefficients for κ≤j≤κ+S−1 (which depend on q).
The setM is a finite subset of ℕ2×ℤ which satisfies in particular the constraint m2<S+κ for all m¯=m1,m2,m3∈M and R is a finite subset of ℕ3 with the property that r2≤S whenever r1,r2,r3∈R.
For allm¯∈M, the coefficients fm¯t,z are polynomial in t and holomorphic w.r.t z the disc DR. The map Gt,z,ur¯r¯∈R is polynomial in the variable t, a polynomial of degree at most 2 in the variables ur¯ for r¯∈R and relies holomorphically on the variable z in the disc DR.Proof.
We introduce the next twoq−difference differential operators
(139)O1t,z,∂t,∂z,σq≔Ptk+1∂t∂zS−∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2,and
(140)O2t,z,∂t,∂z,σqv≔∂zκv−∑h¯=h0,h1,h2,h3∈Cdh¯zth0σq;t−h3tk+1∂th1∂zh2v−at,zv2,where σq;t stands for the dilation operator acting on functions through σq;tft,z=fqt,z. By construction, the holomorphic map vp satisfies
(141)O2t,z,∂t,∂z,σqvpt,z=upt,z,and the map up fulfills
(142)O1t,z,∂t,∂z,σqupt,z=0,for t,z∈Tp×DZ0/2. By coupling (95) and (96), we obtain that
(143)O1t,z,∂t,∂z,σq∘O2t,z,∂t,∂z,σqvpt,z=0,which can be expanded in the more precise shape
(144)Ptk+1∂t∂zS+κvpt,z−∑h¯=h0,h1,h2,h3∈CPtk+1∂t∂zS∘dh¯zth0σq;t−h3tk+1∂th1∂zh2vpt,z−∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2∘∂zκvpt,z+∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2∘∑h¯=h0,h1,h2,h3∈Cdh¯zth0σq;t−h3tk+1∂th1∂zh2vpt,z−Ptk+1∂t∂zSat,zvp2t,z+∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2at,zvp2t,z=0,
and gives rise to the equation (136). Concerning the Cauchy data, the map vp is compelled to the conditions (33) which are rewritten in (93). Furthermore, the fact that upt,z is subjected to the constraints (12) at z=0 can be recast in the form
(145)∂zj∘O2t,z,∂t,∂z,σqvpt,0=φjt,for 0≤j≤S−1 which can be rephrased through the Cauchy conditions (138) for suitably selected polynomials ϕ⌣lt with complex coefficients for κ≤l≤κ+S−1, since j+κ>j+h2 for all h¯=h0,h1,h2,h3∈C provided that 0≤j≤S−1.Example 3.
A model of such an equation (136) can be obtained in coupling the examples 1 and 2 given by (18), (24) and is written in the form
(146)t5∂t−1∂z3vpt,z=5q−5t9∂t∂zvpq−1t,z+q−6t10∂t2∂zvpq−1t,z−q−5t5∂t∂zvpq−1t,z+zt3∂z2vpqt,z−zt8∂tvt,z+2t5∂t∂zvpt,z×vpt,z+∂zvpt,z×∂tvpt,z−2∂zvpt,z×vpt,z−zt3vp2qt,z.


## 3.1. Some Banach Spaces of Analytic Functions

In this subsection, we disclose the definition and properties of the Banach spaces in which we search for solutions to the Cauchy problem (32), (33).Definition 5.
(a)
Letb>1 be a real number. We set
(48)rbβ=∑n=0β1n+1b,for all integers β≥0. We fix σ>0 a real number. We denote Eβ,σ,Upk the vector space of all functions Θτ that are holomorphic on the unbounded sector Up (determined in Theorem 3) for which the norm
(49)Θτβ,σ,Up≔supτ∈Up1+τ2kτexp−σrbβτkΘτ,is finite.(b)
LetZ0>0 be a real number. We set Gσ,Z0,Upk as the vector space of all holomorphic functions
(50)Θτ,z=∑β≥0Θβτzββ!,near z=0 with holomorphic coefficients on Up such that the norm
(51)Θτ,zσ,Z0,Up≔∑β≥0Θβτβ,σ,UpZ0ββ!,is finite. The normed space Gσ,Z0,Upk,.σ,Z0,Up turns out to be a Banach space.It is worth noticing that these Banach spaces are slight modifications of the Banach spaces introduced by Malek and Stenger in the work [11] and by Costin and Tanveer in [12].In the next list of propositions, we analyze the continuity of linear and nonlinear maps acting on these Banach spaces that will show to be useful in the next subsection.Proposition 6.
Leth1,h2,h3≥0 be integers such that
(52)h2≥bh1.
Then, one can find a constantM1>0 (relying on k,q,h1,h2,h3,σ,b) such that
(53)τkh1∂z−h2fq−h3τ,zσ,Z0,Up≤M1Z0h2fτ,zσ,Z0,Up,for all fτ,z∈Gσ,Z0,Upk, where ∂z−h2 stands for the h2-times iteration of the integration map ∂z−1fz=∫0zfsds.Proof.
Letfτ,z=∑β≥0fβτzβ/β! with fβτ∈Eβ,σ,Upk for all β≥0. We check that
(54)τkh1∂z−h2fq−h3τ,z=∑β≥h2τkh1fβ−h2q−h3τzββ!.
In the next lemma, we provide bounds for the coefficients of this last series.Lemma 7.
The next bounds(55)τkh1fβ−h2q−h3τβ,σ,Up≤q2k−1h3h1σh2h1e−h1β+1bh1fβ−h2τβ−h2,σ,Up,hold for all β≥h2.Proof.
We observe from (35) that(56)τkh1fβ−h2q−h3τβ,σ,Up=supτ∈Up1+τ2k∣τ∣exp−σrbβτkτkh1fβ−h2q−h3τ∣supτ∈Up1+q−h3τ2kq−h3τexp−σrbβ−h2q−h3τkfβ−h2q−h3τ×Aτ,β,where
(57)Aτ,β=q−h3ττ1+τ2k1+q−h3τ2kτkh1exp−σrbβτk+σrbβ−h2q−h3τk,for all β≥h2. Besides, we notice that
(58)1+τ2k1+q−h3τ2k=1+q2kh3q−h3τ2k1+q−h3τ2k≤supx≥01+q2h3kx1+x=q2h3k,for all τ∈Up, since the function hx=1+q2kh3x/1+x is increasing on 0,+∞ provided that q>1. We deduce that
(59)Aτ,β≤ψτ,β,where
(60)ψτ,β=q2k−1h3τkh1exp−σrbβ−rbβ−h2τk,for all τ∈Up, all β≥h2. In the next step, we supply bounds for the map ψ. We check that
(61)rbβ−rbβ−h2=∑n=β−h2+1β1n+1b≥h2β+1b,for all β≥h2 and in a row with the classical estimates
(62)supx≥0xm1e−m2x=m1m2m1e−m1,for any given integers m1≥0, m2≥1, we deduce that
(63)ψτ,β≤q2k−1h3τkh1exp−σh2β+1bτk≤q2k−1h3supx≥0xh1exp−σh2β+1bx=q2k−1h3h1σh2h1e−h1β+1bh1,for all τ∈Up, all β≥h2.
At last, collecting (40), (41), and (42) gives rise to the forecast bounds (39).
Owing to the above lemma, we deduce the next bounds(64)τkh1∂z−h2fq−h3τ,zσ,Z0,Up≤∑β≥h2τkh1fβ−h2q−h3τβ,σ,UpZ0ββ!≤q2k−1h3h1σh2h1e−h1∑β≥h2fβ−h2τβ−h2,σ,Upβ+1bh1Z0ββ!=q2k−1h3h1σh2h1e−h1∑β≥h2β+1bh1β−h2!β!Z0h2fβ−h2τβ−h2,σ,UpZ0β−h2β−h2!.
Furthermore, we see that(65)β+1bh1β−h2!β!Z0h2=β+1bh1ββ−1⋯β−h2+1Z0h2≤β+1bh1β−h2+1h2Z0h2≤M⌣1Z0h2,for some constant M⌣1 (relying on b,h1,h2), for all β≥h2 under the constraint (37). Finally, gathering (43) and (44) yields the expected bounds (38).Proposition 8.
There exists a constantM2>0 (depending on k) such that
(66)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssdsσ,Z0,Up≤M2fτ,zσ,Z0,Upgτ,zσ,Z0,Up,for all f,g∈Gσ,Z0,Upk.Proof.
Let(67)fτ,z=∑β≥0fβτzββ!,τ,z=∑β≥0gβτzββ!,where fβτ,gβτ∈Eβ,σ,Upk, for all β≥0. By construction, one can check that
(68)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssds=∑β≥0∑β1+β2=βτk∫0τkβ!fβ1τk−s1/kβ1!gβ2s1/kβ2!1τk−ssdszββ!.In the next lemma, estimates for the coefficients of the latter series are disclosed.Lemma 9.
There exists a constantBk>0 depending on k such that
(69)τk∫0τkfβ1τk−s1/kgβ2s1/k1τk−ssdsβ,σ,Up≤Bkfβ1τβ1,σ,Upgβ2τβ2,σ,Up,for all β≥0, all β1,β2≥0 such that β1+β2=β.Proof.
Departing from the very definition of the norms, we can factorize the upper bounds as follows(70)τk∫0τkfβ1τk−s1/kgβ2s1/k1τk−ssdsβ,σ,Up≤supτ∈Up1+τ2k∣τ∣exp−σrbβτk×τk∫0τk1+τk−s1/k2kτk−s1/kexp−σrbβ1τk−s1/kkfβ1τk−s1/k×1+s1/k2ks1/kexp−σrbβ2s1/kkgβ2s1/k×τk−s1/k1+τk−s1/k2ks1/k1+s1/k2k×expσrbβ1τk−s1/kkexpσrbβ2s1/kk1τk−ssds,≤Bβ1,β2fβ1τβ1,σ,Upgβ2τβ2,σ,Upwhere
(71)Bβ1,β2≔supτ∈Up1+τ2kτexp−σrbβτk×τk∫0τkτk−u1/k1+τk−u2u1/k1+u2expσrbβ1τk−u×expσrbβ2u1τk−uudu,for all β1,β2≥0 with β1+β2=β. Since β↦rbβ is an increasing sequence, we observe that rbβ1≤rbβ together with rbβ2≤rbβ, and we get
(72)σrbβ1τk−u+σrbβ2u≤σrbβτk,for all 0≤u≤τk. We deduce that
(73)Bβ1,β2≤Bk,with
(74)Bk=supτ∈Up1+τ2k∣τ∣τk∫0τkτk−u1/k−11+τk−u2u1/k−11+u2du.
From now on, we perform computations based on the ones already done in our previous work [13]. We first assume that k≥2.
We make the change of variableu=τkx, 0≤x≤1 in the above integral part
(75)∫0τkτk−u1/k−11+τk−u2u1/k−1=du1+u2=∫0111+τ2k1−x211+τ2kx211−x1−1/k1x1−1/kdx×τ−k+2.
Besides, using a partial fraction decomposition, we can split the next integral along0,1 as
(76)∫0111+τ2k1−x211+τ2kx211−x1−1/k1x1−1/kdx=14+τ2k∫013−2x1+τ2k1−x211−x1−1/k1x1−1/kdx+∫012x+11+τ2kx211−x1−1/k1x1−1/kdx,for all τ∈Up. Furthermore, the change of variable x1=τkx, for 0≤x≤1 enables us to reach the bounds
(77)∫012x+11+τ2kx211−x1−1/k1x1−1/kdx≤3∫0111+τ2kx211−x1−1/k1x1−1/kdx=3τ∫0τk11+x1211−x1/τk1−1/k1x11−1/kdx1≤Bk,1τ,for some constant Bk,1>0 provided that τ∈Up with ∣τ∣≥1. In a similar way, one can find a constant Bk,2>0 for which
(78)∫013−2x1+τ2k1−x211−x1−1/k1x1−1/kdx≤Bk,2∣τ∣,as long as τ∈Up with τ≥1.
Gathering (51), (52), (53), and (54) yields that(79)Bkisafinitequantity,provided that k≥2.
It remains to check the casek=1. In that situation, the quantity B1 can be computed explicitly
(80)B1=supτ∈Up1+τ2∫0∣τ∣11+∣τ∣−u21+u2du=supx≥01+x22log1+x2+xarctanxxx2+4,and turns out to be a finite positive real number.
Finally, collecting the intermediate upper estimates (48), (50), (55), and (56) gives rise to Lemma9.
From the expansion (46) along with the lemma9, we obtain the next bounds
(81)τk∫0τkfτk−s1/k,zgs1/k,z1τk−ssdsσ,Z0,Up≤∑β≥0∑β1+β2=ββ!Bkfβ1τβ1,σ,Upβ1!gβ2τβ2,σ,Upβ2!Z0ββ!,
from which Proposition8 follows.Proposition 10.
Leth0≥1 and h1,h2,h3≥0 natural numbers such that
(82)h2≥bh1.
Then, there exists a constantM3>0 (depending upon k,q,h0,h1,h2,h3,σ,b) such that
(83)τk∫0τkτk−sh0/k−1sh1−1∂z−h2fq−h3s1/k,zdsσ,Z0,Up≤M3Z0h2fτ,zσ,Z0,Up,for all f∈Gσ,Z0,Upk.Proof.
We check that Proposition10 is a direct consequence of Proposition 6 and Proposition 8. Indeed, we set
(84)g~τ,z=τkh1∂z−h2fq−h3τ,z,for a given f∈Gσ,Z0,Upk. According to Proposition 6, we observe that g~τ,z belongs to Gσ,Z0,Upk and that
(85)g~τ,zσ,Z0,Up≤M1Z0h2fτ,zσ,Z0,Up,for some constant M1>0 provided that (58) holds. Furthermore, we set f~τ,z=τh0. By construction, we notice that f~τ,z∈Gσ,Z0,Upk. Then, we get that
(86)τk∫0τkτk−sh0/k−1sh1−1∂z−h2fq−h3s1/k,zds=τk∫0τkf~τk−s1/k,zg~s1/k,z1τk−ssds.
Owing to Proposition8, we get the next bounds
(87)τk∫0τkf~τk−s1/k,zg~s1/k,z1τk−ssdsσ,Z0,Up≤M2f~τ,zσ,Z0,Upg~τ,zσ,Z0,Up≤M2M1Z0h2τh0σ,Z0,Upfτ,zσ,Z0,Up,for some constant M2 (relying on k). At last, we set M3=M2M1τh0σ,Z0,Up which yields the result.Proposition 11.
Letaz=∑β≥0aβzβ/β! be a holomorphic function on a disc DR with radius R>Z0. Then, the next bounds
(88)azfτ,zσ,Z0,Up≤aZ0fτ,zσ,Z0,Up,hold for all f∈Gσ,Z0,Upk, where
(89)aZ0=∑β≥0aβZ0ββ!.Proof.
Let(90)fτ,z=∑β≥0fβτzββ!,with fβτ∈Eβ,σ,Upk for all β≥0. We first expand the product
(91)azfτ,z=∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ2!zββ!,which allows to control its norm
(92)azfτ,zσ,Z0,Up≤∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ,σ,Upβ2!Z0ββ!.
Besides, sincerbβ2≤rbβ for β2≤β, we notice that
(93)fβ2τβ,σ,Up≤fβ2τβ2,σ,Up,for 0≤β2≤β. Hence,
(94)azfτ,zσ,Z0,Up≤∑β≥0∑β1+β2=ββ!aβ1β1!fβ2τβ2,σ,Upβ2!Z0ββ!=∑β≥0aββ!Z0β∑β≥0fβτβ,σ,Upβ!Z0β,which confirms the statement of Proposition 11.In the next proposition, we show that the Cauchy data (33) belong to the Banach spaces considered in Definition5.Proposition 12.
We set(95)Ψ⌣τ,z=∑j=0κ−1P⌣jτzjj!,where the polynomials P⌣jτ are given by the Cauchy data (33). Then,
(a)
For all integersh1,h3≥0, 0≤h2≤κ−1, the maps
(96)τ,z↦τkh1∂zh2Ψ⌣q−h3τ,z,belong to Gσ,Z0,Upk, for all σ,Z0>0.(b)
For all integersh0≥1, h1,h3≥0, 0≤h2≤κ−1, the maps
(97)τ,z↦τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zds,appertain to Gσ,Z0,Upk, for all σ,Z0>0.Proof.
We first make the Taylor expansion(98)∂zh2Ψ⌣q−h3τ,z=∑j=0κ−1−h2P⌣j+h2q−h3τzjj!,explicit provided that 0≤h2≤κ−1.
We focus on (a). By the very definition of the norm, we get(99)τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up=∑j=0κ−1−h2τkh1P⌣j+h2q−h3τj,σ,UpZ0jj!,and we can find a constant Δ⌣j (relying on j,h1,h2,h3,k) for which
(100)τkh1P⌣j+h2q−h3τj,σ,Up=supτ∈Up1+τ2kτexp−σrbjτkP⌣j+h2q−h3τ≤supx≥01+x2kxxkh1P⌣j+h2q−h3xexp−σxk≔Δ⌣j,since rbj≥1, for all j≥1, where
(101)P⌣j+h2x=∑h∈J⌣j+h2p⌣j+h2,hxh,and J⌣j+h2⊂ℕ\0. As a consequence,
(102)τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up≤∑j=0κ−1−h2Δ⌣jZ0jj!,which is a finite quantity.
We turn our attention to (b). The definition of the norm yields(103)τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up=∑j=0κ−1−h2τk∫0τkτk−sh0/k−1sh1−1P⌣j+h2q−h3s1/kdsj,σ,UpZ0jj!.
We need bounds for the coefficients of this latter polynomial inZ0. Indeed,
(104)Cj≔τk∫0τkτk−sh0/k−1sh1−1P⌣j+h2q−h3s1/kdsj,σ,Up≤supτ∈Up1+τ2kτexp−σrbjτkτk∫0τkτk−hh0/k−1hh1−1P⌣j+h2q−h3h1/kdh,where ∣P⌣j+h2∣x is defined in (65). We make the change of variable h=τkx, for 0≤x≤1 in the above integral and observe that rbj≥1, j≥1. This gives rise to a constant ∇⌣j (depending on j,h0,h1,h2,h3,k) such that
(105)Cj≤supy≥01+y2kyexp−σykyh0+kh1P⌣j+h2q−h3y∫011−xh0/k−1xh1−1dx≔∇⌣j.
Finally,(106)τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up≤∑j=0κ−1−h2∇⌣jZ0jj!,which represents a positive real number.In the following proposition, we claim that the forcing termwp of the Cauchy problem (22), (23) belong to the Banach space described in Definition 5.Proposition 13.
For all0≤p≤ς−1, the map τ,z↦wpτ,z belongs to the space Gσ,Z0,Upk, for all σ>0, provided that Z0<1/2C4.Proof.
According to the Taylor expansion (16), the very definition of the norm yields(107)wpτ,zσ,Z0,Up=∑β≥0wp,βτβ,σ,UpZ0ββ!.
Then, we need to control the coefficients of this latter series. Namely, one can find a constantC~3>0 (relying on C3,k1,α,u0 given in (17) and on k,σ) with
(108)wp,βτβ,σ,Up≔supτ∈Up1+τ2kτexp−σrbβτkwp,βτ≤C3C4ββ!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0=C~3C4ββ!,since rbβ≥1 for all β≥0. Consequently,
(109)wpτ,zσ,Z0,Up≤C~3∑β≥0C4Z0β≤2C~3,whenever Z0<1/2C4 and Proposition 13 ensues.


## 3.2. Solving the Main Nonlinear Convolution Cauchy Problem

In the following, we search for a solution to the nonlinear convolution Cauchy problem (32), (33) expressed by means of the next shape(110)Θpu,z=∂z−κΞpu,z+Ψ⌣u,z,where the map Ψ⌣u,z is defined in (63), for some expression Ξpu,z.We check thatΘpu,z solves the problem (32), (33) if the quantity Ξpu,z fulfills the next fixed point equation
(111)Ξpu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3ukh1×∂z−κ−h2Ξpq−h3u,z+∂zh2Ψ⌣q−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Ξpq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdss+a0zuk∫0uk∂z−κΞpuk−s1/k,z+Ψ⌣uk−s1/k,z×∂z−κΞps1/k,z+Ψ⌣s1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1×s∫0s∂z−κΞps−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞps11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss+wpu,z.Our next task will be to seek for a solution of this last equation (111) in the Banach space we have discussed in the previous subsection 3.1. We introduce the nonlinear map
(112)BΞτ,z≔∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−h3τkh1×∂z−κ−h2Ξq−h3τ,z+∂zh2Ψ⌣q−h3τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zτkΓh0/k∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Ξq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdss+a0zτk∫0τk∂z−κΞτk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞs1/k,z+Ψ⌣s1/k,z1τk−ssds+∑h=1AahzτkΓh/k∫0τkτk−sh/k−1×s∫0s∂z−κΞs−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss+wpτ,z.In the next proposition, we give sufficient conditions under whichB represents a shrinking map on some small ball centered at 0 in the space Gσ,Z0,Upk.Proposition 14.
Under the assumption (20), there exists some small real numberχ>0 such that if 0<Z0<χ, one can select a radius υ>0 such that B satisfies the next two properties.
LetBυ be the ball centered at 0 in Gσ,Z0,Upk with radius υ.
(1)
B maps Bυ into Bυ, meaning that
(113)BBυ⊂Bυ.(2)
For allΞ1,Ξ2∈Bυ, we have
(114)BΞ1−BΞ2σ,Z0,Up≤12Ξ1−Ξ2σ,Z0,Up.Proof.
We discuss the first point 1. LetΞτ,z belong to Gσ,Z0,Upk with Ξτ,zσ,Z0,Up≤υ.
Under the constraint (20) and owing to Propositions6, 11, and 12, we get a constant M1>0 such that
(115)dh¯zτkh1∂z−κ−h2Ξq−h3τ,z+∂zh2Ψ⌣q−h3τ,zσ,Z0,Up≤dh¯Z0M1Z0κ−h2Ξτ,zσ,Z0,Up+τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up.
Condition (30) and Propositions 10, 11, and 12 allow us to reach a constant M3>0 for which
(116)dh¯zτk∫0τkτk−sh0/k−1sh1−1×∂z−κ−h2Ξq−h3s1/k,z+∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up≤dh¯Z0M3Z0κ−h2Ξτ,zσ,Z0,Up+τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up.
According to Propositions6, 8, 11, and 12, we obtain constants M1,M2>0 with
(117)a0zτk∫0τk∂z−κΞτk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞs1/k,z+Ψ⌣s1/k,z1τk−ssdsσ,Z0,Up≤a0Z0M2∂z−κΞτ,z+Ψ⌣τ,zσ,Z0,Up2≤a0Z0M2∂z−κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2≤a0Z0M2M1Z0κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2.
Propositions6, 8, 10, 11, and 12 grant the existence of constants M1,M2,M3>0 such that
(118)ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞs−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dssσ,Z0,Up≤ahZ0M3τk∫0τk∂z−κΞτk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞs11/k,z+Ψ⌣s11/k,z1τk−s1s1ds1σ,Z0,Up≤ahZ0M3M2∂z−κΞτ,z+Ψ⌣τ,zσ,Z0,Up2≤ahZ0M3M2∂z−κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2≤ahZ0M3M2M1Z0κΞτ,zσ,Z0,Up+Ψ⌣τ,zσ,Z0,Up2.
We first choose the radiusυ>0 in a way that
(119)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0×τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up+a0Z0M2Ψ⌣τ,zσ,Z0,Up2+∑h=1AahZ01Γh/kM3M2Ψ⌣τ,zσ,Z0,Up2+wpτ,zσ,Z0,Up<υ,for some fixed Z0>0. Then, we select χ>0 sufficiently small with 0<Z0<χ in a way that the next inequality holds,
(120)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0M1Z0κ−h2υ+τkh1∂zh2Ψ⌣q−h3τ,zσ,Z0,Up+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0×M3Z0κ−h2υ+τk∫0τkτk−sh0/k−1sh1−1∂zh2Ψ⌣q−h3s1/k,zdsσ,Z0,Up+a0Z0M2M1Z0κυ+Ψ⌣τ,zσ,Z0,Up2+∑h=1AahZ01Γh/kM3M2M1Z0κυ+Ψ⌣τ,zσ,Z0,Up2+wpτ,zσ,Z0,Up≤υ.
This last inequality can be achieved since the quantitiesZ0κ−h2 and Z0κ tend to zero as Z0>0 becomes close to the origin, accordingly to the assumption (20).
Collecting all the above estimates (74), (75), (76), and (77), under the latter constraint (78), sires the awaited property (72).
We now turn to the second feature 2. LetΞjτ,z∈Gσ,Z0,Upk with Ξjτ,zσ,Z0,Up≤υ, for j=1,2.
Under the condition (30), Propositions 6 and 11 yield a constant M1>0 for which
(121)dh¯zτkh1∂z−κ−h2Ξ1−Ξ2q−h3τ,zσ,Z0,Up≤dh¯Z0M1Z0κ−h2Ξ1−Ξ2τ,zσ,Z0,Up,
and Propositions10 and 11 breed a constant M3>0 such that
(122)dh¯zτk∫0τkτk−sh0/k−1sh1−1∂z−κ−h2Ξ1−Ξ2q−h3s1/k,zdsσ,Z0,Up≤dh¯Z0M3Z0κ−h2Ξ1τ,z−Ξ2τ,zσ,Z0,Up.
In order to deal with the nonlinear terms, we use the next identityab−cd=a−cb+cb−d which helps to factorize the next difference
(123)∂z−κΞ1τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1s1/k,z+Ψ⌣s1/k,z−∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ2s1/k,z+Ψ⌣s1/k,z=∂z−κΞ1−Ξ2τk−s1/k,z∂z−κΞ1s1/k,z+Ψ⌣s1/k,z+∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1−Ξ2s1/k,z,which leads, by means of Proposition 6, 8, 11, and 12, to constants M1,M2>0 for which
(124)a0zτk∫0τk∂z−κΞ1τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ1s1/k,z+Ψ⌣s1/k,z1τk−ssds−a0zτk∫0τk∂z−κΞ2τk−s1/k,z+Ψ⌣τk−s1/k,z×∂z−κΞ2s1/k,z+Ψ⌣s1/k,z1τk−ssdsσ,Z0,Up≤a0Z0M2∂z−κΞ1−Ξ2τ,zσ,Z0,Up×∂z−κΞ1τ,z+Ψ⌣τ,zσ,Z0,Up+∂z−κΞ2τ,z+Ψ⌣τ,zσ,Z0,Up×∂z−κΞ1−Ξ2τ,zσ,Z0,Up≤a0Z0M2M1Z0κΞ1τ,zσ,Z0,Up+M1Z0κΞ2τ,zσ,Z0,Up+2Ψ⌣τ,zσ,Z0,Up×M1Z0κΞ1τ,z−Ξ2τ,zσ,Z0,Up.
Furthermore, according to Proposition10, we get a constant M3>0 with
(125)A≔ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞ1s−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞ1s11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dss−ahzτk∫0τkτk−sh/k−1×s∫0s∂z−κΞ2s−s11/k,z+Ψ⌣s−s11/k,z×∂z−κΞ2s11/k,z+Ψ⌣s11/k,z1s−s1s1ds1dssσ,Z0,Up≤ahZ0M3τk∫0τk∂z−κΞ1τk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞ1s11/k,z+Ψ⌣s11/k,z−∂z−κΞ2τk−s11/k,z+Ψ⌣τk−s11/k,z×∂z−κΞ2s11/k,z+Ψ⌣s11/k,z×1τk−s1s1ds1σ,Z0,Up.
Using the factorization (81), Propositions6, 8, 11, and 12 grant the existence of constants M1,M2>0 such that
(126)A≤ahZ0M3M2∂z−κΞ1−Ξ2τ,zσ,Z0,Up×∂z−κΞ1τ,z+Ψ⌣τ,zσ,Z0,Up+∂z−κΞ2τ,z+Ψ⌣τ,zσ,Z0,Up×∂z−κΞ1−Ξ2τ,zσ,Z0,Up≤ahZ0M3M2M1Z0κΞ1τ,zσ,Z0,Up+M1Z0κΞ2τ,zσ,Z0,Up+2Ψ⌣τ,zσ,Z0,Up×M1Z0κΞ1τ,z−Ξ2τ,zσ,Z0,Up.
From now on, we select the radiusυ>0 and χ>0 small enough with 0<Z0<χ such that 0<Z0<χ in order that the next condition holds,
(127)∑h¯=h0,h1,h2,h3∈C;h0=0kq−h3kh1dh¯Z0M1Z0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1kq−h3kh1Γh0/kdh¯Z0M3Z0κ−h2+a0Z0M22M1Z0κυ+2Ψ⌣τ,zσ,Z0,UpM1Z0κ+∑h=1AahZ01Γh/kM3M22M1Z0κυ+2Ψ⌣τ,zσ,Z0,UpM1Z0κ≤12.
Gathering the list of bounds (79), (80), (82), (83), (84), subjected to (85), implies the shrinking condition (114) we are looking for.
Finally, in order to certify both properties (72) and (73), we impose on the constantsχ>0 and υ>0 the conjoint constraints (78) and (85). Proposition 14 follows.In the next proposition, we solve the nonlinear Cauchy problem (32), (33) in the Banach spaces described in Subsection3.1.Proposition 15.
We take for granted that assumption (20) holds. We fix the constantsχ>0 and υ>0 as in proposition 14. Then, the convolution Cauchy problem (32), (33) possesses a solution Θpu,z which belongs to the space Gσ,Z0,Upk, for any given σ>0 provided that 0<Z0<χ. Furthermore, one can single out a constant M1>0 (relying on k,q,κ,σ,b) such that
(128)Θpτ,zσ,Z0,Up≤M1Z0κυ+Ψ⌣τ,zσ,Z0,Up.Proof.
According to Proposition14, we can apply the classical fixed point theorem for shrinking maps in complete metric spaces to the map B:Bυ⟶Bυ. By construction, Bυ,d is a complete metric space for the distance dx,y=x−yσ,Z0,Up since the vector space Gσ,Z0,Upk equipped with the norm .σ,Z0,Up is a Banach space. Furthermore, under the constraints imposed in Proposition 14, the map B:Bυ⟶Bυ is shrinking of Lipschitz type 1/2. As a result, B:Bυ⟶Bυ has a unique fixed point, denoted Ξpτ,z∈Bυ, meaning that
(129)BΞp=Ξp.
This means in particular that we obtain a solution (unique in the ballBυ) Ξpτ,z for the equation (111). Moreover, owing to Proposition 6, we check that
(130)∂z−κΞpτ,zσ,Z0,Up≤M1Z0κΞpτ,zσ,Z0,Up≤M1Z0κυ.
The decomposition (69) then confirms that the mapΘpτ,z=∂z−κΞpτ,z+Ψ⌣τ,z belongs to Gσ,Z0,Upk, solves the convolution Cauchy problem (32), (33), and suffers the bounds (86).


## 3.3. Construction of Genuine Solutions to the Main Cauchy Problem (22), (23)

In this subsection, we state the first main result of this work.Theorem 16.
Assume that condition (30) is granted. Assume that the radius rT of each bounded sector Tp described in Theorem 3 fulfills
(131)rT<Δpσζb1/k,for 0≤p≤ς−1, where Δp is introduced in (14) and where ζb=∑n≥01/n+1b.
Then, the Cauchy problem (22), (23) possesses a holomorphic solutionvpt,z on the product Tp×DZ0/2, for some radius Z0>0 small enough. Furthermore, vpt,z can be expressed by means of a Laplace transform of order k,
(132)vpt,z=k∫LγpΘpu,zexp−utkduu,for t,z∈Tp×DZ0/2 along a halfline Lγp=0,+∞e−1γp which appears in the representation (15). The Borel map Θpu,z represents a holomorphic function on the domain Up×DZ0/2 whose Taylor expansion
(133)Θpu,z=∑β≥0Θp,βuzββ!,is submitted to the following estimates
(134)Θp,βu≤K1β!1Z0βu1+u2kexpσrbβuk,for all β≥0, all u∈Up, for well selected constants K1>0, any given σ>0, where rbβ is the sequence defined by (34). In particular, the Borel map Θpu,z suffers the next bounds estimates
(135)Θpu,z≤2K1uexpσζbuk,for u,z∈Up×DZ0/2.Proof.
The proof is a direct consequence of Proposition15 and of the construction made in Subsection 2.3.In the next corollary, we show thatvpt,z turns out to solve a nonlinear Cauchy problem with analytic coefficients in space z near the origin and polynomial in time t, which involves both differential operators and dilatations/contractions q−difference operators acting on time.Corollary 17.
The holomorphic mapvpt,z solves on the product Tp×DZ0/2 a particular nonlinear Cauchy problem which is polynomial in time t of the form
(136)Ptk+1∂t∂zS+κvpt,z=∑m¯=m1,m2,m3∈Mfm¯t,z∂tm1∂zm2vpqm3t,z+Gt,z,∂tr1∂zr2vpqr3t,zr¯=r1,r2,r3∈R,for given Cauchy data of the form
(137)∂zjvpt,0=φ⌣jt,0≤j≤κ−1,together with
(138)∂zjvpt,0=ϕ⌣jt,κ≤j≤κ+S−1,for well-chosen polynomials ϕ⌣jt with complex coefficients for κ≤j≤κ+S−1 (which depend on q).
The setM is a finite subset of ℕ2×ℤ which satisfies in particular the constraint m2<S+κ for all m¯=m1,m2,m3∈M and R is a finite subset of ℕ3 with the property that r2≤S whenever r1,r2,r3∈R.
For allm¯∈M, the coefficients fm¯t,z are polynomial in t and holomorphic w.r.t z the disc DR. The map Gt,z,ur¯r¯∈R is polynomial in the variable t, a polynomial of degree at most 2 in the variables ur¯ for r¯∈R and relies holomorphically on the variable z in the disc DR.Proof.
We introduce the next twoq−difference differential operators
(139)O1t,z,∂t,∂z,σq≔Ptk+1∂t∂zS−∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2,and
(140)O2t,z,∂t,∂z,σqv≔∂zκv−∑h¯=h0,h1,h2,h3∈Cdh¯zth0σq;t−h3tk+1∂th1∂zh2v−at,zv2,where σq;t stands for the dilation operator acting on functions through σq;tft,z=fqt,z. By construction, the holomorphic map vp satisfies
(141)O2t,z,∂t,∂z,σqvpt,z=upt,z,and the map up fulfills
(142)O1t,z,∂t,∂z,σqupt,z=0,for t,z∈Tp×DZ0/2. By coupling (95) and (96), we obtain that
(143)O1t,z,∂t,∂z,σq∘O2t,z,∂t,∂z,σqvpt,z=0,which can be expanded in the more precise shape
(144)Ptk+1∂t∂zS+κvpt,z−∑h¯=h0,h1,h2,h3∈CPtk+1∂t∂zS∘dh¯zth0σq;t−h3tk+1∂th1∂zh2vpt,z−∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2∘∂zκvpt,z+∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2∘∑h¯=h0,h1,h2,h3∈Cdh¯zth0σq;t−h3tk+1∂th1∂zh2vpt,z−Ptk+1∂t∂zSat,zvp2t,z+∑l¯=l0,l1,l2,l3∈Acl¯ztl0σq;tl3tk+1∂tl1∂zl2at,zvp2t,z=0,
and gives rise to the equation (136). Concerning the Cauchy data, the map vp is compelled to the conditions (33) which are rewritten in (93). Furthermore, the fact that upt,z is subjected to the constraints (12) at z=0 can be recast in the form
(145)∂zj∘O2t,z,∂t,∂z,σqvpt,0=φjt,for 0≤j≤S−1 which can be rephrased through the Cauchy conditions (138) for suitably selected polynomials ϕ⌣lt with complex coefficients for κ≤l≤κ+S−1, since j+κ>j+h2 for all h¯=h0,h1,h2,h3∈C provided that 0≤j≤S−1.Example 3.
A model of such an equation (136) can be obtained in coupling the examples 1 and 2 given by (18), (24) and is written in the form
(146)t5∂t−1∂z3vpt,z=5q−5t9∂t∂zvpq−1t,z+q−6t10∂t2∂zvpq−1t,z−q−5t5∂t∂zvpq−1t,z+zt3∂z2vpqt,z−zt8∂tvt,z+2t5∂t∂zvpt,z×vpt,z+∂zvpt,z×∂tvpt,z−2∂zvpt,z×vpt,z−zt3vp2qt,z.


## 4. Asymptotic Expansions in Time Variable

In order to simplify the notations throughout this section, we rewrite our main nonlinear Cauchy problem (22), (23) in the following form(147)∂zκvpt,z=∑k¯=k1,k2,k3∈Bek¯t,z∂tk1∂zk2vpq−k3t,z+at,zvp2t,z+upt,z,for prescribed Cauchy data
(148)∂zjvpt,0=φ⌣jt,0≤j≤κ−1,where B is a finite subset of ℕ3 and where the coefficients ek¯t,z are polynomial in t and holomorphic relatively to z on the disc DR given in Subsection 2.2.According to the condition (31) imposed on the set C in (22), the next feature holds for the set B,
(149)κ≥k2+1k+1k1,κ>k2,k3≥k1,for all k¯=k1,k2,k3∈B. Indeed, it is straight to check that for any given integer h1≥1, the decomposition
(150)tk+1∂th1=∑j=0h1aj,h1,kt∂tj,holds for suitable polynomials aj,h1,kt with real coefficients that rely on j,h1,k.


### 4.1. Reduction to an Auxiliary Cauchy Problem

We first write down the convergent Taylor expansions of the coefficients of (98) atz=0,
(151)ek¯t,z=∑n≥0ek¯,ntznn!,t,z=∑n≥0antznn!,for all t∈ℂ, all z∈DR. Furthermore, we expand the forcing term upt,z at z=0(152)upt,z=∑n≥0up,ntznn!,that converges provided thatz∈D1/2C4, for all t∈Tp according to Theorem 3. Finally, we recast the analytic solution vpt,z to (98), (99) obtained in Theorem 16 as Taylor series at z=0,
(153)vpt,z=∑n≥0vp,ntznn!,which is convergent provided that z∈DZ0/2 and t∈Tp. The constant Z0>0 is in particular taken small enough in a way that Z0<1/C4 and Z0<2R. By plugging the above expressions in the main equation (147), we check that vp solves (98), (99) if and only if the sequence of functions vp,nt satisfies the next recursion
(154)vp,n+κtn!=∑k¯=k1,k2,k3∈B∑n1+n2=nek¯,n1tn1!∂tk1vp,n2+k2q−k3tn2!+∑n1+n2+n3=nan1tn1!vp,n2tn2!vp,n3tn3!+up,ntn!,for all n≥0, with prescribed conditions
(155)vp,jt=φ⌣jt,0≤j≤κ−1.We plan to reduce this problem, by means of a majorant series approach to an auxiliary problem disclosed in (116), (117) that will be solve in the forthcoming Subsection4.4.At the onset, we apply the general differential operator∂tl for all integers l≥0 to the latter recursion (104) and get
(156)∂tlvp,n+κtn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!∂tl1ek¯,n1tl1!n1!∂tl2+k1vp,n2+k2q−k3tl2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!∂tl1an1tl1!n1!∂tl2vp,n2tl2!n2!∂tl3vp,n3tl3!n3!+∂tlup,ntn!,for all n≥0, all l≥0.We setW⊂Tp as a proper subsector centered at 0. We introduce the next set of sequences
(157)Vp,l,n=supt∈W∂tlvp,nt,where by definition,
(158)Vp,l,j=supt∈W∂tlφ⌣jt,for 0≤j≤κ−1, along with
(159)Ek¯,l,n=supt∈W∂tlek¯,nt,Al,n=supt∈W∂tlant,Up,l,n=supt∈W∂tlup,nt,for all l,n≥0. Since the map σq;t−l3:t↦q−l3t leaves W stable (i.e., σq;t−l3W⊂W) for any integer l3≥0, we deduce from the recursion (106) a sequence of inequalities
(160)Vp,l,n+κn!≤∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!Ek¯,l1,n1l1!n1!Vp,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!Al1,n1l1!n1!Vp,l2,n2l2!n2!Vp,l3,n3l3!n3!+Up,l,nn!,for all l,n≥0.We introduce a sequence denotedvp,l,n which fulfills the next recursion relation
(161)vp,l,n+κn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!Ek¯,l1,n1l1!n1!vp,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!Al1,n1l1!n1!vp,l2,n2l2!n2!vp,l3,n3l3!n3!+Up,l,nn!,with given initial data
(162)vp,l,j=Vp,l,j,0≤j≤κ−1.By construction, in comparing (110) and (111), (112), we observe the crucial fact that(163)Vp,l,n≤vp,l,n,for all l,n≥0. Let us define the next formal generating series(164)vpT,X=∑l,n≥0vp,l,nTll!Xnn!,together with
(165)Ek¯T,X=∑l,n≥0Ek¯,l,nTll!Xnn!,AT,X=∑l,n≥0Al,nTll!Xnn!,UpT,X=∑l,n≥0Up,l,nTll!Xnn!.A direct computation following from the recursion (111), (112) shows that the formal seriesvpT,X solves the next nonlinear Cauchy problem
(166)∂XκvpT,X=∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2vpq−k3T,X+AT,Xvp2T,X+UpT,X,for prescribed Cauchy data
(167)∂XjvpT,0=jj,pT,0≤j≤κ−1,where
(168)jj,pT=∑l≥0Vp,l,jTll!,for 0≤j≤κ−1, whose coefficients are defined by (108). Since φ⌣jt are polynomials in t, we deduce that jj,pT are also polynomials in the variable T, for 0≤j≤κ−1.


### 4.2. Asymptotic Expansions and Bounds for then−th Derivative of the Holomorphic Solutions to the Linear Cauchy Problem Discussed in Section 2.1

In this short subsection, we draw attention to parts of the results obtained in our previous work [1] that will be applied in the next Subsections 4.3 and 4.5. Namely, as a consequence of Theorem 16 from [1], we get the following statement on the asymptotic expansion in time t of the maps upt,z (described in Theorem 3).Theorem 18.
(1)
There exists a formal power seriesu^t,z=∑n≥0unztn with bounded holomorphic coefficients unz on some fixed disc D1/2C6 for a well-chosen constant C6>0, which represent the common asymptotic expansion of all the functions t↦upt,z on Tp, for 0≤p≤ς−1, uniformly relatively to z on D1/2C6. It means that, for each 0≤p≤ς−1, for each proper subsector W⊂Tp, for each integer N≥0, one can single out a constant cN,W>0 with
(169)supz∈D1/2C6upt,z−∑n=0Nunztn≤cN,WtN+1,for all t∈W.(2)
The mapst↦upt,z are infinitely often differentiable at the origin and
(170)∂tlup0,z=l!ulz,for all l≥0, all z∈D1/2C6, given that 0≤p≤ς−1.
The second point 2 of the above result is not mentioned in [1] but is a direct consequence of the first item 1. By application of a classical result in asymptotics mentioned in Proposition 14 p. 66 of [14].
Furthermore, in Corollary17 of Theorem 16 from [1], we derive important asymptotic bounds for the l−th derivative of the partial maps t↦upt,z on bounded sectors. Indeed, the next result holds.Theorem 19.
For each0≤p≤ς−1, for each proper subsector W′⊂Tp, one can select two constants C′,M′>0 for which
(171)supz∈D1/2C6∂tlupt,z≤C′M′ll!Γlkql2/2,for all integers l≥1, all t∈W′, where C6>0 is the well-chosen constant fixed in Theorem 3.


### 4.3. Banach Spaces of Formal Power Series

In this subsection, we unveil the definition and useful features of the Banach spaces in which we plan to seek for solutions to the aforementioned Cauchy problem (116), (117).Definition 20.
LetT0,X0>0 be real numbers. We set s>0 and q>1 as real numbers. We define the space Gs,qT0,X0 as the vector space of formal power series VT,X∈ℂT,X,
(172)VT,X=∑l,n≥0vl,nTll!Xnn!,such that the norm
(173)VT,XT0,X0,s,q≔∑l,n≥0vl,nql2/2T0lX0nΓ1+s+1l+n,is finite. One checks that the vector space Gs,qT0,X0 equipped with the norm .T0,X0,s,q represents a complex Banach space.Remark 21.
In the caseq=1, similar norms have been introduced by Miyake in the work [15] in order to construct formal power series solutions of Gevrey type to linear PDEs with analytic coefficients.
The next proposition is central in order to deal with the nonlinearity of the problem (116), (117).Proposition 22.
LetV1T,X,V2T,X∈Gs,qT0,X0. Then, the product V1T,XV2T,X belongs to Gs,qT0,X0 and one can find a constant L1>0 (depending on s) such that
(174)V1T,XV2T,XT0,X0,s,q≤L1V1T,XT0,X0,s,qV2T,XT0,X0,s,q.
In other words,Gs,qT0,X0,.T0,X0,s,q turns out to be a Banach algebra.Proof.
Let(175)VjT,X=∑l,n≥0vl,njTll!Xnn!,for j=1,2 two elements of Gs,qT0,X0. Their product writes
(176)V1T,XV2T,X=∑l,n≥0∑n1+n2=nl1+l2=ll!n!vl1,n11l1!n1!vl2,n22l2!n2!Tll!Xnn!.
By definition, its norm fufills the bounds(177)V1T,XV2T,X≤∑l,n≥0∑n1+n2=nl1+l2=ll!n!vl1,n11l1!n1!vl2,n22l2!n2!1ql2/21Γ1+s+1l+nT0lX0nT0,X0,s,q.
Sinceq>1, we first observe that
(178)ql12/2ql22/2≤ql2/2,for all integers l1,l2≥0 with l1+l2=l.
The next lemma is essential.Lemma 23.
There exists a constantL1>0 (depending on s) such that
(179)l!n!l1!n1!l2!n2!1Γ1+s+1l+n≤L11Γ1+s+1l1+n1Γ1+s+1l2+n2,for all integers l1,l2,n1,n2≥0 with l1+l2=l and n1+n2=n.Proof.
We depart from the next inequality(180)l!n!l1!n1!l2!n2!≤l+n!l1+n1!l2+n2!=Γ1+l+nΓ1+l1+n1Γ1+l2+n2,for all l1,l2,n1,n2≥0 with l1+l2=l and n1+n2=n. These bounds straightly follow from the binomial expansion for each term of the identity 1+xl1+xn=1+xn+l, for x∈ℝ.
In the next step, we need to prove the next.Lemma 24.
The next inequality(181)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ2+s+1l1+l2+n1+n2≤Γ1+l1+n1Γ1+l2+n2Γ2+l1+l2+n1+n2,holds for all l1,l2,n1,n2≥0 Proof We recall the next identity defining the so-called Beta function (see [14], Appendix B)
(182)ΓαΓβΓα+β=∫011−tα−1tβ−1dt,provided that α,β>0 are real numbers. As a result, observing that
(183)ft=1−tsl1tsl2≤1,provided that t∈0,1, we get that
(184)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ2+s+1l1+l2+n1+n2=∫011−ts+1l1+n1ts+1l2+n2dt≤∫011−tl1+n1tl2+n2dt=Γ1+l1+n1Γ1+l2+n2Γ2+l1+l2+n1+n2,from which Lemma 24 follows.
From the functional propertyΓz+1=zΓz for any z>0, we can factorize
(185)Γ2+s+1l1+l2+n1+n2=1+s+1l1+l2+n1+n2Γ1+s+1l1+l2+n1+n2,Γ2+l1+l2+n1+n2=1+l1+l2+n1+n2Γ1+l1+l2+n1+n2.
Combining the bounds (125) and the expansions (127), we get that(186)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ1+s+1l1+l2+n1+n2≤Cl,nΓ1+l1+n1Γ1+l2+n2Γ1+l1+l2+n1+n2,where
(187)Cl,n=1+s+1l+n1+l+n,where l=l1+l2, n=n1+n2 with l1,l2,n1,n2≥0. Besides, a constant L1>0 (relying on s) can be found such that
(188)Cl,n≤L1,for all l,n≥0. Finally, the collection of the bounds (124), (128), and (129) yields the forecast estimates (123).Owing to the upper bounds (122) and (123) applied to (121), we reach the awaited inequality(189)V1T,XV2T,XT0,X0,s,q≤L1∑l,n≥0∑n1+n2=nl1+l2=lvl1,n11ql12/2Γ1+s+1l1+n1vl2,n22ql22/2Γ1+s+1l2+n2T0lX0n=L1V1T,XT0,X0,s,qV2T,XT0,X0,s,q.□The next proposition helps us in addressing the linear part of the Cauchy problem (116), (117).Proposition 25.
Letk1,k2,k3≥0 be integers under the requirement
(190)k2≥s+1k1,k3≥k1.
Then, the linear operatorσq−k3;T∘∂Tk1∂X−k2 is bounded from the space Gs,qT0,X0 into itself. In other words, there exists a constant L2>0 (depending on q,k1,k2,s) with
(191)∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q≤L2X0k2T0−k1VT,XT0,X0,s,q,for all VT,X∈Gs,qT0,X0.Proof.
LetVT,X=∑l,n≥0vl,nTl/l!Xn/n! be in Gs,qT0,X0. The action of the q−difference differential operator is expressed through
(192)∂Tk1∂X−k2Vq−k3T,X=∑l≥0,n≥k2vl+k1,n−k2q−k3lTll!Xnn!,and enables us to rewrite the norm
(193)T0k1X0−k2∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q=∑l≥0,n≥k2∣vl+k1,n−k2∣ql2/2q−k3lΓ1+s+1l+nT0l+k1X0n−k2=∑l≥0,n≥k2Cl,n∣vl+k1,n−k2∣ql+k12/21Γ1+s+1l+k1+n−k2T0l+k1X0n−k2,where
(194)Cl,n=ql+k12/2ql2/2q−k3lΓ1+s+1l+k1+n−k2Γ1+s+1l+n,for all l≥0, n≥k2. In the continuing part of the proof, we show that the sequence Cl,n is actually bounded by some constant.
Indeed, recall from [14], Appendix B, that for a given a∈ℝ, one can find a constant Ka>0 (relying on a) such that
(195)Γx+aΓx≤Kaxa,provided that x≥1. Consequently, we get two constants Ks,k1,k2>0 and L2.1>0 (depending on s,k1,k2) with
(196)Γ1+s+1l+n+s+1k1−k2Γ1+s+1l+n≤Ks,k1,k211+s+1l+nk2−s+1k1≤L2.1,for all l≥0, n≥k2, provided that (131) holds. On the other hand,
(197)ql+k12/2ql2/2q−k3l=qlk1−k3qk12/2≤qk12/2,for all l≥0 under the assumption (131). As a result, we obtain
(198)Cl,n≤qk12/2L2.1,for all l≥0, n≥k2. Finally, based on (133) and (136), we deduce
(199)T0k1X0−k2∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q≤qk12/2L2.1VT,XT0,X0,s,q,which is tantamount to (132).In the next proposition, we show that the coefficients, the forcing term and Cauchy data of the problem (116), (117) belongs to the Banach spaceGs,qT0,X0 for well-chosen parameters.Proposition 26.
Letk,q chosen as in Section 2.
(1)
The seriesUpT,X appertains to G1/k,qT0,X0 for T0,X0>0 small enough(2)
The seriesEk¯T,X and AT,X belong to G1/k,qT0,X0 for T0,X0>0 small enough(3)
We set up(200)WpT,X≔∑j=0κ−1jj,pTXjj!.
For any nonnegative integersk1,k2,k3≥0, the polynomials ∂Tk1∂Xk2Wpq−k3T,X belong to G1/k,qT0,X0 for any given T0,X0>0.Proof.
We focus on the first point (1). According to Theorem3 stated in Subsection 4.2, with the help of the Cauchy formula, we deduce that the sequence Up,l,n defined in (109) is subjected to bounds of the form
(201)Up,l,n≤U0,pU1lU2nl!Γlkql2/2n!,for some selected constants U0,p,U1,U2>0, whenever l≥1, n≥0. Besides, from the formula (182), we observe in particular that
(202)ΓαΓβ≤Γα+β,provided that α,β≥1. This last inequality combined with the functional relation Γz+1=zΓz for any z>0 begets
(203)l!Γlkn!≤Γ2+1k+1l+n=1+1k+1l+nΓ1+1k+1l+n,for all l≥k, all n≥0. Thereby, we deduce constants U⌣0,p,U⌣1,U⌣2>0 with
(204)Up,l,n≤U⌣0,pU⌣1lU⌣2nΓ1+1k+1l+nql2/2,for all l,n≥0. Bestowing the last expansion of (115) implies the next norm bounds
(205)UpT,XT0,X0,1/k,q=∑l,n≥0∣Up,l,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤U⌣0,p∑l,n≥0T0U⌣1lX0U⌣2n≤4U⌣0,p,provided that T0,X0>0 are constrained to T0U⌣1≤1/2 and X0U⌣2≤1/2.
We address the second point (2). According to the assumptions of Section2, the functions ek¯t,z and at,z are polynomial in t and bounded holomorphic on some disc DR relatively to z, for all k¯∈B. From the Cauchy formula, we deduce that the first two sequences in (109) satisfy the bounds
(206)Ek¯,l,n≤E0,k¯E1lE2nl!n!,Al,n≤A0A1lA2nl!n!,for some well-chosen constants E0,k¯,E1,E2>0 and A0,A1,A2>0, for all l,n≥0. Owing to (139) and since the map x↦Γx is increasing on 2,+∞ (see [14], Appendix B), we check that
(207)l!n!≤Γ2+l+n≤Γ2+1k+1l+nql2/2=1+1k+1l+nΓ1+1k+1l+nql2/2,for all l,n≥0. Whence, constants E⌣0,k¯,E⌣1,E⌣2>0 and A⌣0,A⌣1,A⌣2>0 can be found with
(208)Ek¯,l,n≤E⌣0,k¯E⌣1lE⌣2nΓ1+1k+1l+nql2/2,Al,n≤A⌣0A⌣1lA⌣2nΓ1+1k+1l+nql2/2,for all l,n≥0. Thanks to the first two expansions of (115), we deduce the control of the norms
(209)Ek¯T,XT0,X0,1/k,q=∑l,n≥0∣Ek¯,l,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤E⌣0,k¯∑l,n≥0T0E⌣1lX0E⌣2n≤4E⌣0,k¯,together with
(210)AT,XT0,X0,1/k,q=∑l,n≥0∣Al,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤A⌣0∑l,n≥0T0A⌣1lX0A⌣2n≤4A⌣0,whenever T0,X0>0 are submitted to T0E⌣1≤1/2, T0A⌣1≤1/2 and X0E⌣2≤1/2, X0A⌣2≤1/2.
Finally, the last point (3) is straightforward since for any given integersk1,k2,k3≥0, the quantity
(211)∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q,is a polynomial in the variables T0,X0 and is therefore finite for any fixed T0,X0>0.


### 4.4. Solving the Auxiliary Cauchy Problem

In this subsection, we seek for a formal power series solution to the nonlinear Cauchy problem (116), (117) by means of a decomposition(212)vpT,X=∂X−κYpT,X+WpT,X,where the polynomial WpT,X is displayed in (137), for some formal series YpT,X to be determined.We observe thatvpT,X fulfills the problem (116), (117) if the expression YpT,X solves the next fixed point equation
(213)YpT,X=∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2−κYpq−k3T,X+AT,X∂X−κYpT,X+WpT,X2+UpT,X+∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2Wpq−k3T,X.Our ensuing undertaking is the construction of a solution of this latter equation (149) within the Banach space of formal series discussed in the previous Section4.3. In order to fulfill this objective, we set up the next nonlinear mapping A:ℂT,X⟶ℂT,X defined as
(214)AYT,X≔∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂X−κ−k2Yq−k3T,X+AT,X∂X−κYT,X+WpT,X2+UpT,X+∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2Wpq−k3T,X.In the next proposition, we discuss sufficient conditions under whichA represents a shrinking map acting on some small ball centered at the origin in the space G1/k,qT0,X0.Proposition 27.
Taking for granted the condition (149), some small real number ξ>0 can be singled out such that if 0<X0<ξ, one can choose a radius ϖ>0 such that A suffers the next features.
LetBϖ be the ball centered at 0 with radius ϖ in the space G1/k,qT0,X0.
(1)
The next inclusion(215)ABϖ⊂Bϖ,holds.(2)
For anyY1,Y2∈Bϖ, we have
(216)AY1−AY2T0,X0,1/k,q≤12Y1−Y2T0,X0,1/k,q.Proof.
We behold the first property. LetYT,X belong to the ball Bϖ. Under the constraint (100), Propositions 22, 25, and 26 allow the next inequality
(217)Ek¯T,X∂Tk1∂X−κ−k2Yq−k3T,XT0,X0,1/k,q≤L1Ek¯T,XT0,X0,1/k,q×∂Tk1∂X−κ−k2Yq−k3T,XT0,X0,1/k,q≤4L1L2E⌣0,k¯X0κ−k2T0−k1YT,XT0,X0,1/k,q,to hold true along with
(218)AT,X∂X−κYT,X+WpT,X2T0,X0,1/k,q≤L12AT,XT0,X0,1/k,q×∂X−κYT,X+WpT,XT0,X0,1/k,q2≤4L12A⌣0L2X0κYT,XT0,X0,1/k,q+WpT,XT0,X0,1/k,q2,for some constants L1,L2>0 and E⌣0,k¯,A⌣0>0. Eventually, due to Propositions 22 and 26, we arrive at
(219)Ek¯T,X∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤L1Ek¯T,XT0,X0,1/k,q∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q,for some constants L1>0 and E⌣0,k¯>0.From now on, we take the radiusϖ>0 in a way that
(220)4L12A⌣0WpT,XT0,X0,1/k,q2+4U⌣0,p+∑k¯=k1,k2,k3∈B4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q<ϖ,for some fixed X0>0. Then, we pick up a tiny constant ξ>0 with 0<X0<ξ aiming to the next constraint
(221)∑k¯=k1,k2,k3∈B4L1L2E⌣0,k¯X0κ−k2T0−k1ϖ+4L12A⌣0L2X0κϖ+WpT,XT0,X0,1/k,q2+4U⌣0,p+∑k¯=k1,k2,k3∈B4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤ϖ.This latter inequality can be reached since the quantitiesX0κ−k2 and X0κ are small as X0>0 tends to 0, owing to (100).Piling up the above estimates (153), (154), and (155) under the limitation (156) hints at the due inclusion (151).We address the second property. LetY1T,X,Y2T,X be taken within the ball Bϖ. The above inequality (153) prompts the next bounds for the linear piece
(222)Ek¯T,X∂Tk1∂X−κ−k2Y1−Y2q−k3T,XT0,X0,1/k,q≤4L1L2E⌣0,k¯X0κ−k2T0−k1Y1T,X−Y2T,XT0,X0,1/k,q.With an eye toward the nonlinear block, by means of the basic identitya2−b2=a−ba+b, we factorize the difference of squares as
(223)Δ1,2=∂X−κY1T,X+WpT,X2−∂X−κY2T,X+WpT,X2=∂X−κY1T,X−Y2T,X×∂X−κY1T,X+2WpT,X+∂X−κY2T,X.Consequently, on account of Propositions22, 25, and 26, the next estimates
(224)AT,XΔ1,2T0,X0,1/k,q≤L12AT,XT0,X0,1/k,q×∂X−κY1T,X−Y2T,XT0,X0,1/k,q×∂X−κY1T,X+2WpT,X+∂X−κY2T,XT0,X0,1/k,q≤L124A⌣0L2X0κL2X0κY1T,XT0,X0,1/k,q+2WpT,XT0,X0,1/k,q+L2X0κY2T,XT0,X0,1/k,qY1T,X−Y2T,XT0,X0,1/k,q,follow. Then, we pick out a small constant ξ>0 with 0<X0<ξ intending to the next condition
(225)∑k¯=k1,k2,k3∈B4L1L2E⌣0,k¯X0κ−k2T0−k1+L124A⌣0L2X0κ2L2X0κϖ+2WpT,XT0,X0,1/k,q≤12.Storing up the previous bounds (157) and (158) under the restriction (159) triggers the Lipschitz property (152).At last, we compel the constantsϖ>0 and ξ>0 to fulfill both constraints (156) and (159) in order to guarantee each of the two foretold properties (151) and (152).□.In the upcoming proposition, we solve the auxiliary nonlinear Cauchy problem (116), (117) amidst the Banach spaces introduced in Section4.3.Proposition 28.
Let us assume that the condition (149) holds. Let the constants ϖ,ξ>0 be determined by Proposition 27. Then, the nonlinear Cauchy problem (116), (117) owns a formal power series solution vpT,X that belongs to the space G1/k,qT0,X0 presuming that 0<X0<ξ. Along with it, one can find a constant L2>0 (resting on k,q,κ) such that
(226)vpT,XT0,X0,1/k,q≤L2X0κϖ+WpT,XT0,X0,1/k,q.Proof.
On the basis of Proposition27, the classical fixed point theorem for shrinking maps in complete metric spaces can be applied to the map A:Bϖ⟶Bϖ from the plain observation that Bϖ,d represents a complete metric space for the distance dx,y=x−yT0,X0,1/k,q since the vector space G1/k,qT0,X0 endowed with the norm .T0,X0,1/k,q is a Banach space. Indeed, under the conditions asked in Proposition 27, the map A:Bϖ⟶Bϖ appears to be of Lipschitz type 1/2 and hence shrinking. Whence, A:Bϖ⟶Bϖ has a unique fixed point, denoted YpT,X∈Bϖ, signifying that
(227)AYp=Yp.
In particular, a unique solutionYp for the equation (149) is found in the ball Bϖ. Along with it, due to Proposition 25, norms estimates can be achieved,
(228)∂X−κYpT0,X0,1/k,q≤L2X0κYpT0,X0,1/k,q≤L2X0κϖ.
Consequently, the splitting (148) attests that the mapvpT,X=∂X−κYpT,X+WpT,X appertains to G1/k,qT0,X0 and solves the problem (116), (117) under the restriction (160). □.


### 4.5. Asymptotic Expansions of Mixed Order

In order to describe the type of asymptotic behaviour which arises in our settings, we need the following definition issued from our recent work [16].Definition 29.
LetF,.F be a complex Banach space. We set k≥1 an integer and q>1 a real number. Let f:T⟶F be a holomorphic map, where T stands for a bounded sector in ℂ∗ centered at 0. Then, the map f enjoys the property of having the formal series
(229)f^t=∑n≥0antn∈Ft,as Gevrey asymptotic expansion of mixed order 1/k;q,1 on T if for each closed proper subsector W of T centered at 0, two constants C,M>0 can be distinguished with
(230)ft−∑n=0NantnF≤CMN+1ΓN+1kqN+12/2tN+1,for all integers N≥0 and any t∈W.
We are ready to enunciate the second main outcome of the work.Theorem 30.
We consider the setvp0≤p≤ς−1 of solutions to the main Cauchy problem (22), (23) established in Theorem 16. If one sets FZ1 as the Banach space of bounded holomorphic functions on the disc DZ1 equipped with the sup norm, then each partial map t↦vpt,z can be viewed as a holomorphic map from the sector Tp into FZ1, for 0≤p≤ς−1, as long as 0<Z1<Z0/2, where Z0 is set up in Theorem 16.
Hence, provided thatZ1>0 is taken small enough, for all 0≤p≤ς−1, the maps t↦vpt,z share a common formal power series
(231)v^t,z=∑l≥0hlztl∈FZ1t,as Gevrey asymptotic expansion of mixed order 1/k;q,1 on Tp. To rephrase it, for each 0≤p≤ς−1, for each proper subsector W⊂Tp, two constants C,M>0 can be singled out with
(232)supz∈DZ1∣vpt,z−∑l=0Nhlztl∣≤CMN+1ΓN+1kqN+12/2tN+1,for all t∈W, all integers N≥0.Proof.
According to the bounds (160) from Proposition28 and Definition 20, the unique formal series vpT,X∈ℂT,X solution of the Cauchy problem (116), (117) given by the expansion (114) is compelled to satisfy the next bounds
(233)vp,l,n≤L2X0κϖ+WpT,XT0,X0,1/k,q1T0l1X0nΓ1+1k+1l+nql2/2,for all l,n≥0. On the other hand, the next lemma will be required.Lemma 31.
There exist constantsKj>0, j=1,2,3 such that
(234)Γα+β≤K1K2αK3βΓαΓβ,for all real numbers α,β≥1.Proof.
We provide a complete proof of this classical result since it is not contained in our reference [14] on special functions. We depart from the Stirling formula (see [14] Appendix B), which ensures the existence of two constants A1,A2>0 for which
(235)A1xxx−1/2e−x≤Γx≤A2xxx−1/2e−x,for all x≥1. Therefrom,
(236)Γα+βΓαΓβ≤A2A12αβα+β1/21+βαα1+αββ.
Besides, one can pinpoint two constantsB1>0 and B2>1 such that
(237)αβα+β1/2≤β1/2≤B1B2β,for all β≥1 and from the classical estimates log1+x≤x for x>0, we deduce that
(238)1+βαα=expαlog1+βα≤eβ,1+αββ≤eα,for all real numbers α,β≥1. Lastly, piling up the above bounds (165), (166) and (167) yields the awaited estimates (164).In view of the inequality (113), we deduce from (163) together with (164) the next decisive bounds for the higher order time derivatives(239)supt∈Wvp,nt≤L2X0κϖ+WpT,XT0,X0,1/k,q1X0nn!≤V~01X0nn!,for all n≥0, together with
(240)supt∈W∂tlvp,nt≤L2X0κϖ+WpT,XT0,X0,1/k,q1T0l1X0nΓ1+1k+1l+nql2/2≤V^0V∧1lV∧2nΓ1k+1lΓ1+nql2/2=V^0V∧1lV∧2n11/k+1lΓ1+1k+1ln!ql2/2≤V^0V∧1lV∧2nΓ1k+1lΓ1+nql2/2=V^0V∧1lV∧2n11/k+1lΓ1+1k+1ln!ql2/2,for all l≥1, n≥0, for some constants V~0>0, V^0,V^1,V^2>0, and V⌣0,V⌣1,V⌣2>0 (relying k,q,κ,T0,X0,ϖ and Vp,l,j for 0≤j≤κ−1), where W stands for a fixed subsector of Tp.From the Taylor formula with integral remainder, for each0≤p≤ς−1, n≥0, we can expand the function vp,nt as follows
(241)vp,nt=∑l=0N∂tlvp,n0l!tl+RN+1,p,nt,where
(242)RN+1,p,nt=∫0t∂hN+1vp,nht−hNN!dh,for all t∈W, all integers N≥0. The remainder RN+1,p,nt can be estimated from above by means of the latter key bounds (169),
(243)∣RN+1,p,nt∣≤∫0∣t∣∣t∣−uNduV⌣0V⌣1N+1N+1V⌣2nn!Γ1kN+1qN+12/2≤V⌣0V⌣2nn!V⌣1N+1Γ1kN+1qN+12/2tN+1,for all t∈W.For each0≤p≤ς−1, we define the formal power series
(244)v^pt,z=∑l,n≥0∂tlvp,n0tll!znn!∈ℂt,z.It turns out thatv^pt,z can be rewritten in the form
(245)v^pt,z=∑l≥0hp,lztl,where the expressions hp,lz are holomorphic on a disc DZ1 provided that its radius Z1>0 is taken small enough. Indeed, the first term hp,0z is expressed through the series
(246)hp,0z=∑n≥0vp,n0znn!,which represents a holomorphic function on DZ1 for Z1<X0/2 since
(247)∣hp,0z∣≤V~0∑n≥0∣z∣X0n≤2V~0,owing to the bounds (168), whenever ∣z∣<X0/2. Besides, for all l≥1, the coefficients hp,lz are given by the expansions
(248)hp,lz=1l!∑n≥0∂tlvp,n0znn!,which define holomorphic maps on DZ1, when Z1<1/2V⌣2 since
(249)∣hp,lz∣≤V⌣0V⌣1lΓlkql2/2∑n≥0V⌣2nzn≤2V⌣0V⌣1lΓlkql2/2,according to the bounds (169), given that ∣z∣<1/2V⌣2.Consequently, in view of both expansions (170) and (173) along with the remainder estimates (171), we deduce the next error bounds(250)∣vpt,z−∑l=0Nhp,lztl∣=∣∑n≥0vp,ntznn!−∑n≥0∑l=0N∂tlvp,n0tll!znn!∣=∣∑n≥0RN+1,p,ntznn!∣≤V⌣0V⌣1N+1Γ1kN+1qN+12/2tN+1∑n≥0∣z∣V⌣2n≤2V⌣0V⌣1N+1Γ1kN+1qN+12/2tN+1,for all t∈W, all integers N≥0, given that ∣z∣<Z1, for Z1>0 chosen as above.In the last part of the proof disclosed within the next lemma, we show that the coefficientshp,lz do not depend on p for all l≥0.Lemma 32.
The coefficients(251)v^p,l,n=∂tlvp,n0,of the formal series v^pt,z given by (172) do not depend on p, for all l,n≥0. As a result, the coefficients hp,lz of the formal expansion (173) do not depend on p for all l≥0 and thereby each formal series v^pt,z turns out to be written as a single formal series
(252)v^t,z=∑l≥0hlztl,with holomorphic coefficients hlz on DZ1, for l≥0.Proof.
Paying regard to the relation (156), we get in particular the next recursion for the sequence v^p,l,n,
(253)v^p,l,n+κn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!∂tl1ek¯,n10l1!n1!v^p,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!∂tl1an10l1!n1!v^p,l2,n2l2!n2!v^p,l3,n3l3!n3!+∂tlup,n0n!,with prescribed set of data
(254)v^p,l,j=∂tlφ⌣j0,0≤j≤κ−1,l≥0.
Now, we observe that the above prescribed quantitiesv^p,l,j for 0≤j≤κ−1 and l≥0 along with the forcing terms ∂tlup,n0/n! for all l≥0, all n≥0, do not depend on p, according to our assumption (23) and Theorem 3, 2.. As a result, we deduce by induction from the above recursion (175) that the whose sequence v^p,l,n do not depend on p for all l,n≥0. □.
At last, the statement of Theorem30 issues from the above lemma 32 and the error bounds (174).


## 4.1. Reduction to an Auxiliary Cauchy Problem

We first write down the convergent Taylor expansions of the coefficients of (98) atz=0,
(151)ek¯t,z=∑n≥0ek¯,ntznn!,t,z=∑n≥0antznn!,for all t∈ℂ, all z∈DR. Furthermore, we expand the forcing term upt,z at z=0(152)upt,z=∑n≥0up,ntznn!,that converges provided thatz∈D1/2C4, for all t∈Tp according to Theorem 3. Finally, we recast the analytic solution vpt,z to (98), (99) obtained in Theorem 16 as Taylor series at z=0,
(153)vpt,z=∑n≥0vp,ntznn!,which is convergent provided that z∈DZ0/2 and t∈Tp. The constant Z0>0 is in particular taken small enough in a way that Z0<1/C4 and Z0<2R. By plugging the above expressions in the main equation (147), we check that vp solves (98), (99) if and only if the sequence of functions vp,nt satisfies the next recursion
(154)vp,n+κtn!=∑k¯=k1,k2,k3∈B∑n1+n2=nek¯,n1tn1!∂tk1vp,n2+k2q−k3tn2!+∑n1+n2+n3=nan1tn1!vp,n2tn2!vp,n3tn3!+up,ntn!,for all n≥0, with prescribed conditions
(155)vp,jt=φ⌣jt,0≤j≤κ−1.We plan to reduce this problem, by means of a majorant series approach to an auxiliary problem disclosed in (116), (117) that will be solve in the forthcoming Subsection4.4.At the onset, we apply the general differential operator∂tl for all integers l≥0 to the latter recursion (104) and get
(156)∂tlvp,n+κtn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!∂tl1ek¯,n1tl1!n1!∂tl2+k1vp,n2+k2q−k3tl2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!∂tl1an1tl1!n1!∂tl2vp,n2tl2!n2!∂tl3vp,n3tl3!n3!+∂tlup,ntn!,for all n≥0, all l≥0.We setW⊂Tp as a proper subsector centered at 0. We introduce the next set of sequences
(157)Vp,l,n=supt∈W∂tlvp,nt,where by definition,
(158)Vp,l,j=supt∈W∂tlφ⌣jt,for 0≤j≤κ−1, along with
(159)Ek¯,l,n=supt∈W∂tlek¯,nt,Al,n=supt∈W∂tlant,Up,l,n=supt∈W∂tlup,nt,for all l,n≥0. Since the map σq;t−l3:t↦q−l3t leaves W stable (i.e., σq;t−l3W⊂W) for any integer l3≥0, we deduce from the recursion (106) a sequence of inequalities
(160)Vp,l,n+κn!≤∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!Ek¯,l1,n1l1!n1!Vp,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!Al1,n1l1!n1!Vp,l2,n2l2!n2!Vp,l3,n3l3!n3!+Up,l,nn!,for all l,n≥0.We introduce a sequence denotedvp,l,n which fulfills the next recursion relation
(161)vp,l,n+κn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!Ek¯,l1,n1l1!n1!vp,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!Al1,n1l1!n1!vp,l2,n2l2!n2!vp,l3,n3l3!n3!+Up,l,nn!,with given initial data
(162)vp,l,j=Vp,l,j,0≤j≤κ−1.By construction, in comparing (110) and (111), (112), we observe the crucial fact that(163)Vp,l,n≤vp,l,n,for all l,n≥0. Let us define the next formal generating series(164)vpT,X=∑l,n≥0vp,l,nTll!Xnn!,together with
(165)Ek¯T,X=∑l,n≥0Ek¯,l,nTll!Xnn!,AT,X=∑l,n≥0Al,nTll!Xnn!,UpT,X=∑l,n≥0Up,l,nTll!Xnn!.A direct computation following from the recursion (111), (112) shows that the formal seriesvpT,X solves the next nonlinear Cauchy problem
(166)∂XκvpT,X=∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2vpq−k3T,X+AT,Xvp2T,X+UpT,X,for prescribed Cauchy data
(167)∂XjvpT,0=jj,pT,0≤j≤κ−1,where
(168)jj,pT=∑l≥0Vp,l,jTll!,for 0≤j≤κ−1, whose coefficients are defined by (108). Since φ⌣jt are polynomials in t, we deduce that jj,pT are also polynomials in the variable T, for 0≤j≤κ−1.


## 4.2. Asymptotic Expansions and Bounds for then−th Derivative of the Holomorphic Solutions to the Linear Cauchy Problem Discussed in Section 2.1

In this short subsection, we draw attention to parts of the results obtained in our previous work [1] that will be applied in the next Subsections 4.3 and 4.5. Namely, as a consequence of Theorem 16 from [1], we get the following statement on the asymptotic expansion in time t of the maps upt,z (described in Theorem 3).Theorem 18.
(1)
There exists a formal power seriesu^t,z=∑n≥0unztn with bounded holomorphic coefficients unz on some fixed disc D1/2C6 for a well-chosen constant C6>0, which represent the common asymptotic expansion of all the functions t↦upt,z on Tp, for 0≤p≤ς−1, uniformly relatively to z on D1/2C6. It means that, for each 0≤p≤ς−1, for each proper subsector W⊂Tp, for each integer N≥0, one can single out a constant cN,W>0 with
(169)supz∈D1/2C6upt,z−∑n=0Nunztn≤cN,WtN+1,for all t∈W.(2)
The mapst↦upt,z are infinitely often differentiable at the origin and
(170)∂tlup0,z=l!ulz,for all l≥0, all z∈D1/2C6, given that 0≤p≤ς−1.
The second point 2 of the above result is not mentioned in [1] but is a direct consequence of the first item 1. By application of a classical result in asymptotics mentioned in Proposition 14 p. 66 of [14].
Furthermore, in Corollary17 of Theorem 16 from [1], we derive important asymptotic bounds for the l−th derivative of the partial maps t↦upt,z on bounded sectors. Indeed, the next result holds.Theorem 19.
For each0≤p≤ς−1, for each proper subsector W′⊂Tp, one can select two constants C′,M′>0 for which
(171)supz∈D1/2C6∂tlupt,z≤C′M′ll!Γlkql2/2,for all integers l≥1, all t∈W′, where C6>0 is the well-chosen constant fixed in Theorem 3.


## 4.3. Banach Spaces of Formal Power Series

In this subsection, we unveil the definition and useful features of the Banach spaces in which we plan to seek for solutions to the aforementioned Cauchy problem (116), (117).Definition 20.
LetT0,X0>0 be real numbers. We set s>0 and q>1 as real numbers. We define the space Gs,qT0,X0 as the vector space of formal power series VT,X∈ℂT,X,
(172)VT,X=∑l,n≥0vl,nTll!Xnn!,such that the norm
(173)VT,XT0,X0,s,q≔∑l,n≥0vl,nql2/2T0lX0nΓ1+s+1l+n,is finite. One checks that the vector space Gs,qT0,X0 equipped with the norm .T0,X0,s,q represents a complex Banach space.Remark 21.
In the caseq=1, similar norms have been introduced by Miyake in the work [15] in order to construct formal power series solutions of Gevrey type to linear PDEs with analytic coefficients.
The next proposition is central in order to deal with the nonlinearity of the problem (116), (117).Proposition 22.
LetV1T,X,V2T,X∈Gs,qT0,X0. Then, the product V1T,XV2T,X belongs to Gs,qT0,X0 and one can find a constant L1>0 (depending on s) such that
(174)V1T,XV2T,XT0,X0,s,q≤L1V1T,XT0,X0,s,qV2T,XT0,X0,s,q.
In other words,Gs,qT0,X0,.T0,X0,s,q turns out to be a Banach algebra.Proof.
Let(175)VjT,X=∑l,n≥0vl,njTll!Xnn!,for j=1,2 two elements of Gs,qT0,X0. Their product writes
(176)V1T,XV2T,X=∑l,n≥0∑n1+n2=nl1+l2=ll!n!vl1,n11l1!n1!vl2,n22l2!n2!Tll!Xnn!.
By definition, its norm fufills the bounds(177)V1T,XV2T,X≤∑l,n≥0∑n1+n2=nl1+l2=ll!n!vl1,n11l1!n1!vl2,n22l2!n2!1ql2/21Γ1+s+1l+nT0lX0nT0,X0,s,q.
Sinceq>1, we first observe that
(178)ql12/2ql22/2≤ql2/2,for all integers l1,l2≥0 with l1+l2=l.
The next lemma is essential.Lemma 23.
There exists a constantL1>0 (depending on s) such that
(179)l!n!l1!n1!l2!n2!1Γ1+s+1l+n≤L11Γ1+s+1l1+n1Γ1+s+1l2+n2,for all integers l1,l2,n1,n2≥0 with l1+l2=l and n1+n2=n.Proof.
We depart from the next inequality(180)l!n!l1!n1!l2!n2!≤l+n!l1+n1!l2+n2!=Γ1+l+nΓ1+l1+n1Γ1+l2+n2,for all l1,l2,n1,n2≥0 with l1+l2=l and n1+n2=n. These bounds straightly follow from the binomial expansion for each term of the identity 1+xl1+xn=1+xn+l, for x∈ℝ.
In the next step, we need to prove the next.Lemma 24.
The next inequality(181)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ2+s+1l1+l2+n1+n2≤Γ1+l1+n1Γ1+l2+n2Γ2+l1+l2+n1+n2,holds for all l1,l2,n1,n2≥0 Proof We recall the next identity defining the so-called Beta function (see [14], Appendix B)
(182)ΓαΓβΓα+β=∫011−tα−1tβ−1dt,provided that α,β>0 are real numbers. As a result, observing that
(183)ft=1−tsl1tsl2≤1,provided that t∈0,1, we get that
(184)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ2+s+1l1+l2+n1+n2=∫011−ts+1l1+n1ts+1l2+n2dt≤∫011−tl1+n1tl2+n2dt=Γ1+l1+n1Γ1+l2+n2Γ2+l1+l2+n1+n2,from which Lemma 24 follows.
From the functional propertyΓz+1=zΓz for any z>0, we can factorize
(185)Γ2+s+1l1+l2+n1+n2=1+s+1l1+l2+n1+n2Γ1+s+1l1+l2+n1+n2,Γ2+l1+l2+n1+n2=1+l1+l2+n1+n2Γ1+l1+l2+n1+n2.
Combining the bounds (125) and the expansions (127), we get that(186)Γ1+s+1l1+n1Γ1+s+1l2+n2Γ1+s+1l1+l2+n1+n2≤Cl,nΓ1+l1+n1Γ1+l2+n2Γ1+l1+l2+n1+n2,where
(187)Cl,n=1+s+1l+n1+l+n,where l=l1+l2, n=n1+n2 with l1,l2,n1,n2≥0. Besides, a constant L1>0 (relying on s) can be found such that
(188)Cl,n≤L1,for all l,n≥0. Finally, the collection of the bounds (124), (128), and (129) yields the forecast estimates (123).Owing to the upper bounds (122) and (123) applied to (121), we reach the awaited inequality(189)V1T,XV2T,XT0,X0,s,q≤L1∑l,n≥0∑n1+n2=nl1+l2=lvl1,n11ql12/2Γ1+s+1l1+n1vl2,n22ql22/2Γ1+s+1l2+n2T0lX0n=L1V1T,XT0,X0,s,qV2T,XT0,X0,s,q.□The next proposition helps us in addressing the linear part of the Cauchy problem (116), (117).Proposition 25.
Letk1,k2,k3≥0 be integers under the requirement
(190)k2≥s+1k1,k3≥k1.
Then, the linear operatorσq−k3;T∘∂Tk1∂X−k2 is bounded from the space Gs,qT0,X0 into itself. In other words, there exists a constant L2>0 (depending on q,k1,k2,s) with
(191)∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q≤L2X0k2T0−k1VT,XT0,X0,s,q,for all VT,X∈Gs,qT0,X0.Proof.
LetVT,X=∑l,n≥0vl,nTl/l!Xn/n! be in Gs,qT0,X0. The action of the q−difference differential operator is expressed through
(192)∂Tk1∂X−k2Vq−k3T,X=∑l≥0,n≥k2vl+k1,n−k2q−k3lTll!Xnn!,and enables us to rewrite the norm
(193)T0k1X0−k2∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q=∑l≥0,n≥k2∣vl+k1,n−k2∣ql2/2q−k3lΓ1+s+1l+nT0l+k1X0n−k2=∑l≥0,n≥k2Cl,n∣vl+k1,n−k2∣ql+k12/21Γ1+s+1l+k1+n−k2T0l+k1X0n−k2,where
(194)Cl,n=ql+k12/2ql2/2q−k3lΓ1+s+1l+k1+n−k2Γ1+s+1l+n,for all l≥0, n≥k2. In the continuing part of the proof, we show that the sequence Cl,n is actually bounded by some constant.
Indeed, recall from [14], Appendix B, that for a given a∈ℝ, one can find a constant Ka>0 (relying on a) such that
(195)Γx+aΓx≤Kaxa,provided that x≥1. Consequently, we get two constants Ks,k1,k2>0 and L2.1>0 (depending on s,k1,k2) with
(196)Γ1+s+1l+n+s+1k1−k2Γ1+s+1l+n≤Ks,k1,k211+s+1l+nk2−s+1k1≤L2.1,for all l≥0, n≥k2, provided that (131) holds. On the other hand,
(197)ql+k12/2ql2/2q−k3l=qlk1−k3qk12/2≤qk12/2,for all l≥0 under the assumption (131). As a result, we obtain
(198)Cl,n≤qk12/2L2.1,for all l≥0, n≥k2. Finally, based on (133) and (136), we deduce
(199)T0k1X0−k2∂Tk1∂X−k2Vq−k3T,XT0,X0,s,q≤qk12/2L2.1VT,XT0,X0,s,q,which is tantamount to (132).In the next proposition, we show that the coefficients, the forcing term and Cauchy data of the problem (116), (117) belongs to the Banach spaceGs,qT0,X0 for well-chosen parameters.Proposition 26.
Letk,q chosen as in Section 2.
(1)
The seriesUpT,X appertains to G1/k,qT0,X0 for T0,X0>0 small enough(2)
The seriesEk¯T,X and AT,X belong to G1/k,qT0,X0 for T0,X0>0 small enough(3)
We set up(200)WpT,X≔∑j=0κ−1jj,pTXjj!.
For any nonnegative integersk1,k2,k3≥0, the polynomials ∂Tk1∂Xk2Wpq−k3T,X belong to G1/k,qT0,X0 for any given T0,X0>0.Proof.
We focus on the first point (1). According to Theorem3 stated in Subsection 4.2, with the help of the Cauchy formula, we deduce that the sequence Up,l,n defined in (109) is subjected to bounds of the form
(201)Up,l,n≤U0,pU1lU2nl!Γlkql2/2n!,for some selected constants U0,p,U1,U2>0, whenever l≥1, n≥0. Besides, from the formula (182), we observe in particular that
(202)ΓαΓβ≤Γα+β,provided that α,β≥1. This last inequality combined with the functional relation Γz+1=zΓz for any z>0 begets
(203)l!Γlkn!≤Γ2+1k+1l+n=1+1k+1l+nΓ1+1k+1l+n,for all l≥k, all n≥0. Thereby, we deduce constants U⌣0,p,U⌣1,U⌣2>0 with
(204)Up,l,n≤U⌣0,pU⌣1lU⌣2nΓ1+1k+1l+nql2/2,for all l,n≥0. Bestowing the last expansion of (115) implies the next norm bounds
(205)UpT,XT0,X0,1/k,q=∑l,n≥0∣Up,l,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤U⌣0,p∑l,n≥0T0U⌣1lX0U⌣2n≤4U⌣0,p,provided that T0,X0>0 are constrained to T0U⌣1≤1/2 and X0U⌣2≤1/2.
We address the second point (2). According to the assumptions of Section2, the functions ek¯t,z and at,z are polynomial in t and bounded holomorphic on some disc DR relatively to z, for all k¯∈B. From the Cauchy formula, we deduce that the first two sequences in (109) satisfy the bounds
(206)Ek¯,l,n≤E0,k¯E1lE2nl!n!,Al,n≤A0A1lA2nl!n!,for some well-chosen constants E0,k¯,E1,E2>0 and A0,A1,A2>0, for all l,n≥0. Owing to (139) and since the map x↦Γx is increasing on 2,+∞ (see [14], Appendix B), we check that
(207)l!n!≤Γ2+l+n≤Γ2+1k+1l+nql2/2=1+1k+1l+nΓ1+1k+1l+nql2/2,for all l,n≥0. Whence, constants E⌣0,k¯,E⌣1,E⌣2>0 and A⌣0,A⌣1,A⌣2>0 can be found with
(208)Ek¯,l,n≤E⌣0,k¯E⌣1lE⌣2nΓ1+1k+1l+nql2/2,Al,n≤A⌣0A⌣1lA⌣2nΓ1+1k+1l+nql2/2,for all l,n≥0. Thanks to the first two expansions of (115), we deduce the control of the norms
(209)Ek¯T,XT0,X0,1/k,q=∑l,n≥0∣Ek¯,l,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤E⌣0,k¯∑l,n≥0T0E⌣1lX0E⌣2n≤4E⌣0,k¯,together with
(210)AT,XT0,X0,1/k,q=∑l,n≥0∣Al,n∣ql2/2T0lX0nΓ1+1/k+1l+n≤A⌣0∑l,n≥0T0A⌣1lX0A⌣2n≤4A⌣0,whenever T0,X0>0 are submitted to T0E⌣1≤1/2, T0A⌣1≤1/2 and X0E⌣2≤1/2, X0A⌣2≤1/2.
Finally, the last point (3) is straightforward since for any given integersk1,k2,k3≥0, the quantity
(211)∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q,is a polynomial in the variables T0,X0 and is therefore finite for any fixed T0,X0>0.


## 4.4. Solving the Auxiliary Cauchy Problem

In this subsection, we seek for a formal power series solution to the nonlinear Cauchy problem (116), (117) by means of a decomposition(212)vpT,X=∂X−κYpT,X+WpT,X,where the polynomial WpT,X is displayed in (137), for some formal series YpT,X to be determined.We observe thatvpT,X fulfills the problem (116), (117) if the expression YpT,X solves the next fixed point equation
(213)YpT,X=∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2−κYpq−k3T,X+AT,X∂X−κYpT,X+WpT,X2+UpT,X+∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2Wpq−k3T,X.Our ensuing undertaking is the construction of a solution of this latter equation (149) within the Banach space of formal series discussed in the previous Section4.3. In order to fulfill this objective, we set up the next nonlinear mapping A:ℂT,X⟶ℂT,X defined as
(214)AYT,X≔∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂X−κ−k2Yq−k3T,X+AT,X∂X−κYT,X+WpT,X2+UpT,X+∑k¯=k1,k2,k3∈BEk¯T,X∂Tk1∂Xk2Wpq−k3T,X.In the next proposition, we discuss sufficient conditions under whichA represents a shrinking map acting on some small ball centered at the origin in the space G1/k,qT0,X0.Proposition 27.
Taking for granted the condition (149), some small real number ξ>0 can be singled out such that if 0<X0<ξ, one can choose a radius ϖ>0 such that A suffers the next features.
LetBϖ be the ball centered at 0 with radius ϖ in the space G1/k,qT0,X0.
(1)
The next inclusion(215)ABϖ⊂Bϖ,holds.(2)
For anyY1,Y2∈Bϖ, we have
(216)AY1−AY2T0,X0,1/k,q≤12Y1−Y2T0,X0,1/k,q.Proof.
We behold the first property. LetYT,X belong to the ball Bϖ. Under the constraint (100), Propositions 22, 25, and 26 allow the next inequality
(217)Ek¯T,X∂Tk1∂X−κ−k2Yq−k3T,XT0,X0,1/k,q≤L1Ek¯T,XT0,X0,1/k,q×∂Tk1∂X−κ−k2Yq−k3T,XT0,X0,1/k,q≤4L1L2E⌣0,k¯X0κ−k2T0−k1YT,XT0,X0,1/k,q,to hold true along with
(218)AT,X∂X−κYT,X+WpT,X2T0,X0,1/k,q≤L12AT,XT0,X0,1/k,q×∂X−κYT,X+WpT,XT0,X0,1/k,q2≤4L12A⌣0L2X0κYT,XT0,X0,1/k,q+WpT,XT0,X0,1/k,q2,for some constants L1,L2>0 and E⌣0,k¯,A⌣0>0. Eventually, due to Propositions 22 and 26, we arrive at
(219)Ek¯T,X∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤L1Ek¯T,XT0,X0,1/k,q∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q,for some constants L1>0 and E⌣0,k¯>0.From now on, we take the radiusϖ>0 in a way that
(220)4L12A⌣0WpT,XT0,X0,1/k,q2+4U⌣0,p+∑k¯=k1,k2,k3∈B4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q<ϖ,for some fixed X0>0. Then, we pick up a tiny constant ξ>0 with 0<X0<ξ aiming to the next constraint
(221)∑k¯=k1,k2,k3∈B4L1L2E⌣0,k¯X0κ−k2T0−k1ϖ+4L12A⌣0L2X0κϖ+WpT,XT0,X0,1/k,q2+4U⌣0,p+∑k¯=k1,k2,k3∈B4L1E⌣0,k¯∂Tk1∂Xk2Wpq−k3T,XT0,X0,1/k,q≤ϖ.This latter inequality can be reached since the quantitiesX0κ−k2 and X0κ are small as X0>0 tends to 0, owing to (100).Piling up the above estimates (153), (154), and (155) under the limitation (156) hints at the due inclusion (151).We address the second property. LetY1T,X,Y2T,X be taken within the ball Bϖ. The above inequality (153) prompts the next bounds for the linear piece
(222)Ek¯T,X∂Tk1∂X−κ−k2Y1−Y2q−k3T,XT0,X0,1/k,q≤4L1L2E⌣0,k¯X0κ−k2T0−k1Y1T,X−Y2T,XT0,X0,1/k,q.With an eye toward the nonlinear block, by means of the basic identitya2−b2=a−ba+b, we factorize the difference of squares as
(223)Δ1,2=∂X−κY1T,X+WpT,X2−∂X−κY2T,X+WpT,X2=∂X−κY1T,X−Y2T,X×∂X−κY1T,X+2WpT,X+∂X−κY2T,X.Consequently, on account of Propositions22, 25, and 26, the next estimates
(224)AT,XΔ1,2T0,X0,1/k,q≤L12AT,XT0,X0,1/k,q×∂X−κY1T,X−Y2T,XT0,X0,1/k,q×∂X−κY1T,X+2WpT,X+∂X−κY2T,XT0,X0,1/k,q≤L124A⌣0L2X0κL2X0κY1T,XT0,X0,1/k,q+2WpT,XT0,X0,1/k,q+L2X0κY2T,XT0,X0,1/k,qY1T,X−Y2T,XT0,X0,1/k,q,follow. Then, we pick out a small constant ξ>0 with 0<X0<ξ intending to the next condition
(225)∑k¯=k1,k2,k3∈B4L1L2E⌣0,k¯X0κ−k2T0−k1+L124A⌣0L2X0κ2L2X0κϖ+2WpT,XT0,X0,1/k,q≤12.Storing up the previous bounds (157) and (158) under the restriction (159) triggers the Lipschitz property (152).At last, we compel the constantsϖ>0 and ξ>0 to fulfill both constraints (156) and (159) in order to guarantee each of the two foretold properties (151) and (152).□.In the upcoming proposition, we solve the auxiliary nonlinear Cauchy problem (116), (117) amidst the Banach spaces introduced in Section4.3.Proposition 28.
Let us assume that the condition (149) holds. Let the constants ϖ,ξ>0 be determined by Proposition 27. Then, the nonlinear Cauchy problem (116), (117) owns a formal power series solution vpT,X that belongs to the space G1/k,qT0,X0 presuming that 0<X0<ξ. Along with it, one can find a constant L2>0 (resting on k,q,κ) such that
(226)vpT,XT0,X0,1/k,q≤L2X0κϖ+WpT,XT0,X0,1/k,q.Proof.
On the basis of Proposition27, the classical fixed point theorem for shrinking maps in complete metric spaces can be applied to the map A:Bϖ⟶Bϖ from the plain observation that Bϖ,d represents a complete metric space for the distance dx,y=x−yT0,X0,1/k,q since the vector space G1/k,qT0,X0 endowed with the norm .T0,X0,1/k,q is a Banach space. Indeed, under the conditions asked in Proposition 27, the map A:Bϖ⟶Bϖ appears to be of Lipschitz type 1/2 and hence shrinking. Whence, A:Bϖ⟶Bϖ has a unique fixed point, denoted YpT,X∈Bϖ, signifying that
(227)AYp=Yp.
In particular, a unique solutionYp for the equation (149) is found in the ball Bϖ. Along with it, due to Proposition 25, norms estimates can be achieved,
(228)∂X−κYpT0,X0,1/k,q≤L2X0κYpT0,X0,1/k,q≤L2X0κϖ.
Consequently, the splitting (148) attests that the mapvpT,X=∂X−κYpT,X+WpT,X appertains to G1/k,qT0,X0 and solves the problem (116), (117) under the restriction (160). □.


## 4.5. Asymptotic Expansions of Mixed Order

In order to describe the type of asymptotic behaviour which arises in our settings, we need the following definition issued from our recent work [16].Definition 29.
LetF,.F be a complex Banach space. We set k≥1 an integer and q>1 a real number. Let f:T⟶F be a holomorphic map, where T stands for a bounded sector in ℂ∗ centered at 0. Then, the map f enjoys the property of having the formal series
(229)f^t=∑n≥0antn∈Ft,as Gevrey asymptotic expansion of mixed order 1/k;q,1 on T if for each closed proper subsector W of T centered at 0, two constants C,M>0 can be distinguished with
(230)ft−∑n=0NantnF≤CMN+1ΓN+1kqN+12/2tN+1,for all integers N≥0 and any t∈W.
We are ready to enunciate the second main outcome of the work.Theorem 30.
We consider the setvp0≤p≤ς−1 of solutions to the main Cauchy problem (22), (23) established in Theorem 16. If one sets FZ1 as the Banach space of bounded holomorphic functions on the disc DZ1 equipped with the sup norm, then each partial map t↦vpt,z can be viewed as a holomorphic map from the sector Tp into FZ1, for 0≤p≤ς−1, as long as 0<Z1<Z0/2, where Z0 is set up in Theorem 16.
Hence, provided thatZ1>0 is taken small enough, for all 0≤p≤ς−1, the maps t↦vpt,z share a common formal power series
(231)v^t,z=∑l≥0hlztl∈FZ1t,as Gevrey asymptotic expansion of mixed order 1/k;q,1 on Tp. To rephrase it, for each 0≤p≤ς−1, for each proper subsector W⊂Tp, two constants C,M>0 can be singled out with
(232)supz∈DZ1∣vpt,z−∑l=0Nhlztl∣≤CMN+1ΓN+1kqN+12/2tN+1,for all t∈W, all integers N≥0.Proof.
According to the bounds (160) from Proposition28 and Definition 20, the unique formal series vpT,X∈ℂT,X solution of the Cauchy problem (116), (117) given by the expansion (114) is compelled to satisfy the next bounds
(233)vp,l,n≤L2X0κϖ+WpT,XT0,X0,1/k,q1T0l1X0nΓ1+1k+1l+nql2/2,for all l,n≥0. On the other hand, the next lemma will be required.Lemma 31.
There exist constantsKj>0, j=1,2,3 such that
(234)Γα+β≤K1K2αK3βΓαΓβ,for all real numbers α,β≥1.Proof.
We provide a complete proof of this classical result since it is not contained in our reference [14] on special functions. We depart from the Stirling formula (see [14] Appendix B), which ensures the existence of two constants A1,A2>0 for which
(235)A1xxx−1/2e−x≤Γx≤A2xxx−1/2e−x,for all x≥1. Therefrom,
(236)Γα+βΓαΓβ≤A2A12αβα+β1/21+βαα1+αββ.
Besides, one can pinpoint two constantsB1>0 and B2>1 such that
(237)αβα+β1/2≤β1/2≤B1B2β,for all β≥1 and from the classical estimates log1+x≤x for x>0, we deduce that
(238)1+βαα=expαlog1+βα≤eβ,1+αββ≤eα,for all real numbers α,β≥1. Lastly, piling up the above bounds (165), (166) and (167) yields the awaited estimates (164).In view of the inequality (113), we deduce from (163) together with (164) the next decisive bounds for the higher order time derivatives(239)supt∈Wvp,nt≤L2X0κϖ+WpT,XT0,X0,1/k,q1X0nn!≤V~01X0nn!,for all n≥0, together with
(240)supt∈W∂tlvp,nt≤L2X0κϖ+WpT,XT0,X0,1/k,q1T0l1X0nΓ1+1k+1l+nql2/2≤V^0V∧1lV∧2nΓ1k+1lΓ1+nql2/2=V^0V∧1lV∧2n11/k+1lΓ1+1k+1ln!ql2/2≤V^0V∧1lV∧2nΓ1k+1lΓ1+nql2/2=V^0V∧1lV∧2n11/k+1lΓ1+1k+1ln!ql2/2,for all l≥1, n≥0, for some constants V~0>0, V^0,V^1,V^2>0, and V⌣0,V⌣1,V⌣2>0 (relying k,q,κ,T0,X0,ϖ and Vp,l,j for 0≤j≤κ−1), where W stands for a fixed subsector of Tp.From the Taylor formula with integral remainder, for each0≤p≤ς−1, n≥0, we can expand the function vp,nt as follows
(241)vp,nt=∑l=0N∂tlvp,n0l!tl+RN+1,p,nt,where
(242)RN+1,p,nt=∫0t∂hN+1vp,nht−hNN!dh,for all t∈W, all integers N≥0. The remainder RN+1,p,nt can be estimated from above by means of the latter key bounds (169),
(243)∣RN+1,p,nt∣≤∫0∣t∣∣t∣−uNduV⌣0V⌣1N+1N+1V⌣2nn!Γ1kN+1qN+12/2≤V⌣0V⌣2nn!V⌣1N+1Γ1kN+1qN+12/2tN+1,for all t∈W.For each0≤p≤ς−1, we define the formal power series
(244)v^pt,z=∑l,n≥0∂tlvp,n0tll!znn!∈ℂt,z.It turns out thatv^pt,z can be rewritten in the form
(245)v^pt,z=∑l≥0hp,lztl,where the expressions hp,lz are holomorphic on a disc DZ1 provided that its radius Z1>0 is taken small enough. Indeed, the first term hp,0z is expressed through the series
(246)hp,0z=∑n≥0vp,n0znn!,which represents a holomorphic function on DZ1 for Z1<X0/2 since
(247)∣hp,0z∣≤V~0∑n≥0∣z∣X0n≤2V~0,owing to the bounds (168), whenever ∣z∣<X0/2. Besides, for all l≥1, the coefficients hp,lz are given by the expansions
(248)hp,lz=1l!∑n≥0∂tlvp,n0znn!,which define holomorphic maps on DZ1, when Z1<1/2V⌣2 since
(249)∣hp,lz∣≤V⌣0V⌣1lΓlkql2/2∑n≥0V⌣2nzn≤2V⌣0V⌣1lΓlkql2/2,according to the bounds (169), given that ∣z∣<1/2V⌣2.Consequently, in view of both expansions (170) and (173) along with the remainder estimates (171), we deduce the next error bounds(250)∣vpt,z−∑l=0Nhp,lztl∣=∣∑n≥0vp,ntznn!−∑n≥0∑l=0N∂tlvp,n0tll!znn!∣=∣∑n≥0RN+1,p,ntznn!∣≤V⌣0V⌣1N+1Γ1kN+1qN+12/2tN+1∑n≥0∣z∣V⌣2n≤2V⌣0V⌣1N+1Γ1kN+1qN+12/2tN+1,for all t∈W, all integers N≥0, given that ∣z∣<Z1, for Z1>0 chosen as above.In the last part of the proof disclosed within the next lemma, we show that the coefficientshp,lz do not depend on p for all l≥0.Lemma 32.
The coefficients(251)v^p,l,n=∂tlvp,n0,of the formal series v^pt,z given by (172) do not depend on p, for all l,n≥0. As a result, the coefficients hp,lz of the formal expansion (173) do not depend on p for all l≥0 and thereby each formal series v^pt,z turns out to be written as a single formal series
(252)v^t,z=∑l≥0hlztl,with holomorphic coefficients hlz on DZ1, for l≥0.Proof.
Paying regard to the relation (156), we get in particular the next recursion for the sequence v^p,l,n,
(253)v^p,l,n+κn!=∑k¯=k1,k2,k3∈B∑n1+n2=n∑l1+l2=ll!∂tl1ek¯,n10l1!n1!v^p,l2+k1,n2+k2l2!n2!q−k3l2+∑n1+n2+n3=n∑l1+l2+l3=ll!∂tl1an10l1!n1!v^p,l2,n2l2!n2!v^p,l3,n3l3!n3!+∂tlup,n0n!,with prescribed set of data
(254)v^p,l,j=∂tlφ⌣j0,0≤j≤κ−1,l≥0.
Now, we observe that the above prescribed quantitiesv^p,l,j for 0≤j≤κ−1 and l≥0 along with the forcing terms ∂tlup,n0/n! for all l≥0, all n≥0, do not depend on p, according to our assumption (23) and Theorem 3, 2.. As a result, we deduce by induction from the above recursion (175) that the whose sequence v^p,l,n do not depend on p for all l,n≥0. □.
At last, the statement of Theorem30 issues from the above lemma 32 and the error bounds (174).


## 5. Confluence asq Tends to 1

Throughout this section, the notations introduced in the earlier sections of the work are lightly modified. Our objective is now to keep track of the dependence of the family of solutionsvpt,z0≤p≤ς−1 to the initial problem (22), (23) relatively to the parameter q>1, constructed in Theorem 16. On that account, we denote vp;qt,z the function vpt,z. We also attach a second index q to the Borel map Θpu,z by setting Θpu,z=Θp;qu,z within the integral representation (88). From now on, the real parameter q is chosen inside an interval 1,q0 for some fixed real number q0>1.


### 5.1. A Limit Singular Linear Cauchy Problem

In this subsection, we call attention to parts of the results reached in our previous work [1] that will be applied within the next subsection. We keep the same notations as the ones introduced in Section 2.1.We consider the next singular Cauchy problem(255)Ptk+1∂t∂zSu;1t,z=∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2u;1t,z,for given Cauchy data
(256)∂zju;1t,0=φjt,0≤j≤S−1.where all the data k,S,P,A and the coefficients cl¯z with l¯∈A along with the initial data φjt for 0≤j≤S−1 are already declared in Section 2.1.In Section 8.1 of the paper [1], the next statement is outlined.Proposition 33.
LetD¯=T¯,U¯ be an admissible set of sectors as chosen in Definition 2. Let U be one sector belonging to the family of unbounded sectors U¯.
One can build up a solutionu;1t,z to the Cauchy problem (177), (178) which is bounded holomorphic on a domain T×D1/2C8 where the bounded sector T belongs to the family T¯ of bounded sectors from D¯ and corresponds to U under the requirement (2) of Definition 2 and where C8>0 is some well-chosen constant. The map u;1t,z can be expressed by means of a Laplace transform of order k,
(257)u;1t,z=k∫Lγw;1u,zexp−utkduu,along a halfline Lγ⊂U∪0 described in Definition 2 (2), for all t,z∈T×D1/2C8.
The Borel mapw;1u,z represents a holomorphic function on U×D1/2C8. Its Taylor expansion
(258)w;1u,z=∑n≥0wn;1uznn!,suffers the next bounds
(259)wn;1u≤C7C8nn!uexpk1log2u+u0+αlogu+u0,for all n≥0, all u∈U∪0, for fittingly chosen constants C7,k1>0 and u0>1, α≥0.


### 5.2. A Limit Nonlinear Cauchy Problem

In this subsection, a novel Cauchy problem is introduced that we call the limit problem asq>1 tends to 1. Its shape is displayed as follows
(260)∂zκv;1t,z=∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2v;1t,z+at,zv;12t,z+u;1t,z,for given Cauchy data
(261)∂zjv;1t,0=φ⌣jt,0≤j≤κ−1,where the forcing term u;1t,z is the holomorphic solution of the singular linear Cauchy problem (177), (178) unveiled in Proposition 33 of the former subsection. Besides, all the items κ,k,C along with the coefficients dh¯z, at,z for h¯∈C and the Cauchy data φ⌣jt for 0≤j≤κ−1 are those already introduced in Section 2.2.We aim for a solution to (182), (183) having the profile of a Laplace transform of orderk,
(262)v;1t,z=k∫LγΘ;1u,zexp−utkduu,along the halfline Lγ given in (179), where the Borel map Θ;1u,z is holomorphic with respect to u on U and analytic w.r.t z on some small disc Dr centered at 0 with radius r>0.The same computations as the ones of Section2.3 by means of Proposition 4 shows that the Borel map Θ;1u,z solves that next auxiliary Cauchy problem
(263)∂zκΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1∂zh2Θ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1ksh1∂zh2Θ;1s1/k,zdss+a0zuk∫0ukΘ;1uk−s1/k,zΘ;1s1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,z1s−s1s1ds1dss+w;1u,z,for given Cauchy data
(264)∂zjΘ;1u,0=P⌣ju,0≤j≤κ−1.For later need, we make the change of variabless=ukx and s1=sx1 for 0≤x,x1≤1 in the integrals involved in (185). As a result, Θ;1u,z solves that next auxiliary Cauchy problem
(265)∂zκΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1∂zh2Θ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zuh0+kh1Γh0/k∫011−xh0/k−1kh1xh1∂zh2Θ;1ux1/k,zdxx+a0z∫01Θ;1u1−x1/k,zΘ;1ux1/k,zdx1−xx+∑h=1AahzuhΓh/k∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+w;1u,z,for given Cauchy data (186).The next proposition comes along with the same steps as in the proofs of Proposition15 and Theorem 16.Proposition 34.
LetD¯=T¯,U¯ be the admissible set of sectors distinguished in Proposition 33. Let U be the infinite sector belonging to U¯ selected in Proposition 33.
A solutionv;1t,z to the Cauchy problem (182), (183) can be reached, that is bounded and holomorphic on a domain T×DZ0;1/2 for some small radius Z0;1>0, where T is the bounded sector from T¯ singled out in Proposition 33. Moreover, v;1t,z is described using a Laplace transform of order k,
(266)v;1t,z=k∫LγΘ;1u,zexp−utkduu,for all t,z∈T×DZ0;1/2, along the halfline Lγ that appears in the representation (179). The Borel map Θ;1u,z stands for a holomorphic map on the domain U×DZ0;1/2 which solves the above auxiliary Cauchy problem (185), (187), and (186). Besides, the map Θ;1u,z belongs to the space Gσ,Z0;1,Uk and enjoys a decomposition of the form
(267)Θ;1u,z=∂z−κΞ;1u,z+Ψ⌣u,z,where Ψ⌣u,z is defined in (63) and Ξ;1u,z belongs to Gσ,Z0;1,Uk and satisfies
(268)Ξ;1u,zσ,Z0;1,U≤υ;1,for some suitable constant υ;1>0.In the ensuing corollary, we observe thatv;1t,z actually solves a limit nonlinear Cauchy problem with analytic coefficients in space z in the vicinity of the origin and polynomial in time t.Corollary 35.
The analytic mapv;1t,z solves on the product T×DZ0;1/2 a particular Cauchy problem relying polynomially on time t with the shape
(269)Ptk+1∂t∂zS+κv;1t,z=G⌣t,z,∂tr1∂zr2v;1t,zr¯=r1,r2∈R⌣,for given Cauchy data of the form
(270)∂zjv;1t,0=φ⌣jt,0≤j≤κ−1,together with
(271)∂zjv;1t,0=ϕ⌣j;1t,κ≤j≤κ+S−1,for well selected polynomials ϕ⌣j;1t, κ≤j≤κ+S−1 (that are independent from q). The set R⌣ is a finite subset of ℕ2 for r1,r2∈R⌣ with the property that r2<S+κ for any r1,r2∈R⌣. The map G⌣t,z,ur¯r¯∈R⌣ is polynomial in time t and in its arguments ur¯ with r¯∈R⌣ and analytic w.r.t z on DZ0;1/2.Proof.
We set up the differential operators(272)D1t,z,∂t,∂z≔Ptk+1∂t∂zS−∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2,and
(273)D2t,z,∂t,∂zv≔∂zκ−∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2v−at,zv2.
According to Proposition34, v;1t,z satisfies
(274)D2t,z,∂t,∂zv;1t,z=u;1t,z,and based on Proposition 33, the map u;1t,z solves
(275)D1t,z,∂t,∂zu;1t,z=0,provided that t,z∈T×DZ0;1/2. By pairing (194) together with (195), we obtain
(276)D1t,z,∂t,∂z∘D2t,z,∂t,∂zv;1t,z=0,that is represented by the equation (269). Regarding the Cauchy data, the map v;1 suffers the conditions (261) which are replicated in (192). Moreover, the constraints (178) put on u;1t,z can be reworded as
(277)∂zj∘D2t,z,∂t,∂zv;1t,0=φjt,for 0≤j≤S−1 which begets the assumptions (193) for properly chosen polynomials ϕ⌣l;1t with complex coefficients for κ≤l≤κ+S−1.


### 5.3. Analytic Solutions to the Limit Singular Linear Cauchy Problem under the Action of aq−Difference Operator

In this subsection, we take heed of a technical result achieved in our foregoing work [1] that will be called upon in the next subsection. We keep the notations of Subsection 5.1.In Section 8.2 of [1], the next result is established.Proposition 36.
Letβ≥1 be an integer and let q∈1,q0. Two constants C9,C10>0 (that are unrelated to q) can be found such that
(278)∣wn;1u−wn;1q−βu∣≤∣q−β−1∣C9C10nn!∣u∣expk1log2∣u∣+u0+αlog∣u∣+u0,for all u∈U∪0, for all integers n≥0, where the constants k1>0, u0>1, α≥0, and the unbounded sector U appear in Proposition 33.


### 5.4. Analytic Solutions to the Limit Nonlinear Cauchy Problem under the Action of aq−Difference Operator

This subsection is dedicated to the proof of the next technical proposition.Proposition 37.
Letβ≥1 be an integer and set q∈1,q0. Then, one can find some constants υ;1>0 and M~1;1>0 (unrelated to q) such that for any given σ>0, if Z~0>0 is taken small enough, the next bounds
(279)Θ;1u,z−Θ;1q−βu,zσ,Z~0,U≤M~1;11−q−β,hold.Proof.
The proof is rather lengthy and is made up with several steps.
In the first step of the proof, we formulate a Cauchy problem, stated below in (208), (209), that the difference(280)ΔβΘ;1u,z≔Θ;1u,z−Θ;1q−βu,z,is compelled to solve. We first state a Cauchy problem satisfied by the quantity Θ;1q−βu,z by substituting u by q−βu in the equations (265), (186).
(281)∂zκΘ;1q−βu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−βukh1∂zh2Θ;1q−βu,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zq−βuh0+kh1Γh0/k∫011−xh0/k−1kh1xh1∂zh2Θ;1q−βux1/k,zdxx+a0z∫01Θ;1q−βu1−x1/k,zΘ;1q−βux1/k,zdx1−xx+∑h=1Aahzq−βuhΓh/k∫011−xh/k−1∫01Θ;1q−βux1/k1−x11/k,zΘ;1q−βux1/kx11/k,zdx11−x1x1dxx+w;1q−βu,z,∂zjΘ;1q−βu,0=P⌣jq−βu,0≤j≤κ−1.
On the way, we need to perform some practical computations. Namely, using the basic identityab−cd=a−cb+cb−d, we recast the next list of pieces in a suitable manner:
(282)kukh1∂zh2Θ;1u,z−kq−βukh1∂zh2Θ;1q−βu,z=kukh1−kq−βukh1∂zh2Θ;1u,z+kq−βukh1∂zh2Θ;1u,z−∂zh2Θ;1q−βu,z,and
(283)uh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx−q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1q−βux1/k,zdxx=uh0+kh1−q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx+q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,z−∂zh2Θ;1q−βux1/k,zdxx,along with
(284)∫01Θ;1u1−x1/k,zΘ;1ux1/k,zdx1−xx−∫01Θ;1q−βu1−x1/k,zΘ;1q−βux1/k,zdx1−xx=∫01Θ;1u1−x1/k,z−Θ;1q−βu1−x1/k,zΘ;1ux1/k,z+Θ;1q−βu1−x1/k,zΘ;1ux1/k,z−Θ;1q−βux1/k,zdx1−xx,and
(285)uh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx−q−βuh∫011−xh/k−1∫01Θ;1q−βux1/k1−x11/k,zΘ;1q−βux1/kx11/k,zdx11−x1x1dxx=uh−q−βuh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+q−βuh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,z−Θ;1q−βux1/k1−x11/k,z×Θ;1ux1/kx11/k,z+Θ;1q−βux1/k1−x11/k,z×Θ;1ux1/kx11/k,z−Θ;1q−βux1/kx11/k,zdx11−x1x1dxx.
Based on the two equations (265), (200) together with the prescribed data (186), (201) and thanks to the above computations (202), (203), (204), (205), we can now exhibit a Cauchy problem fulfilled by the difference ΔβΘ;1u,z given in (199) owning the following shape:
(286)∂zκΔβΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−βukh1∂zh2Θ;1u,z+kq−βukh1∂zh2ΔβΘ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/kuh0+kh1−q−βuh0+kh1×∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx+q−βuh0+kh1∫011−xh0/k−1xh1∂zh2ΔβΘ;1ux1/k,zdxx+a0z∫01ΔβΘ;1u1−x1/k,zΘ;1ux1/k,z+Θ;1q−βu1−x1/k,zΔβΘ;1ux1/k,zdx1−xx+∑h=1Aahz1Γh/kuh−q−βuh∫011−xh/k−1×∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+q−βuh∫011−xh/k−1∫01ΔβΘ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,z+Θ;1q−βux1/k1−x11/k,zΔβΘ;1ux1/kx11/k,zdx11−x1x1dxx+w;1u,z−w;1q−βu,z,for given Cauchy data
(287)∂zjΔβΘ;1u,0=P⌣ju−P⌣jq−βu,0≤j≤κ−1.
In the upcoming step, we plan to solve the problem (206), (207) within the Banach space described in Subsection3.1. In the first instance, we need to rephrase the integral operators involved in (206) in terms of those appearing in Subsection 3.1 by means of the parametrization s=ukx, s1=sx1 for 0≤x,x1≤1. Indeed, we deduce that ΔβΘ;1u,z turns out to solve the next linear Cauchy problem
(288)∂zκΔβΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1ukh1∂zh2Θ;1u,z+kq−βukh1∂zh2ΔβΘ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×uk∫0ukuk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1uk∫0ukuk−sh0/k−1sh1∂zh2ΔβΘ;1s1/k,zdss+a0zuk∫0ukΔβΘ;1uk−s1/k,zΘ;1s1/k,z+Θ;1q−βuk−s1/k,zΔβΘ;1s1/k,zdsuk−ss+∑h=1Aahz1Γh/k1−q−βhuk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhuk∫0ukuk−sh/k−1s∫0sΔβΘ;1s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,zΔβΘ;1s11/k,zds1s−s1s1dss+w;1u,z−w;1q−βu,z,for assigned Cauchy data
(289)∂zjΔβΘ;1u,0=P⌣ju−P⌣jq−βu,0≤j≤κ−1.
In the second step of the proof, we seek for a solution to the above problem (208), (209) expressed by means of the next splitting(290)ΔβΘ;1u,z=∂z−κΔβΞ;1u,z+ΔβΨ⌣u,z,for some expression ΔβΞ;1u,z to be determined, where
(291)ΔβΨ⌣u,z=∑j=0κ−1ΔβP⌣juzjj!,for ΔβP⌣ju=P⌣ju−P⌣jq−βu, with 0≤j≤κ−1.
We pinpoint the crucial fact thatΔβΘ;1u,z solves the problem (208), (209) if the quantity ΔβΞ;1u,z satisfies the next fixed point equation
(292)ΔβΞ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1ukh1∂zh2Θ;1u,z+kq−βukh1∂z−κ−h2ΔβΞ;1u,z+∂zh2ΔβΨ⌣u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×uk∫0ukuk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1uk∫0ukuk−sh0/k−1sh1∂z−κ−h2ΔβΞ;1s1/k,z+∂zh2ΔβΨ⌣s1/k,zdss+a0zuk∫0uk∂z−κΔβΞ;1uk−s1/k,z+ΔβΨ⌣uk−s1/k,zΘ;1s1/k,z+Θ;1q−βuk−s1/k,z∂z−κΔβΞ;1s1/k,z+ΔβΨ⌣s1/k,zdsuk−ss+∑h=1Aahz1Γh/k1−q−βhuk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhuk∫0ukuk−sh/k−1s∫0s∂z−κΔβΞ;1s−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞ;1s11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dss+w;1u,z−w;1q−βu,z.
Our next duty will be to search for a solution and provide bounds relatively toq for a solution to (212) in the Banach space Gσ,Z~0,Uk, for any given σ>0, provided that Z~0>0 is chosen small enough. With that in mind, we introduce the linear map
(293)DΔβΞτ,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1τkh1∂zh2Θ;1τ,z+kq−βτkh1∂z−κ−h2ΔβΞτ,z+∂zh2ΔβΨ⌣τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1τk∫0τkτk−sh0/k−1sh1∂z−κ−h2ΔβΞs1/k,z+∂zh2ΔβΨ⌣s1/k,zdss+a0zτk∫0τk∂z−κΔβΞτk−s1/k,z+ΔβΨ⌣τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞs1/k,z+ΔβΨ⌣s1/k,zdsτk−ss+∑h=1Aahz1Γh/k1−q−βhτk∫0τkτk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhτk∫0τkτk−sh/k−1s∫0s∂z−κΔβΞs−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dss+w;1τ,z−w;1q−βτ,z.
In the next lemma, we discuss sufficient conditions under whichD acts as a shrinking map on a small ball centered at 0, whose radius depends on q, in the space Gσ,Z~0,Uk.Lemma 38.
We take for granted that the conditions (30) hold. Then, one can single out a small real number χ~>0 in a way that if 0<Z~0<χ~, one can select a radius υ~>0 which is independent of q (but relies on q0 such that 1<q≤q0), such that D possesses the next two qualities: let Bυ~1−q−β be a ball centered at 0 with radius υ~1−q−β(1)
D maps Bυ~1−q−β into itself, meaning that
(294)DBυ~1−q−β⊂Bυ~1−q−β.(2)
For anyΔβΞ1,ΔβΞ2∈Bυ~1−q−β, we have
(295)DΔβΞ1−DΔβΞ2σ,Z~0,U≤12ΔβΞ1−ΔβΞ2σ,Z~0,U.Proof.
As a prefatory material, norm bounds estimates are required for some pieces of the mapD.
(a)
Indeed, at first we need estimates for the normw;1τ,z−w;1q−βτ,zσ,Z~0,U. According to the expansion (180) and the definition of the norm, we get
(296)w;1τ,z−w;1q−βτ,zσ,Z~0,U=∑n≥0wn;1τ−wn;1q−βτn,σ,UZ~0nn!,and owing to the bounds (197) brought to mind in Proposition 36, we can upper bound the coefficients of this latter series by
(297)wn;1τ−wn;1q−βτn,σ,U=supτ∈U1+τ2k∣τ∣exp−σrbnτkwn;1τ−wn;1q−βτ≤1−q−βC9C10nn!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0≤1−q−βC~9C10nn!,for some constant C~9 (unrelated to q), for all integers n≥0. Thereupon, it leads to
(298)w;1τ,z−w;1q−βτ,zσ,Z~0,U≤1−q−βC~9∑n≥0C10Z~0n≤21−q−βC~9,provided that Z~0<1/2C10.(b)
We focus on the quantitiesτkh1∂zh2Θ;1τ,zσ,Z~0,U for integers h1,h2≥0 under the constraint (20) and on Θ;1q−βτ,zσ,Z~0,U
According to the decomposition (189) with the bounds (190) and owing to Propositions6 and 12, we obtain
(299)τkh1∂zh2Θ;1τ,zσ,Z~0,U≤τkh1∂zh2Θ;1τ,zσ,Z0;1,U≤τkh1∂z−κ−h2Ξ;1τ,zσ,Z0;1,U+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U≤M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U,along with
(300)Θ;1q−βτ,zσ,Z~0,U≤Θ;1q−βτ,zσ,Z0;1,U≤∂z−κΞ;1q−βτ,zσ,Z0;1,U+Ψ⌣q−βτ,zσ,Z0;1,U≤M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,U,for some constant M1>0 (independent of q but relying on q0 such that 1<q≤q0) provided that Z~0<Z0;1.
(c)
We ask for sharp bounds for the normsτkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U, for integers h1,h2≥0 under the constraint (20). Departing from the expansion (211), we observe that(301)τkh1∂zh2ΔβΨ⌣τ,z=∑j=0κ−1−h2τkh1ΔβP⌣j+h2τzjj!,and its norm writes
(302)τkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U=∑j=0κ−1−h2τkh1ΔβP⌣j+h2τj,σ,UZ~0jj!,where
(303)τkh1ΔβP⌣j+h2τj,σ,U=supτ∈U1+τ2kτexp−σrbjτkτkh1ΔβP⌣j+h2τ≤supτ∈U1+τ2k∣τ∣exp−στkτkh1ΔβP⌣j+h2τ,since rbj≥1, for all j≥0. Furthermore, we can recast the difference as an integral
(304)ΔβP⌣j+h2τ=∫q−βττP⌣j+h2′sds=1−q−βτ∫01P⌣j+h2′τh+q−βτ1−hdh,by means of the parametrization s=τh+q−βτ1−h with 0≤h≤1. Since P⌣j+h2 is a polynomial with complex coefficients, its derivative can be written in the form
(305)P⌣j+h2′s=∑h∈J⌣j+h2′P⌣j+h2,h′sh,for some finite subset J⌣j+h2′ of ℕ and complex coefficients P⌣j+h2,h′. We set
(306)∣P⌣j+h2′∣x=∑h∈J⌣j+h2′∣P⌣j+h2,h′∣xh,and since
(307)τh+q−βτ1−h≤τh+q−βτ1−h≤τ,for all τ∈ℂ, all 0≤h≤1, we get in particular that
(308)∣P⌣j+h2′τh+q−βτ1−h∣≤∣P⌣j+h2′∣∣τ∣,whenever τ∈ℂ and 0≤h≤1. We deduce from (221) and (222) that
(309)ΔβPj+h2∨τ≤1−q−βτPj+h2∨′τfor all τ∈ℂ. Thereupon, gathering (220) and (223) yields a constant Mk,h1,j,h2>0 for which
(310)τkh1ΔβP⌣j+h2τj,σ,U≤1−q−βsupx≥01+x2kexp−σxkxkh1∣P⌣j+h2′∣x≤1−q−βMk,h1,j,h2.
Keeping in mind (219), we finally get a constantNk,h1,h2,κ,Z~0>0 such that
(311)τkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U≤1−q−βNk,h1,h2,κ,Z~0.(d)
Bounds for the quantityτk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zds/sσ,Z~0,U are also required, for integers h0,h1,h2≥0 under the constraint (20). Based on the decomposition (189) with the bounds (190), Propositions 10 and 12 allow to get a constant M3>0 with(312)τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z~0,U≤τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z0;1,U≤τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Ξ;1s1/k,zdssσ,Z0;1,U+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U≤M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U,presuming that Z~0<Z0;1.
(e)
We demanded also accurate estimates for the norms(313)ΔI≔τk∫0τkτk−sh0/k−1sh1∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U,for integers h0,h1,h2≥0 under the constraint (20). From the expansion (211), we check that
(314)∂zh2ΔβΨ⌣s1/k,z=∑j=0κ−1−h2ΔβP⌣j+h2s1/kzjj!,and therefore,
(315)ΔI=∑j=0κ−1−h2τk∫0τkτk−sh0/k−1sh1−1ΔβP⌣j+h2s1/kdsj,σ,UZ~0jj!.
On the basis of the upper bounds (223), by means of the change of variableh=τkx with 0≤x≤1 and keeping in mind the lower bounds rbj≥1, we deduce a constant M¯k,h0,h1,h2,j>0 for which
(316)τk∫0τkτk−sh0/k−1sh1−1ΔβP⌣j+h2s1/kdsj,σ,U≤supτ∈U1+τ2k∣τ∣exp−σrbjτkτk∫0τkτk−hh0/k−1hh1−11−q−βh1/k∣P⌣j+h2′∣h1/kdh≤1−q−βsupy≥01+y2kexp−σykyh0+kh1∣P⌣j+h2′∣y∫011−xh0/k−1xh1−1+1/kdx≤1−q−βM¯k,h0,h1,h2,j.
At last, pairing (227) with (228) gives rise to a constantN¯k,h0,h1,h2,κ,Z~0>0 such that
(317)τk∫0τkτk−sh0/k−1sh1∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U≤1−q−βN¯k,h0,h1,h2,κ,Z~0..
We are now ready to come to the core of the proof. We fix our attention on the first item 1.
LetΔβΞ belong to Gσ,Z~0,Uk such that
(318)ΔβΞτ,zσ,Z~0,U≤υ~1−q−β.
We provide explicit bounds for each block of the mapDΔβΞ.
Proposition11 and (217) return
(319)dh¯zkh11−q−βkh1τkh1∂zh2Θ;1τ,zσ,Z~0,U≤kh11−q−βkh1dh¯Z~0M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U.
Propositions6 and 11 together with (225) trigger
(320)dh¯zkh1q−βkh1τkh1∂z−κ−h2ΔβΞτ,z+∂zh2ΔβΨ⌣τ,zσ,Z~0,U≤dh¯Z~0kh1q−βkh1M1Z~0κ−h2ΔβΞτ,zσ,Z~0,U+1−q−βNk,h1,h2,κ,Z~0.
Proposition11 and (226) afford
(321)dh¯zkh1Γh0/k1−q−βh0+kh1τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/k1−q−βh0+kh1×M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U.
Propositions10 and 11 along with (229) prompt
(322)dh¯zkh1Γh0/kq−βh0+kh1τk∫0τkτk−sh0/k−1sh1×∂z−κ−h2ΔβΞs1/k,z+∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2ΔβΞτ,zσ,Z~0,U+1−q−βN¯k,h0,h1,h2,κ,Z~0.
Propositions6, 8, and 11 with the help of (217), (218) and (225) furnish
(323)a0zτk∫0τk∂z−κΔβΞτk−s1/k,z+ΔβΨ⌣τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞs1/k,z+ΔβΨ⌣s1/k,zdsτk−ssσ,Z~0,U≤a0Z~0M2∂z−κΔβΞτ,z+ΔβΨ⌣τ,zσ,Z~0,UΘ;1τ,zσ,Z~0,U+Θ;1q−βτ,zσ,Z~0,U∂z−κΔβΞτ,z+ΔβΨ⌣τ,zσ,Z~0,U≤a0Z~0M2M1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0.
Propositions8, 10, and 11 together with (217) beget
(324)ahz1Γh/k1−q−βhτk∫0τkτk−sh/k−1×s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/k1−q−βhM3τk∫0τkΘ;1τk−s11/k,zΘ;1s11/k,zds1τk−s1s1σ,Z~0,U≤ahZ~01Γh/k1−q−βhM3M2Θ;1τ,zσ,Z~0,U2≤ahZ~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2≤ahZ~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2.
Proposition6, 8, 10, and 11 coupled with (217), (218), and (225) breed
(325)ahz1Γh/kq−βhτk∫0τkτk−sh/k−1×s∫0s∂z−κΔβΞs−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/kq−βhM3τk∫0τk∂z−κΔβΞτk−s11/k,z+ΔβΨ⌣τk−s11/k,zΘ;1s11/k,z+Θ;1q−βτk−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1τk−s1s1σ,Z~0,U≤ahZ~01Γh/kq−βhM3M2∂z−κΔβΞτ,z+ΔβΨ⌣Θ;1s11/k,zσ,Z~0,UΘ;1τ,zσ,Z~0,U+Θ;1q−βτ,zσ,Z~0,U∂z−κΔβΞτ,z+ΔβΨ⌣Θ;1s11/k,zσ,Z~0,U≤ahZ~01Γh/kq−βhM3M2M1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0.
From now on, we select a small sized quantityZ~0>0 and suitable υ~>0 (taken independently of q∈1,q0) in a way that the next inequality holds
(326)∑h¯=h0,h1,h2,h3∈C;h0=0∣dh¯∣Z~0kh11−q−βkh1M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U+kh1q−βkh1M1Z~0κ−h2υ~1−q−β+1−q−βNk,h1,h2,κ,Z~0+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z~0kh1Γh0/k1−q−βh0+kh1×M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U+q−βh0+kh1M3Z~0κ−h2υ~1−q−β+1−q−βN¯k,h0,h1,h2,κ,Z~0+∣a0∣Z~0M2M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0+∑h=1A∣ah∣Z~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2+q−βhM3M2M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0+21−q−βC~9≤υ~1−q−β.
Notice that the above constraint (238) can be achieved since all the quantities(327)1−q−βkh11−q−β=∑j=0kh1−1q−βj,1−q−βh0+kh11−q−β=∑j=0h0+kh1−1q−βj,1−q−βkh1−q−β=∑j=0kh−1q−βj,remain bounded for q∈1,q0 provided that kh1≥1, h0+kh1≥1, and h≥1.
At last, the collection of all the inequalities (216), (231), (232), (233), (234), (235), (236), and (237) listed overhead under the condition (326) yields the first expected item (214).
We concentrate on the second feature 2. of the mapD. Let ΔβΞ1,ΔβΞ2 be elements of Gσ,Z~0,Uk with
(328)ΔβΞjτ,zσ,Z~0,U≤υ~1−q−β,for j=1,2.
According to the computations made to treat the first property 1 ofD, we deduce forthrightly the next list of inequalities
(329)dh¯zkh1q−βkh1τkh1∂z−κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U≤dh¯Z~0kh1q−βkh1M1Z~0κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,and
(330)dh¯zkh1Γh0/kq−βh0+kh1τk∫0τkτk−sh0/k−1sh1×∂z−κ−h2ΔβΞ1s1/k,z−ΔβΞ2s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,along with
(331)a0zτk∫0τk∂z−κΔβΞ1τk−s1/k,z−ΔβΞ2τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞ1s1/k,z−ΔβΞ2s1/k,zdsτk−ssσ,Z~0,U≤a0Z~0M2M1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,and
(332)ahz1Γh/kq−βhτk∫0τkτk−sh/k−1×s∫0s∂z−κΔβΞ1s−s11/k,z−ΔβΞ2s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞ1s11/k,z−ΔβΞ2s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/kq−βhM3M2M1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U.
Hereafter, we chooseZ~0>0 small enough in a way that
(333)∑h¯=h0,h1,h2,h3∈C;h0=0dh¯Z~0kh1q−βh1kM1Z~0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2+a0Z~0M2M1Z~0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κ+∑h=1AahZ~01Γh/kq−βhM3M2M1Z~0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κ≤12,and gathering (239), (240), (241), and (242) gives rise to the shrinking property (215).
Eventually, we select the constantsZ~0>0 and υ~>0 in a way that both constraints (238) and (243) hold concomitantly. Lemma 38 follows. □.
In the upcoming lemma, we solve the linear Cauchy problem (208), (209) within the Banach spaceGσ,Z~0,Uk.Lemma 39.
Let us presume that condition (30) holds. We single out the constants Z~0>0 and υ~>0 (independently of q in 1,q0) as in Lemma 38. Then, the linear Cauchy problem (208), (209) owns a solution ΔβΘ;1u,z that belong to the Banach space Gσ,Z~0,Uk for any given σ>0. Along with it, a constant M~1;1>0 (unrelated to q∈1,q0) can be found with the next feature
(334)ΔβΘ;1u,zσ,Z~0,U≤M~1;11−q−β.Proof.
Derived from Lemma38, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map D:Bυ~1−q−β⟶Bυ~1−q−β according to the fact Bυ~1−q−β,d stands for a complete metric space for the distance dx,y=x−yσ,Z~0,U. Whence, D:Bυ~1−q−β⟶Bυ~1−q−β carries a unique fixed point denoted ΔβΞ;1u,z inside the ball Bυ~1−q−β, meaning that
(335)DΔβΞ;1=ΔβΞ;1.
As a result, a unique solutionΔβΞ;1 for the equation (292) is established in the ball Bυ~1−q−β. Furthermore, proposition 6 conjointly with the decomposition (210) and the bounds (225) certify that the map
(336)ΔβΘ;1u,z=∂z−κΔβΞ;1u,z+ΔβΨ⌣u,z,belongs to Gσ,Z~0,Uk, solves the problem (208), (209), and suffers the next upper estimates
(337)ΔβΘ;1u,zσ,Z~0,U≤M1Z~0κΔβΞ;1u,zσ,Z~0,U+ΔβΨ⌣u,zσ,Z~0,U≤M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0≤M~1;11−q−β,for some constant M~1;1>0 which is unattached to q in the range 1,q0.□.
According to Lemma39, it turns out that the unique formal series in z with holomorphic coefficients on U solution of (208), (209) is subjected to the bounds (244). Since the difference Θ;1u,z−Θ;1q−βu,z, which represents in particular a formal power series in z with holomorphic coefficients on U, is shown to solve (208), (209) in the first step of Proposition 37, it must coincide with the solution ΔβΘ;1u,z constructed above in the second step of the proof, and we deduce conclusively that the for sought estimates (198) hold for it. This completes the proof of the proposition.


### 5.5. Error Bounds between the Borel Maps of the Analytic Solutions to the Linear Cauchy Problems (11), (12) and (177), (178)

In this subsection, we remind the reader of a result obtained in our previous work [1] related to the dependence of the family of solutions upt,z0≤p≤ς−1 to the linear Cauchy problem (11), (12) relatively to the parameter q>1 set up in Theorem 3. This result will be applied in the ensuing subsection.We consider the admissible setD¯=T¯,U¯ of sectors chosen in Proposition 33, where an unbounded sector U and corresponding bounded sector T are distinguished. We denote
(338)ut,z=k∫Lγw;qu,zexp−u/tkduu,for t,z∈T×D1/2C4, the solution of the problem (11), (12) displayed in Theorem 3, along the halfline Lγ taken in Proposition 33, where the Borel map w;qu,z has now an attached index q in order to keep track of the dependence in q>1. According to Theorem 3, w;qu,z represents a holomorphic function on the domain U×D1/2C4 with a Taylor expansion of the form
(339)w;qu,z=∑n≥0wn;quznn!,whenever u∈U, z∈D1/2C4. We keep the notations of Section 5.1. In Section 8.3 of [1], the next result is stated.Proposition 40.
Letq∈1,q0. Two constants C11,C12>0 (that are unconnected to q) can be singled out such that
(340)∣wn;qu−wn;1u∣≤q−1C11C12nn!∣u∣expk1log2∣u∣+u0+αlog∣u∣+u0,for all u∈U∪0, all integers n≥0, where the constants k1>0, u0>1, α≥0 are fixed in Proposition 33.


### 5.6. Error Bounds between the Analytic Solutions of the Nonlinear Auxiliary Cauchy Problems (32), (33) and (185), (186)

This subsection is devoted to the expounding of the next proposition which plays a central role in the upcoming third main result of this work discussed in Theorem44.Proposition 41.
Letq∈1,q0. Then, one can select two constants Z⌣0>0 and M⌣1>0 that are unrelated to q, such that for any given σ>0, the next bounds
(341)Θ;1u,z−Θ;qu,zσ,Z⌣0,U≤M⌣1Z⌣0κq−1,hold, where Θ;q stands for the solution to the Cauchy problem (32), (33) that belongs to the space Gσ,Z0,Uk for some Z0>0 (relying on q0 but independent of q) built up in Proposition 15 and where Θ;1 solves the problem (185), (186) and is exhibited in Proposition 34.Proof.
The proof is split up is two main parts.
In the first part, we frame a Cauchy problem specified later on in (253), (254) which is fulfilled by the difference(342)Δ1;qΘu,z≔Θ;1u,z−Θ;qu,z,for which we seek upper bounds.
To that end, we need some prelusive computations. The basic identityab−cd=a−cb+cb−d will help us in recasting in an appropriate manner the next list of differences appearing as building blocks of the pending equation (347). The differences dealing with the linear terms can be reorganized by inserting some auxiliary terms as
(343)kukh1∂zh2Θ;1u,z−kq−h3ukh1∂zh2Θ;qq−h3u,z=kukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Θ;1u,z−∂zh2Θ;qq−h3u,z=kukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Θ;1u,z−∂zh2Θ;1q−h3u,z+∂zh2Θ;1q−h3u,z−∂zh2Θ;qq−h3u,z,and
(344)uk∫0ukuk−sh0/k−1ksh1∂zh2Θ;1s1/k,zdss−uk∫0ukuk−sh0/k−1kq−h3s1/kkh1∂zh2Θ;qq−h3s1/k,zdss=uk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Θ;1s1/k,z−∂zh2Θ;qq−h3s1/k,zdss=uk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Θ;1s1/k,z−∂zh2Θ;1q−h3s1/k,z+∂zh2Θ;1q−h3s1/k,z−∂zh2Θ;qq−h3s1/k,zdss,along with the differences for the nonlinear terms that can be reshaped as
(345)uk∫0ukΘ;1uk−s1/k,zΘ;1s1/k,zdsuk−ss−uk∫0ukΘ;quk−s1/k,zΘ;qs1/k,zdsuk−ss=uk∫0ukΘ;1uk−s1/k,z−Θ;quk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,zΘ;1s1/k,z−Θ;qs1/k,zdsuk−ss,and
(346)uk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss−uk∫0ukuk−sh/k−1s∫0sΘ;qs−s11/k,zΘ;qs11/k,zds1s−s1s1dss=uk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,z−Θ;qs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,zΘ;1s11/k,z−Θ;qs11/k,zds1s−s1s1dss.
Owing to the fact thatΘ;1 (resp. Θ;q) solves the linear convolution Cauchy problem (185), (186) (resp. (32), (33)), paying regard to the redrawn expressions (343), (344), (345), and (346) and keeping in mind the notation (199) of the previous Section 5.4, we can state the linear convolution Cauchy problem fulfilled by the difference Δ1;qΘu,z as follows
(347)∂zκΔ1;qΘu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Δh3Θ;1u,z+∂zh2Δ1;qΘq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kuk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂zh2Δ1;qΘq−h3s1/k,zdss+a0zuk∫0ukΔ1;qΘuk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,zΔ1;qΘs1/k,zdsuk−ss+∑h=1Aahz1Γh/kuk∫0ukuk−sh/k−1s∫0sΔ1;qΘs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,zΔ1;qΘs11/k,zds1s−s1s1dss+w;1u,z−w;qu,z,for prescribed vanishing Cauchy data
(348)∂zjΔ1;qΘu,0≡0,0≤j≤κ−1
In the ensuing part, we intend to solve the above problem (253), (254) by way of the Banach spaces introduced in Subsection3.1. Namely, we search for a solution to (253), (254) shaped as
(349)Δ1;qΘu,z=∂z−κΔ1;qΞu,z,for some expression Δ1;qΞu,z to be specified. One checks that Δ1;qΘu,z matches the problem (253), (254) if the map Δ1;qΞu,z is subjected to the next fixed point equation
(350)Δ1;qΞu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Δh3Θ;1u,z+∂z−κ−h2Δ1;qΞq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kuk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂z−κ−h2Δ1;qΞq−h3s1/k,zdss+a0zuk∫0uk∂z−κΔ1;qΞuk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,z∂z−κΔ1;qΞs1/k,zdsuk−ss+∑h=1Aahz1Γh/kuk∫0ukuk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dss+w;1u,z−w;qu,z.
In the sequel, we seek for a solution, for which sharp bounds relatively toq are exhibited, to (256) inside the Banach space Gσ,Z⌣0,Uk, for any prescribed σ>0 in the condition that Z⌣0>0 remains small enough. In order to meet this objective, we set up the next linear mapping
(351)EΔ1;qΞτ,z≔∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkτkh1−kq−h3τkh1∂zh2Θ;1τ,z+kq−h3τkh1∂zh2Δh3Θ;1τ,z+∂z−κ−h2Δ1;qΞq−h3τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kτk∫0τkτk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂z−κ−h2Δ1;qΞq−h3s1/k,zdss+a0zτk∫0τk∂z−κΔ1;qΞτk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞs1/k,zdsτk−ss+∑h=1Aahz1Γh/kτk∫0τkτk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dss+w;1τ,z−w;qτ,z.
In the following lemma, sufficient conditions are enunciated under whichE becomes a shrinking map on a tiny ball whose radius hinges on q centered at the origin in the space Gσ,Z⌣0,Uk.Lemma 42.
Assume that the condition (30) holds. Then, a small sized real number χ⌣>0 can be singled out in a way that if 0<Z⌣0<χ⌣, a radius υ⌣>0 can be distinguished (in an unrelated manner to q, but depending on q0 for which 1<q≤q0), such that E acquires the next two hallmarks: let us denote Bυ⌣q−1 the ball centered at 0 with radius υ⌣q−1 in Gσ,Z⌣0,Uk,
(1)
E maps Bυ⌣q−1 into itself, signifying that
(352)EBυ⌣q−1⊂Bυ⌣q−1.(2)
The inequality(353)EΔ1;qΞ1−EΔ1;qΞ2σ,Z⌣0,U≤12Δ1;qΞ1−Δ1;qΞ2σ,Z⌣0,U,holds as long as Δ1;qΞ1,Δ1;qΞ2∈Bυ⌣q−1.Proof.
We first supply norm upper estimates for some parts of the mapE.
(a)
Upper bounds are established for the normw;1τ,z−w;qτ,zσ,Z⌣0,U. Departing from the expressions (258) and (339), we arrive at
(354)w;1τ,z−w;qτ,zσ,Z⌣0,U=∑n≥0wn;1τ−wn;qτn,σ,UZ⌣0nn!,and owing to the bounds (247) stirred up in Proposition 40, the coefficients of the above series can be upper bounded by
(355)wn;1τ−wn;qτn,σ,U=supτ∈U1+τ2k∣τ∣exp−σrbnτk∣wn;1τ−wn;qu∣≤q−1C11C12nn!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0≤q−1C⌣11C12nn!,for some constant C⌣11>0 (unrelated to q), for any integers n≥0. Thereby, it brings in
(356)w;1τ,z−w;qτ,zσ,Z⌣0,U≤q−1C⌣11∑n≥0C12Z⌣0n≤2q−1C⌣11,on the condition that Z⌣0<1/2C12.(b)
Bearing in mind (217), we already know that(357)τkh1∂zh2Θ;1τ,zσ,Z⌣0,U≤M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U,for some constant M1>0 (independent of q but relying on q0), where Z0;1,υ;1>0 are defined in Proposition 34, as long as Z⌣0<Z0;1.(c)
We need accurate bounds for the normsτkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U, for nonnegative integers h1,h2,h3 submitted to the condition (30). Calling to mind Lemma 39, the next splitting
(358)Δh3Θ;1τ,z=∂z−κΔh3Ξ;1τ,z+Δh3Ψ⌣τ,z,holds where Δh3Ξ;1τ,z belongs to the ball Bυ~1−q−h3 in Gσ,Z~0,Uk for some constants Z~0,υ~>0 and where Δh3Ψ⌣τ,z is defined in (211). Owing to (225), we can set a constant Nk,h1,h2,κ,Z~0>0 with
(359)τkh1∂zh2Δh3Ψ⌣τ,zσ,Z~0,U≤1−q−h3Nk,h1,h2,κ,Z~0.
Besides, according to Proposition6, we get a constant M1>0 (independent of q) such that
(360)τkh1∂z−κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M1Z~0κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M1Z~0κ−h2υ~1−q−h3.
Thereupon, we deduce from (263) and (264) the awaited bounds(361)τkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U≤τkh1∂z−κ−h2Δh3Ξ;1τ,zσ,Z~0,U+τkh1∂zh2Δh3Ψ⌣τ,zσ,Z~0,U≤M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0,provided that Z⌣0<Z~0.
(d)
Keeping in mind (226), we observe that(362)τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z⌣0,U≤M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U,as long as that Z⌣0<Z~0.(e)
Precise bounds for the norms(363)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U,are required. Owing to the bounds (229), we can exhibit a constant N¯k,h0,h1,h2,κ,Z~0>0 with
(364)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Ψ⌣s1/k,zdssσ,Z~0,U≤1−q−h3N¯k,h0,h1,h2,κ,Z~0,and according to Proposition 10, we get a constant M3>0 (unrelated to q) with
(365)τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Δh3Ξ;1s1/k,zdssσ,Z~0,U≤M3Z~0κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M3Z~0κ−h2υ~1−q−h3.
Thus, on grounds of (267) and (268), we reach the due bounds(366)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U≤τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Δh3Ξ;1s1/k,zdssσ,Z~0,U+τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Ψ⌣s1/k,zdssσ,Z~0,U≤M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0,whenever Z⌣0<Z~0.
(f)
We also need the bounds(367)Θ;1τ,zσ,Z⌣0,U≤M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U,which is a particular case of (261) and according to Proposition 15, one can select constants M1,Z0,υ>0 (unrelated to q but depending on q0) with
(368)Θ;qτ,zσ,Z⌣0,U≤Θ;qτ,zσ,Z0,U≤M1Z0κυ+Ψ⌣τ,zσ,Z0,U,whenever Z⌣0<Z0.
We are now in position to reach the heart of the proof. We focus on the first attribute 1. We setΔ1;qΞτ,z in Gσ,Z⌣0,Uk with
(369)Δ1;qΞτ,zσ,Z⌣0,U≤υ⌣q−1.
We display explicit bounds for each piece of the mapEΔ1;qΞ.
Proposition11 and (261) beget
(370)dh¯zkτkh1−kq−h3τkh1∂zh2Θ;1τ,zσ,Z⌣0,U≤kh11−q−h3kh1dh¯Z⌣0×M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U.
Proposition11 and (265) yield
(371)dh¯zkq−h3τkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1×M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0.
Propositions6 and 11 trigger
(372)dh¯zkq−h3τkh1∂z−κ−h2Δ1;qΞq−h3τ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1M1Z⌣0κ−h2Δ1;qΞτ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1M1Z⌣0κ−h2υ⌣q−1.
Proposition11 and (266) prompt
(373)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh11−q−h3kh1M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U.
Proposition11 and (269) return
(374)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0.
Propositions10 and 11 furnish
(375)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Δ1;qΞq−h3s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2Δ1;qΞτ,zσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2υ⌣q−1.
Propositions6, 8, and 11 with the help of (270) and (271) promote
(376)a0zτk∫0τk∂z−κΔ1;qΞτk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞs1/k,zdsτk−ssσ,Z⌣0,U≤∣a0∣Z⌣0M2∂z−κΔ1;qΞτ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞτ,zσ,Z⌣0,U≤∣a0∣Z⌣0M2M1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,U≤∣a0∣Z⌣0M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κυ⌣q−1.
Propositions6, 8, 10, and 11 together with (270), (271) spark off
(377)ahz1Γh/kτk∫0τkτk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dssσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3τk∫0τk∂z−κΔ1;qΞτk−s11/k,zΘ;1s11/k,z+Θ;qτk−s11/k,z∂z−κΔ1;qΞs11/k,zds1τk−s1s1σ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2∂z−κΔ1;qΞτ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞτ,zσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κυ⌣q−1.
From now onwards, we choose a small sized quantityZ⌣0>0 and proper υ⌣>0 (taken freely from q∈1,q0) in a way that the next inequality holds
(378)∑h¯=h0,h1,h2,h3∈C;h0=0kh11−q−h3kh1∣dh¯∣Z⌣0×M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U+∣dh¯∣Z⌣0kh1q−h3kh1×M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0+∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2υ⌣q−1+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z⌣01Γh0/kkh11−q−h3kh1M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U+∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0+∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2υ⌣q−1+∣a0∣Z⌣0M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+∑h=1A∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κυ⌣q−1×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+2q−1C⌣11≤υ⌣q−1.
Observe that the above constraint (281) is achievable since all the quantities(379)1−q−h3q−1=1q1−1/qh31−1/q=1q∑j=0h3−11/qj,1−q−h3kh1q−1=1q1−1/qh3kh11−1/q=1q∑j=0h3kh1−11qj,remain bounded whenever q∈1,q0, for given integers h3≥1, h3kh1≥1.
Eventually, piling up the inequalities (260), (273), (274), (275), (276), (277), (278), (279), and (280) reached above under the condition (378) produces the awaited item (258).
We focalize on the second aspect 2. of the mapE. Let Δ1;qΞ1,Δ1;qΞ2 be elements of Gσ,Z⌣0,Uk such that
(380)Δ1;qΞjτ,zσ,Z⌣0,U≤υ⌣q−1,for j=1,2.
The estimates involved in the first property 1. ofE enable us to write down the next set of inequalities
(381)dh¯zkq−h3τkh1∂z−κ−h2Δ1;qΞ1q−h3τ,z−Δ1;qΞ2q−h3τ,zσ,Z⌣0,U≤∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2Δ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,and
(382)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Δ1;qΞ1q−h3s1/k,z−Δ1;qΞ2q−h3s1/k,zdssσ,Z⌣0,U≤∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2Δ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,
along with(383)a0zτk∫0τk∂z−κΔ1;qΞ1τk−s1/k,z−Δ1;qΞ2τk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞ1s1/k,z−Δ1;qΞ2s1/k,zdsτk−ssσ,Z⌣0,U≤a0Z⌣0M2∂z−κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U≤a0Z⌣0M2M1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,and
(384)ahz1Γh/kτk∫0τkτk−sh/k−1×s∫0s∂z−κΔ1;qΞ1s−s11/k,z−Δ1;qΞ2s−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞ1s11/k,z−Δ1;qΞ2s11/k,zds1s−s1s1dssσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U.
From this point forward, we pick upZ⌣0>0 close enough to 0 in a way that
(385)∑h¯=h0,h1,h2,h3∈C;h0=0∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2+∣a0∣Z⌣0M2M1Z⌣0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+∑h=1A∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U≤12.
The collection of bounds (282), (283), (284), and (285) contingent upon (286) sparks off the second feature (259).
At last, we single out the constantsZ⌣0 and υ⌣>0 in a way that both constraints (281), (286) occur unitedly. Lemma 42 follows.
In the lemma to come, the linear Cauchy problem (253), (254) is solved within the Banach spaceGσ,Z⌣0,Uk.Lemma 43.
Take for granted that the condition (30) hold. We fix the constants Z⌣0>0 and υ⌣>0 (independently of q in 1,q0) as in Lemma 42. Then, the linear Cauchy problem (253), (254) carries a solution Δ1;qΘu,z that belongs to the Banach space Gσ,Z⌣0,Uk for any given σ>0. Along with it, a constant M⌣1>0 (unrelated to q∈1,q0) can be distinguished with the next property
(386)Δ1;qΘu,zσ,Z⌣0,U≤M⌣1Z⌣0κq−1.Proof.
Based on Lemma42, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map E:Bυ⌣q−1⟶Bυ⌣q−1 due to the fact that Bυ⌣q−1,d represents a complete metric space for the distance dx,y=x−yσ,Z⌣0,U. Thus, E:Bυ⌣q−1⟶Bυ⌣q−1 gets a unique fixed point denoted Δ1;qΞu,z inside the ball Bυ⌣q−1, meaning that
(387)EΔ1;qΞ=Δ1;qΞ.
Thereby, a unique solutionΔ1;qΞ for the equation (350) is confirmed in the ball Bυ⌣q−1. Furthermore, Proposition 6 conjointly with the decomposition (255) warrant that the map
(388)Δ1;qΘu,z=∂z−κΔ1;qΞu,z,belongs to Gσ,Z⌣0,Uk, solves the problem (253), (254) and is submitted to the next upper estimates
(389)Δ1;qΘu,zσ,Z⌣0,U≤M1Z⌣0κΔ1;qΞu,zσ,Z⌣0,U≤M1Z⌣0κυ⌣q−1≤M⌣1Z⌣0κq−1,for some constant M⌣1>0 which is unattached to q in the range 1,q0.□.
According to Lemma43, it turns out that the unique formal series in z with holomorphic coefficients on U solution of (253), (254) suffers the bounds (287). Since the difference Θ;1u,z−Θ;qu,z, which represents in particular a formal power series in z with holomorphic coefficients on U, is shown to solve (253), (254) in the first step of Proposition 41, it must coincide with the solution Δ1;qΘu,z constructed above in the second step and we arrive at the conclusion that the for sought estimates (248) necessarily hold for it. The proof of the proposition is completed.


### 5.7. Confluence for the Analytic Solutions of the Problem (22), (23) asq⟶1

In this subsection, we unveil the third and last main result of the work.Theorem 44.
LetD¯=T¯,U¯ be the admissible set of sectors distinguished in Proposition 33. We denote U the unbounded sector singled out in Proposition 33 and T its corresponding bounded sector in accordance with the requirement (2) of Definition 2.
We denotev;qt,z the bounded holomorphic solution to (22), (23) on the product T×DZ0/2, given by a Laplace transform of order k, see (88), constructed in Theorem 16. Besides, we consider the bounded holomorphic solution v;1t,z of the nonlinear limit problem (182), (183) on the domain T×DZ0;1/2 expressed through a Laplace transform of order k, see (188), built up in Proposition 34.
Then, a constantC>0 (unrelated to q∈1,q0) can be found such that
(390)supt∈T,z∈DZ⌣0/2∣v;1t,z−v;qt,z∣≤Cq−1,for all q∈1,q0. In other words, the solution v;qt,z of (22), (23) merges uniformly on T×DZ⌣0/2 to the solution v;1t,z of (182), (183) as q⟶1.Proof.
We express both solutionsv;qt,z and v;1t,z as Laplace transforms
(391)v;qt,z=k∫LγΘ;qu,zexp−utkduu,v;1t,z=k∫LγΘ;1u,zexp−utkduu.along a halfline Lγ=0,+∞e−1γ⊂U∪0 assigned to the condition
(392)coskγ−argt>Δ,for some fixed constant Δ>0, provided that t∈T, where the Borel map Θ;qu,z is outlined in (89) and Θ;1u,z is described in (189) whose Taylor expansion can be displayed as
(393)Θ;1u,z=∑β≥0Θ;1,βuzββ!,for u,z∈U×DZ0;1/2.
The deviations bounds reached in Proposition41 can be rephrased in the next explicit way
(394)Θ;1u,z−Θ;qu,z≤∑β≥0Θ;1,βu−Θ;q,βuzββ!≤M⌣1Z⌣0κq−1∣u∣1+u2k∑β≥0expσrbβuk∣z∣Z⌣0β≤2M⌣1Z⌣0κq−1∣u∣expσζbuk,for all u∈U∪0, provided that z∈DZ⌣0/2.
Thereby, we can govern the differencev;1t,z−v;qt,z in the following manner
(395)v;1t,z−v;qt,z=k∫LγΘ;1u,z−Θ;qu,zexp−utkduu≤2M⌣1Z⌣0κq−1∫0+∞expσζbrkexp−rrTkΔdr,for all t∈T, all z∈DZ⌣0/2, bearing in mind that radius rT of T fulfills
(396)rT<Δσζb1/k,according to (87). This achieves the expected bounds (289).


## 5.1. A Limit Singular Linear Cauchy Problem

In this subsection, we call attention to parts of the results reached in our previous work [1] that will be applied within the next subsection. We keep the same notations as the ones introduced in Section 2.1.We consider the next singular Cauchy problem(255)Ptk+1∂t∂zSu;1t,z=∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2u;1t,z,for given Cauchy data
(256)∂zju;1t,0=φjt,0≤j≤S−1.where all the data k,S,P,A and the coefficients cl¯z with l¯∈A along with the initial data φjt for 0≤j≤S−1 are already declared in Section 2.1.In Section 8.1 of the paper [1], the next statement is outlined.Proposition 33.
LetD¯=T¯,U¯ be an admissible set of sectors as chosen in Definition 2. Let U be one sector belonging to the family of unbounded sectors U¯.
One can build up a solutionu;1t,z to the Cauchy problem (177), (178) which is bounded holomorphic on a domain T×D1/2C8 where the bounded sector T belongs to the family T¯ of bounded sectors from D¯ and corresponds to U under the requirement (2) of Definition 2 and where C8>0 is some well-chosen constant. The map u;1t,z can be expressed by means of a Laplace transform of order k,
(257)u;1t,z=k∫Lγw;1u,zexp−utkduu,along a halfline Lγ⊂U∪0 described in Definition 2 (2), for all t,z∈T×D1/2C8.
The Borel mapw;1u,z represents a holomorphic function on U×D1/2C8. Its Taylor expansion
(258)w;1u,z=∑n≥0wn;1uznn!,suffers the next bounds
(259)wn;1u≤C7C8nn!uexpk1log2u+u0+αlogu+u0,for all n≥0, all u∈U∪0, for fittingly chosen constants C7,k1>0 and u0>1, α≥0.


## 5.2. A Limit Nonlinear Cauchy Problem

In this subsection, a novel Cauchy problem is introduced that we call the limit problem asq>1 tends to 1. Its shape is displayed as follows
(260)∂zκv;1t,z=∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2v;1t,z+at,zv;12t,z+u;1t,z,for given Cauchy data
(261)∂zjv;1t,0=φ⌣jt,0≤j≤κ−1,where the forcing term u;1t,z is the holomorphic solution of the singular linear Cauchy problem (177), (178) unveiled in Proposition 33 of the former subsection. Besides, all the items κ,k,C along with the coefficients dh¯z, at,z for h¯∈C and the Cauchy data φ⌣jt for 0≤j≤κ−1 are those already introduced in Section 2.2.We aim for a solution to (182), (183) having the profile of a Laplace transform of orderk,
(262)v;1t,z=k∫LγΘ;1u,zexp−utkduu,along the halfline Lγ given in (179), where the Borel map Θ;1u,z is holomorphic with respect to u on U and analytic w.r.t z on some small disc Dr centered at 0 with radius r>0.The same computations as the ones of Section2.3 by means of Proposition 4 shows that the Borel map Θ;1u,z solves that next auxiliary Cauchy problem
(263)∂zκΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1∂zh2Θ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zukΓh0/k∫0ukuk−sh0/k−1ksh1∂zh2Θ;1s1/k,zdss+a0zuk∫0ukΘ;1uk−s1/k,zΘ;1s1/k,z1uk−ssds+∑h=1AahzukΓh/k∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,z1s−s1s1ds1dss+w;1u,z,for given Cauchy data
(264)∂zjΘ;1u,0=P⌣ju,0≤j≤κ−1.For later need, we make the change of variabless=ukx and s1=sx1 for 0≤x,x1≤1 in the integrals involved in (185). As a result, Θ;1u,z solves that next auxiliary Cauchy problem
(265)∂zκΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1∂zh2Θ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zuh0+kh1Γh0/k∫011−xh0/k−1kh1xh1∂zh2Θ;1ux1/k,zdxx+a0z∫01Θ;1u1−x1/k,zΘ;1ux1/k,zdx1−xx+∑h=1AahzuhΓh/k∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+w;1u,z,for given Cauchy data (186).The next proposition comes along with the same steps as in the proofs of Proposition15 and Theorem 16.Proposition 34.
LetD¯=T¯,U¯ be the admissible set of sectors distinguished in Proposition 33. Let U be the infinite sector belonging to U¯ selected in Proposition 33.
A solutionv;1t,z to the Cauchy problem (182), (183) can be reached, that is bounded and holomorphic on a domain T×DZ0;1/2 for some small radius Z0;1>0, where T is the bounded sector from T¯ singled out in Proposition 33. Moreover, v;1t,z is described using a Laplace transform of order k,
(266)v;1t,z=k∫LγΘ;1u,zexp−utkduu,for all t,z∈T×DZ0;1/2, along the halfline Lγ that appears in the representation (179). The Borel map Θ;1u,z stands for a holomorphic map on the domain U×DZ0;1/2 which solves the above auxiliary Cauchy problem (185), (187), and (186). Besides, the map Θ;1u,z belongs to the space Gσ,Z0;1,Uk and enjoys a decomposition of the form
(267)Θ;1u,z=∂z−κΞ;1u,z+Ψ⌣u,z,where Ψ⌣u,z is defined in (63) and Ξ;1u,z belongs to Gσ,Z0;1,Uk and satisfies
(268)Ξ;1u,zσ,Z0;1,U≤υ;1,for some suitable constant υ;1>0.In the ensuing corollary, we observe thatv;1t,z actually solves a limit nonlinear Cauchy problem with analytic coefficients in space z in the vicinity of the origin and polynomial in time t.Corollary 35.
The analytic mapv;1t,z solves on the product T×DZ0;1/2 a particular Cauchy problem relying polynomially on time t with the shape
(269)Ptk+1∂t∂zS+κv;1t,z=G⌣t,z,∂tr1∂zr2v;1t,zr¯=r1,r2∈R⌣,for given Cauchy data of the form
(270)∂zjv;1t,0=φ⌣jt,0≤j≤κ−1,together with
(271)∂zjv;1t,0=ϕ⌣j;1t,κ≤j≤κ+S−1,for well selected polynomials ϕ⌣j;1t, κ≤j≤κ+S−1 (that are independent from q). The set R⌣ is a finite subset of ℕ2 for r1,r2∈R⌣ with the property that r2<S+κ for any r1,r2∈R⌣. The map G⌣t,z,ur¯r¯∈R⌣ is polynomial in time t and in its arguments ur¯ with r¯∈R⌣ and analytic w.r.t z on DZ0;1/2.Proof.
We set up the differential operators(272)D1t,z,∂t,∂z≔Ptk+1∂t∂zS−∑l¯=l0,l1,l2,l3∈Acl¯ztl0tk+1∂tl1∂zl2,and
(273)D2t,z,∂t,∂zv≔∂zκ−∑h¯=h0,h1,h2,h3∈Cdh¯zth0tk+1∂th1∂zh2v−at,zv2.
According to Proposition34, v;1t,z satisfies
(274)D2t,z,∂t,∂zv;1t,z=u;1t,z,and based on Proposition 33, the map u;1t,z solves
(275)D1t,z,∂t,∂zu;1t,z=0,provided that t,z∈T×DZ0;1/2. By pairing (194) together with (195), we obtain
(276)D1t,z,∂t,∂z∘D2t,z,∂t,∂zv;1t,z=0,that is represented by the equation (269). Regarding the Cauchy data, the map v;1 suffers the conditions (261) which are replicated in (192). Moreover, the constraints (178) put on u;1t,z can be reworded as
(277)∂zj∘D2t,z,∂t,∂zv;1t,0=φjt,for 0≤j≤S−1 which begets the assumptions (193) for properly chosen polynomials ϕ⌣l;1t with complex coefficients for κ≤l≤κ+S−1.


## 5.3. Analytic Solutions to the Limit Singular Linear Cauchy Problem under the Action of aq−Difference Operator

In this subsection, we take heed of a technical result achieved in our foregoing work [1] that will be called upon in the next subsection. We keep the notations of Subsection 5.1.In Section 8.2 of [1], the next result is established.Proposition 36.
Letβ≥1 be an integer and let q∈1,q0. Two constants C9,C10>0 (that are unrelated to q) can be found such that
(278)∣wn;1u−wn;1q−βu∣≤∣q−β−1∣C9C10nn!∣u∣expk1log2∣u∣+u0+αlog∣u∣+u0,for all u∈U∪0, for all integers n≥0, where the constants k1>0, u0>1, α≥0, and the unbounded sector U appear in Proposition 33.


## 5.4. Analytic Solutions to the Limit Nonlinear Cauchy Problem under the Action of aq−Difference Operator

This subsection is dedicated to the proof of the next technical proposition.Proposition 37.
Letβ≥1 be an integer and set q∈1,q0. Then, one can find some constants υ;1>0 and M~1;1>0 (unrelated to q) such that for any given σ>0, if Z~0>0 is taken small enough, the next bounds
(279)Θ;1u,z−Θ;1q−βu,zσ,Z~0,U≤M~1;11−q−β,hold.Proof.
The proof is rather lengthy and is made up with several steps.
In the first step of the proof, we formulate a Cauchy problem, stated below in (208), (209), that the difference(280)ΔβΘ;1u,z≔Θ;1u,z−Θ;1q−βu,z,is compelled to solve. We first state a Cauchy problem satisfied by the quantity Θ;1q−βu,z by substituting u by q−βu in the equations (265), (186).
(281)∂zκΘ;1q−βu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkq−βukh1∂zh2Θ;1q−βu,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zq−βuh0+kh1Γh0/k∫011−xh0/k−1kh1xh1∂zh2Θ;1q−βux1/k,zdxx+a0z∫01Θ;1q−βu1−x1/k,zΘ;1q−βux1/k,zdx1−xx+∑h=1Aahzq−βuhΓh/k∫011−xh/k−1∫01Θ;1q−βux1/k1−x11/k,zΘ;1q−βux1/kx11/k,zdx11−x1x1dxx+w;1q−βu,z,∂zjΘ;1q−βu,0=P⌣jq−βu,0≤j≤κ−1.
On the way, we need to perform some practical computations. Namely, using the basic identityab−cd=a−cb+cb−d, we recast the next list of pieces in a suitable manner:
(282)kukh1∂zh2Θ;1u,z−kq−βukh1∂zh2Θ;1q−βu,z=kukh1−kq−βukh1∂zh2Θ;1u,z+kq−βukh1∂zh2Θ;1u,z−∂zh2Θ;1q−βu,z,and
(283)uh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx−q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1q−βux1/k,zdxx=uh0+kh1−q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx+q−βuh0+kh1∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,z−∂zh2Θ;1q−βux1/k,zdxx,along with
(284)∫01Θ;1u1−x1/k,zΘ;1ux1/k,zdx1−xx−∫01Θ;1q−βu1−x1/k,zΘ;1q−βux1/k,zdx1−xx=∫01Θ;1u1−x1/k,z−Θ;1q−βu1−x1/k,zΘ;1ux1/k,z+Θ;1q−βu1−x1/k,zΘ;1ux1/k,z−Θ;1q−βux1/k,zdx1−xx,and
(285)uh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx−q−βuh∫011−xh/k−1∫01Θ;1q−βux1/k1−x11/k,zΘ;1q−βux1/kx11/k,zdx11−x1x1dxx=uh−q−βuh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+q−βuh∫011−xh/k−1∫01Θ;1ux1/k1−x11/k,z−Θ;1q−βux1/k1−x11/k,z×Θ;1ux1/kx11/k,z+Θ;1q−βux1/k1−x11/k,z×Θ;1ux1/kx11/k,z−Θ;1q−βux1/kx11/k,zdx11−x1x1dxx.
Based on the two equations (265), (200) together with the prescribed data (186), (201) and thanks to the above computations (202), (203), (204), (205), we can now exhibit a Cauchy problem fulfilled by the difference ΔβΘ;1u,z given in (199) owning the following shape:
(286)∂zκΔβΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−βukh1∂zh2Θ;1u,z+kq−βukh1∂zh2ΔβΘ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/kuh0+kh1−q−βuh0+kh1×∫011−xh0/k−1xh1∂zh2Θ;1ux1/k,zdxx+q−βuh0+kh1∫011−xh0/k−1xh1∂zh2ΔβΘ;1ux1/k,zdxx+a0z∫01ΔβΘ;1u1−x1/k,zΘ;1ux1/k,z+Θ;1q−βu1−x1/k,zΔβΘ;1ux1/k,zdx1−xx+∑h=1Aahz1Γh/kuh−q−βuh∫011−xh/k−1×∫01Θ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,zdx11−x1x1dxx+q−βuh∫011−xh/k−1∫01ΔβΘ;1ux1/k1−x11/k,zΘ;1ux1/kx11/k,z+Θ;1q−βux1/k1−x11/k,zΔβΘ;1ux1/kx11/k,zdx11−x1x1dxx+w;1u,z−w;1q−βu,z,for given Cauchy data
(287)∂zjΔβΘ;1u,0=P⌣ju−P⌣jq−βu,0≤j≤κ−1.
In the upcoming step, we plan to solve the problem (206), (207) within the Banach space described in Subsection3.1. In the first instance, we need to rephrase the integral operators involved in (206) in terms of those appearing in Subsection 3.1 by means of the parametrization s=ukx, s1=sx1 for 0≤x,x1≤1. Indeed, we deduce that ΔβΘ;1u,z turns out to solve the next linear Cauchy problem
(288)∂zκΔβΘ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1ukh1∂zh2Θ;1u,z+kq−βukh1∂zh2ΔβΘ;1u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×uk∫0ukuk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1uk∫0ukuk−sh0/k−1sh1∂zh2ΔβΘ;1s1/k,zdss+a0zuk∫0ukΔβΘ;1uk−s1/k,zΘ;1s1/k,z+Θ;1q−βuk−s1/k,zΔβΘ;1s1/k,zdsuk−ss+∑h=1Aahz1Γh/k1−q−βhuk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhuk∫0ukuk−sh/k−1s∫0sΔβΘ;1s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,zΔβΘ;1s11/k,zds1s−s1s1dss+w;1u,z−w;1q−βu,z,for assigned Cauchy data
(289)∂zjΔβΘ;1u,0=P⌣ju−P⌣jq−βu,0≤j≤κ−1.
In the second step of the proof, we seek for a solution to the above problem (208), (209) expressed by means of the next splitting(290)ΔβΘ;1u,z=∂z−κΔβΞ;1u,z+ΔβΨ⌣u,z,for some expression ΔβΞ;1u,z to be determined, where
(291)ΔβΨ⌣u,z=∑j=0κ−1ΔβP⌣juzjj!,for ΔβP⌣ju=P⌣ju−P⌣jq−βu, with 0≤j≤κ−1.
We pinpoint the crucial fact thatΔβΘ;1u,z solves the problem (208), (209) if the quantity ΔβΞ;1u,z satisfies the next fixed point equation
(292)ΔβΞ;1u,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1ukh1∂zh2Θ;1u,z+kq−βukh1∂z−κ−h2ΔβΞ;1u,z+∂zh2ΔβΨ⌣u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×uk∫0ukuk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1uk∫0ukuk−sh0/k−1sh1∂z−κ−h2ΔβΞ;1s1/k,z+∂zh2ΔβΨ⌣s1/k,zdss+a0zuk∫0uk∂z−κΔβΞ;1uk−s1/k,z+ΔβΨ⌣uk−s1/k,zΘ;1s1/k,z+Θ;1q−βuk−s1/k,z∂z−κΔβΞ;1s1/k,z+ΔβΨ⌣s1/k,zdsuk−ss+∑h=1Aahz1Γh/k1−q−βhuk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhuk∫0ukuk−sh/k−1s∫0s∂z−κΔβΞ;1s−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞ;1s11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dss+w;1u,z−w;1q−βu,z.
Our next duty will be to search for a solution and provide bounds relatively toq for a solution to (212) in the Banach space Gσ,Z~0,Uk, for any given σ>0, provided that Z~0>0 is chosen small enough. With that in mind, we introduce the linear map
(293)DΔβΞτ,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkh11−q−βkh1τkh1∂zh2Θ;1τ,z+kq−βτkh1∂z−κ−h2ΔβΞτ,z+∂zh2ΔβΨ⌣τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯zkh1Γh0/k1−q−βh0+kh1×τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdss+q−βh0+kh1τk∫0τkτk−sh0/k−1sh1∂z−κ−h2ΔβΞs1/k,z+∂zh2ΔβΨ⌣s1/k,zdss+a0zτk∫0τk∂z−κΔβΞτk−s1/k,z+ΔβΨ⌣τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞs1/k,z+ΔβΨ⌣s1/k,zdsτk−ss+∑h=1Aahz1Γh/k1−q−βhτk∫0τkτk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss+q−βhτk∫0τkτk−sh/k−1s∫0s∂z−κΔβΞs−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dss+w;1τ,z−w;1q−βτ,z.
In the next lemma, we discuss sufficient conditions under whichD acts as a shrinking map on a small ball centered at 0, whose radius depends on q, in the space Gσ,Z~0,Uk.Lemma 38.
We take for granted that the conditions (30) hold. Then, one can single out a small real number χ~>0 in a way that if 0<Z~0<χ~, one can select a radius υ~>0 which is independent of q (but relies on q0 such that 1<q≤q0), such that D possesses the next two qualities: let Bυ~1−q−β be a ball centered at 0 with radius υ~1−q−β(1)
D maps Bυ~1−q−β into itself, meaning that
(294)DBυ~1−q−β⊂Bυ~1−q−β.(2)
For anyΔβΞ1,ΔβΞ2∈Bυ~1−q−β, we have
(295)DΔβΞ1−DΔβΞ2σ,Z~0,U≤12ΔβΞ1−ΔβΞ2σ,Z~0,U.Proof.
As a prefatory material, norm bounds estimates are required for some pieces of the mapD.
(a)
Indeed, at first we need estimates for the normw;1τ,z−w;1q−βτ,zσ,Z~0,U. According to the expansion (180) and the definition of the norm, we get
(296)w;1τ,z−w;1q−βτ,zσ,Z~0,U=∑n≥0wn;1τ−wn;1q−βτn,σ,UZ~0nn!,and owing to the bounds (197) brought to mind in Proposition 36, we can upper bound the coefficients of this latter series by
(297)wn;1τ−wn;1q−βτn,σ,U=supτ∈U1+τ2k∣τ∣exp−σrbnτkwn;1τ−wn;1q−βτ≤1−q−βC9C10nn!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0≤1−q−βC~9C10nn!,for some constant C~9 (unrelated to q), for all integers n≥0. Thereupon, it leads to
(298)w;1τ,z−w;1q−βτ,zσ,Z~0,U≤1−q−βC~9∑n≥0C10Z~0n≤21−q−βC~9,provided that Z~0<1/2C10.(b)
We focus on the quantitiesτkh1∂zh2Θ;1τ,zσ,Z~0,U for integers h1,h2≥0 under the constraint (20) and on Θ;1q−βτ,zσ,Z~0,U
According to the decomposition (189) with the bounds (190) and owing to Propositions6 and 12, we obtain
(299)τkh1∂zh2Θ;1τ,zσ,Z~0,U≤τkh1∂zh2Θ;1τ,zσ,Z0;1,U≤τkh1∂z−κ−h2Ξ;1τ,zσ,Z0;1,U+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U≤M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U,along with
(300)Θ;1q−βτ,zσ,Z~0,U≤Θ;1q−βτ,zσ,Z0;1,U≤∂z−κΞ;1q−βτ,zσ,Z0;1,U+Ψ⌣q−βτ,zσ,Z0;1,U≤M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,U,for some constant M1>0 (independent of q but relying on q0 such that 1<q≤q0) provided that Z~0<Z0;1.
(c)
We ask for sharp bounds for the normsτkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U, for integers h1,h2≥0 under the constraint (20). Departing from the expansion (211), we observe that(301)τkh1∂zh2ΔβΨ⌣τ,z=∑j=0κ−1−h2τkh1ΔβP⌣j+h2τzjj!,and its norm writes
(302)τkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U=∑j=0κ−1−h2τkh1ΔβP⌣j+h2τj,σ,UZ~0jj!,where
(303)τkh1ΔβP⌣j+h2τj,σ,U=supτ∈U1+τ2kτexp−σrbjτkτkh1ΔβP⌣j+h2τ≤supτ∈U1+τ2k∣τ∣exp−στkτkh1ΔβP⌣j+h2τ,since rbj≥1, for all j≥0. Furthermore, we can recast the difference as an integral
(304)ΔβP⌣j+h2τ=∫q−βττP⌣j+h2′sds=1−q−βτ∫01P⌣j+h2′τh+q−βτ1−hdh,by means of the parametrization s=τh+q−βτ1−h with 0≤h≤1. Since P⌣j+h2 is a polynomial with complex coefficients, its derivative can be written in the form
(305)P⌣j+h2′s=∑h∈J⌣j+h2′P⌣j+h2,h′sh,for some finite subset J⌣j+h2′ of ℕ and complex coefficients P⌣j+h2,h′. We set
(306)∣P⌣j+h2′∣x=∑h∈J⌣j+h2′∣P⌣j+h2,h′∣xh,and since
(307)τh+q−βτ1−h≤τh+q−βτ1−h≤τ,for all τ∈ℂ, all 0≤h≤1, we get in particular that
(308)∣P⌣j+h2′τh+q−βτ1−h∣≤∣P⌣j+h2′∣∣τ∣,whenever τ∈ℂ and 0≤h≤1. We deduce from (221) and (222) that
(309)ΔβPj+h2∨τ≤1−q−βτPj+h2∨′τfor all τ∈ℂ. Thereupon, gathering (220) and (223) yields a constant Mk,h1,j,h2>0 for which
(310)τkh1ΔβP⌣j+h2τj,σ,U≤1−q−βsupx≥01+x2kexp−σxkxkh1∣P⌣j+h2′∣x≤1−q−βMk,h1,j,h2.
Keeping in mind (219), we finally get a constantNk,h1,h2,κ,Z~0>0 such that
(311)τkh1∂zh2ΔβΨ⌣τ,zσ,Z~0,U≤1−q−βNk,h1,h2,κ,Z~0.(d)
Bounds for the quantityτk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zds/sσ,Z~0,U are also required, for integers h0,h1,h2≥0 under the constraint (20). Based on the decomposition (189) with the bounds (190), Propositions 10 and 12 allow to get a constant M3>0 with(312)τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z~0,U≤τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z0;1,U≤τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Ξ;1s1/k,zdssσ,Z0;1,U+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U≤M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U,presuming that Z~0<Z0;1.
(e)
We demanded also accurate estimates for the norms(313)ΔI≔τk∫0τkτk−sh0/k−1sh1∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U,for integers h0,h1,h2≥0 under the constraint (20). From the expansion (211), we check that
(314)∂zh2ΔβΨ⌣s1/k,z=∑j=0κ−1−h2ΔβP⌣j+h2s1/kzjj!,and therefore,
(315)ΔI=∑j=0κ−1−h2τk∫0τkτk−sh0/k−1sh1−1ΔβP⌣j+h2s1/kdsj,σ,UZ~0jj!.
On the basis of the upper bounds (223), by means of the change of variableh=τkx with 0≤x≤1 and keeping in mind the lower bounds rbj≥1, we deduce a constant M¯k,h0,h1,h2,j>0 for which
(316)τk∫0τkτk−sh0/k−1sh1−1ΔβP⌣j+h2s1/kdsj,σ,U≤supτ∈U1+τ2k∣τ∣exp−σrbjτkτk∫0τkτk−hh0/k−1hh1−11−q−βh1/k∣P⌣j+h2′∣h1/kdh≤1−q−βsupy≥01+y2kexp−σykyh0+kh1∣P⌣j+h2′∣y∫011−xh0/k−1xh1−1+1/kdx≤1−q−βM¯k,h0,h1,h2,j.
At last, pairing (227) with (228) gives rise to a constantN¯k,h0,h1,h2,κ,Z~0>0 such that
(317)τk∫0τkτk−sh0/k−1sh1∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U≤1−q−βN¯k,h0,h1,h2,κ,Z~0..
We are now ready to come to the core of the proof. We fix our attention on the first item 1.
LetΔβΞ belong to Gσ,Z~0,Uk such that
(318)ΔβΞτ,zσ,Z~0,U≤υ~1−q−β.
We provide explicit bounds for each block of the mapDΔβΞ.
Proposition11 and (217) return
(319)dh¯zkh11−q−βkh1τkh1∂zh2Θ;1τ,zσ,Z~0,U≤kh11−q−βkh1dh¯Z~0M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U.
Propositions6 and 11 together with (225) trigger
(320)dh¯zkh1q−βkh1τkh1∂z−κ−h2ΔβΞτ,z+∂zh2ΔβΨ⌣τ,zσ,Z~0,U≤dh¯Z~0kh1q−βkh1M1Z~0κ−h2ΔβΞτ,zσ,Z~0,U+1−q−βNk,h1,h2,κ,Z~0.
Proposition11 and (226) afford
(321)dh¯zkh1Γh0/k1−q−βh0+kh1τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/k1−q−βh0+kh1×M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U.
Propositions10 and 11 along with (229) prompt
(322)dh¯zkh1Γh0/kq−βh0+kh1τk∫0τkτk−sh0/k−1sh1×∂z−κ−h2ΔβΞs1/k,z+∂zh2ΔβΨ⌣s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2ΔβΞτ,zσ,Z~0,U+1−q−βN¯k,h0,h1,h2,κ,Z~0.
Propositions6, 8, and 11 with the help of (217), (218) and (225) furnish
(323)a0zτk∫0τk∂z−κΔβΞτk−s1/k,z+ΔβΨ⌣τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞs1/k,z+ΔβΨ⌣s1/k,zdsτk−ssσ,Z~0,U≤a0Z~0M2∂z−κΔβΞτ,z+ΔβΨ⌣τ,zσ,Z~0,UΘ;1τ,zσ,Z~0,U+Θ;1q−βτ,zσ,Z~0,U∂z−κΔβΞτ,z+ΔβΨ⌣τ,zσ,Z~0,U≤a0Z~0M2M1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0.
Propositions8, 10, and 11 together with (217) beget
(324)ahz1Γh/k1−q−βhτk∫0τkτk−sh/k−1×s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/k1−q−βhM3τk∫0τkΘ;1τk−s11/k,zΘ;1s11/k,zds1τk−s1s1σ,Z~0,U≤ahZ~01Γh/k1−q−βhM3M2Θ;1τ,zσ,Z~0,U2≤ahZ~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2≤ahZ~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2.
Proposition6, 8, 10, and 11 coupled with (217), (218), and (225) breed
(325)ahz1Γh/kq−βhτk∫0τkτk−sh/k−1×s∫0s∂z−κΔβΞs−s11/k,z+ΔβΨ⌣s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/kq−βhM3τk∫0τk∂z−κΔβΞτk−s11/k,z+ΔβΨ⌣τk−s11/k,zΘ;1s11/k,z+Θ;1q−βτk−s11/k,z∂z−κΔβΞs11/k,z+ΔβΨ⌣s11/k,zds1τk−s1s1σ,Z~0,U≤ahZ~01Γh/kq−βhM3M2∂z−κΔβΞτ,z+ΔβΨ⌣Θ;1s11/k,zσ,Z~0,UΘ;1τ,zσ,Z~0,U+Θ;1q−βτ,zσ,Z~0,U∂z−κΔβΞτ,z+ΔβΨ⌣Θ;1s11/k,zσ,Z~0,U≤ahZ~01Γh/kq−βhM3M2M1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞτ,zσ,Z~0,U+1−q−βNk,0,0,κ,Z~0.
From now on, we select a small sized quantityZ~0>0 and suitable υ~>0 (taken independently of q∈1,q0) in a way that the next inequality holds
(326)∑h¯=h0,h1,h2,h3∈C;h0=0∣dh¯∣Z~0kh11−q−βkh1M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U+kh1q−βkh1M1Z~0κ−h2υ~1−q−β+1−q−βNk,h1,h2,κ,Z~0+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z~0kh1Γh0/k1−q−βh0+kh1×M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U+q−βh0+kh1M3Z~0κ−h2υ~1−q−β+1−q−βN¯k,h0,h1,h2,κ,Z~0+∣a0∣Z~0M2M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0+∑h=1A∣ah∣Z~01Γh/k1−q−βhM3M2M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U2+q−βhM3M2M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0+21−q−βC~9≤υ~1−q−β.
Notice that the above constraint (238) can be achieved since all the quantities(327)1−q−βkh11−q−β=∑j=0kh1−1q−βj,1−q−βh0+kh11−q−β=∑j=0h0+kh1−1q−βj,1−q−βkh1−q−β=∑j=0kh−1q−βj,remain bounded for q∈1,q0 provided that kh1≥1, h0+kh1≥1, and h≥1.
At last, the collection of all the inequalities (216), (231), (232), (233), (234), (235), (236), and (237) listed overhead under the condition (326) yields the first expected item (214).
We concentrate on the second feature 2. of the mapD. Let ΔβΞ1,ΔβΞ2 be elements of Gσ,Z~0,Uk with
(328)ΔβΞjτ,zσ,Z~0,U≤υ~1−q−β,for j=1,2.
According to the computations made to treat the first property 1 ofD, we deduce forthrightly the next list of inequalities
(329)dh¯zkh1q−βkh1τkh1∂z−κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U≤dh¯Z~0kh1q−βkh1M1Z~0κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,and
(330)dh¯zkh1Γh0/kq−βh0+kh1τk∫0τkτk−sh0/k−1sh1×∂z−κ−h2ΔβΞ1s1/k,z−ΔβΞ2s1/k,zdssσ,Z~0,U≤dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2ΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,along with
(331)a0zτk∫0τk∂z−κΔβΞ1τk−s1/k,z−ΔβΞ2τk−s1/k,zΘ;1s1/k,z+Θ;1q−βτk−s1/k,z∂z−κΔβΞ1s1/k,z−ΔβΞ2s1/k,zdsτk−ssσ,Z~0,U≤a0Z~0M2M1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U,and
(332)ahz1Γh/kq−βhτk∫0τkτk−sh/k−1×s∫0s∂z−κΔβΞ1s−s11/k,z−ΔβΞ2s−s11/k,zΘ;1s11/k,z+Θ;1q−βs−s11/k,z∂z−κΔβΞ1s11/k,z−ΔβΞ2s11/k,zds1s−s1s1dssσ,Z~0,U≤ahZ~01Γh/kq−βhM3M2M1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κΔβΞ1τ,z−ΔβΞ2τ,zσ,Z~0,U.
Hereafter, we chooseZ~0>0 small enough in a way that
(333)∑h¯=h0,h1,h2,h3∈C;h0=0dh¯Z~0kh1q−βh1kM1Z~0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯Z~0kh1Γh0/kq−βh0+kh1M3Z~0κ−h2+a0Z~0M2M1Z~0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κ+∑h=1AahZ~01Γh/kq−βhM3M2M1Z~0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0;1κυ;1+Ψ⌣q−βτ,zσ,Z0;1,UM1Z~0κ≤12,and gathering (239), (240), (241), and (242) gives rise to the shrinking property (215).
Eventually, we select the constantsZ~0>0 and υ~>0 in a way that both constraints (238) and (243) hold concomitantly. Lemma 38 follows. □.
In the upcoming lemma, we solve the linear Cauchy problem (208), (209) within the Banach spaceGσ,Z~0,Uk.Lemma 39.
Let us presume that condition (30) holds. We single out the constants Z~0>0 and υ~>0 (independently of q in 1,q0) as in Lemma 38. Then, the linear Cauchy problem (208), (209) owns a solution ΔβΘ;1u,z that belong to the Banach space Gσ,Z~0,Uk for any given σ>0. Along with it, a constant M~1;1>0 (unrelated to q∈1,q0) can be found with the next feature
(334)ΔβΘ;1u,zσ,Z~0,U≤M~1;11−q−β.Proof.
Derived from Lemma38, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map D:Bυ~1−q−β⟶Bυ~1−q−β according to the fact Bυ~1−q−β,d stands for a complete metric space for the distance dx,y=x−yσ,Z~0,U. Whence, D:Bυ~1−q−β⟶Bυ~1−q−β carries a unique fixed point denoted ΔβΞ;1u,z inside the ball Bυ~1−q−β, meaning that
(335)DΔβΞ;1=ΔβΞ;1.
As a result, a unique solutionΔβΞ;1 for the equation (292) is established in the ball Bυ~1−q−β. Furthermore, proposition 6 conjointly with the decomposition (210) and the bounds (225) certify that the map
(336)ΔβΘ;1u,z=∂z−κΔβΞ;1u,z+ΔβΨ⌣u,z,belongs to Gσ,Z~0,Uk, solves the problem (208), (209), and suffers the next upper estimates
(337)ΔβΘ;1u,zσ,Z~0,U≤M1Z~0κΔβΞ;1u,zσ,Z~0,U+ΔβΨ⌣u,zσ,Z~0,U≤M1Z~0κυ~1−q−β+1−q−βNk,0,0,κ,Z~0≤M~1;11−q−β,for some constant M~1;1>0 which is unattached to q in the range 1,q0.□.
According to Lemma39, it turns out that the unique formal series in z with holomorphic coefficients on U solution of (208), (209) is subjected to the bounds (244). Since the difference Θ;1u,z−Θ;1q−βu,z, which represents in particular a formal power series in z with holomorphic coefficients on U, is shown to solve (208), (209) in the first step of Proposition 37, it must coincide with the solution ΔβΘ;1u,z constructed above in the second step of the proof, and we deduce conclusively that the for sought estimates (198) hold for it. This completes the proof of the proposition.


## 5.5. Error Bounds between the Borel Maps of the Analytic Solutions to the Linear Cauchy Problems (11), (12) and (177), (178)

In this subsection, we remind the reader of a result obtained in our previous work [1] related to the dependence of the family of solutions upt,z0≤p≤ς−1 to the linear Cauchy problem (11), (12) relatively to the parameter q>1 set up in Theorem 3. This result will be applied in the ensuing subsection.We consider the admissible setD¯=T¯,U¯ of sectors chosen in Proposition 33, where an unbounded sector U and corresponding bounded sector T are distinguished. We denote
(338)ut,z=k∫Lγw;qu,zexp−u/tkduu,for t,z∈T×D1/2C4, the solution of the problem (11), (12) displayed in Theorem 3, along the halfline Lγ taken in Proposition 33, where the Borel map w;qu,z has now an attached index q in order to keep track of the dependence in q>1. According to Theorem 3, w;qu,z represents a holomorphic function on the domain U×D1/2C4 with a Taylor expansion of the form
(339)w;qu,z=∑n≥0wn;quznn!,whenever u∈U, z∈D1/2C4. We keep the notations of Section 5.1. In Section 8.3 of [1], the next result is stated.Proposition 40.
Letq∈1,q0. Two constants C11,C12>0 (that are unconnected to q) can be singled out such that
(340)∣wn;qu−wn;1u∣≤q−1C11C12nn!∣u∣expk1log2∣u∣+u0+αlog∣u∣+u0,for all u∈U∪0, all integers n≥0, where the constants k1>0, u0>1, α≥0 are fixed in Proposition 33.


## 5.6. Error Bounds between the Analytic Solutions of the Nonlinear Auxiliary Cauchy Problems (32), (33) and (185), (186)

This subsection is devoted to the expounding of the next proposition which plays a central role in the upcoming third main result of this work discussed in Theorem44.Proposition 41.
Letq∈1,q0. Then, one can select two constants Z⌣0>0 and M⌣1>0 that are unrelated to q, such that for any given σ>0, the next bounds
(341)Θ;1u,z−Θ;qu,zσ,Z⌣0,U≤M⌣1Z⌣0κq−1,hold, where Θ;q stands for the solution to the Cauchy problem (32), (33) that belongs to the space Gσ,Z0,Uk for some Z0>0 (relying on q0 but independent of q) built up in Proposition 15 and where Θ;1 solves the problem (185), (186) and is exhibited in Proposition 34.Proof.
The proof is split up is two main parts.
In the first part, we frame a Cauchy problem specified later on in (253), (254) which is fulfilled by the difference(342)Δ1;qΘu,z≔Θ;1u,z−Θ;qu,z,for which we seek upper bounds.
To that end, we need some prelusive computations. The basic identityab−cd=a−cb+cb−d will help us in recasting in an appropriate manner the next list of differences appearing as building blocks of the pending equation (347). The differences dealing with the linear terms can be reorganized by inserting some auxiliary terms as
(343)kukh1∂zh2Θ;1u,z−kq−h3ukh1∂zh2Θ;qq−h3u,z=kukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Θ;1u,z−∂zh2Θ;qq−h3u,z=kukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Θ;1u,z−∂zh2Θ;1q−h3u,z+∂zh2Θ;1q−h3u,z−∂zh2Θ;qq−h3u,z,and
(344)uk∫0ukuk−sh0/k−1ksh1∂zh2Θ;1s1/k,zdss−uk∫0ukuk−sh0/k−1kq−h3s1/kkh1∂zh2Θ;qq−h3s1/k,zdss=uk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Θ;1s1/k,z−∂zh2Θ;qq−h3s1/k,zdss=uk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Θ;1s1/k,z−∂zh2Θ;1q−h3s1/k,z+∂zh2Θ;1q−h3s1/k,z−∂zh2Θ;qq−h3s1/k,zdss,along with the differences for the nonlinear terms that can be reshaped as
(345)uk∫0ukΘ;1uk−s1/k,zΘ;1s1/k,zdsuk−ss−uk∫0ukΘ;quk−s1/k,zΘ;qs1/k,zdsuk−ss=uk∫0ukΘ;1uk−s1/k,z−Θ;quk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,zΘ;1s1/k,z−Θ;qs1/k,zdsuk−ss,and
(346)uk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,zΘ;1s11/k,zds1s−s1s1dss−uk∫0ukuk−sh/k−1s∫0sΘ;qs−s11/k,zΘ;qs11/k,zds1s−s1s1dss=uk∫0ukuk−sh/k−1s∫0sΘ;1s−s11/k,z−Θ;qs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,zΘ;1s11/k,z−Θ;qs11/k,zds1s−s1s1dss.
Owing to the fact thatΘ;1 (resp. Θ;q) solves the linear convolution Cauchy problem (185), (186) (resp. (32), (33)), paying regard to the redrawn expressions (343), (344), (345), and (346) and keeping in mind the notation (199) of the previous Section 5.4, we can state the linear convolution Cauchy problem fulfilled by the difference Δ1;qΘu,z as follows
(347)∂zκΔ1;qΘu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Δh3Θ;1u,z+∂zh2Δ1;qΘq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kuk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂zh2Δ1;qΘq−h3s1/k,zdss+a0zuk∫0ukΔ1;qΘuk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,zΔ1;qΘs1/k,zdsuk−ss+∑h=1Aahz1Γh/kuk∫0ukuk−sh/k−1s∫0sΔ1;qΘs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,zΔ1;qΘs11/k,zds1s−s1s1dss+w;1u,z−w;qu,z,for prescribed vanishing Cauchy data
(348)∂zjΔ1;qΘu,0≡0,0≤j≤κ−1
In the ensuing part, we intend to solve the above problem (253), (254) by way of the Banach spaces introduced in Subsection3.1. Namely, we search for a solution to (253), (254) shaped as
(349)Δ1;qΘu,z=∂z−κΔ1;qΞu,z,for some expression Δ1;qΞu,z to be specified. One checks that Δ1;qΘu,z matches the problem (253), (254) if the map Δ1;qΞu,z is subjected to the next fixed point equation
(350)Δ1;qΞu,z=∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkukh1−kq−h3ukh1∂zh2Θ;1u,z+kq−h3ukh1∂zh2Δh3Θ;1u,z+∂z−κ−h2Δ1;qΞq−h3u,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kuk∫0ukuk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂z−κ−h2Δ1;qΞq−h3s1/k,zdss+a0zuk∫0uk∂z−κΔ1;qΞuk−s1/k,zΘ;1s1/k,z+Θ;quk−s1/k,z∂z−κΔ1;qΞs1/k,zdsuk−ss+∑h=1Aahz1Γh/kuk∫0ukuk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dss+w;1u,z−w;qu,z.
In the sequel, we seek for a solution, for which sharp bounds relatively toq are exhibited, to (256) inside the Banach space Gσ,Z⌣0,Uk, for any prescribed σ>0 in the condition that Z⌣0>0 remains small enough. In order to meet this objective, we set up the next linear mapping
(351)EΔ1;qΞτ,z≔∑h¯=h0,h1,h2,h3∈C;h0=0dh¯zkτkh1−kq−h3τkh1∂zh2Θ;1τ,z+kq−h3τkh1∂zh2Δh3Θ;1τ,z+∂z−κ−h2Δ1;qΞq−h3τ,z+∑h¯=h0,h1,h2,h3∈C;h0≥1dh¯z1Γh0/kτk∫0τkτk−sh0/k−1ksh1−kq−h3s1/kkh1×∂zh2Θ;1s1/k,z+kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,z+∂z−κ−h2Δ1;qΞq−h3s1/k,zdss+a0zτk∫0τk∂z−κΔ1;qΞτk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞs1/k,zdsτk−ss+∑h=1Aahz1Γh/kτk∫0τkτk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dss+w;1τ,z−w;qτ,z.
In the following lemma, sufficient conditions are enunciated under whichE becomes a shrinking map on a tiny ball whose radius hinges on q centered at the origin in the space Gσ,Z⌣0,Uk.Lemma 42.
Assume that the condition (30) holds. Then, a small sized real number χ⌣>0 can be singled out in a way that if 0<Z⌣0<χ⌣, a radius υ⌣>0 can be distinguished (in an unrelated manner to q, but depending on q0 for which 1<q≤q0), such that E acquires the next two hallmarks: let us denote Bυ⌣q−1 the ball centered at 0 with radius υ⌣q−1 in Gσ,Z⌣0,Uk,
(1)
E maps Bυ⌣q−1 into itself, signifying that
(352)EBυ⌣q−1⊂Bυ⌣q−1.(2)
The inequality(353)EΔ1;qΞ1−EΔ1;qΞ2σ,Z⌣0,U≤12Δ1;qΞ1−Δ1;qΞ2σ,Z⌣0,U,holds as long as Δ1;qΞ1,Δ1;qΞ2∈Bυ⌣q−1.Proof.
We first supply norm upper estimates for some parts of the mapE.
(a)
Upper bounds are established for the normw;1τ,z−w;qτ,zσ,Z⌣0,U. Departing from the expressions (258) and (339), we arrive at
(354)w;1τ,z−w;qτ,zσ,Z⌣0,U=∑n≥0wn;1τ−wn;qτn,σ,UZ⌣0nn!,and owing to the bounds (247) stirred up in Proposition 40, the coefficients of the above series can be upper bounded by
(355)wn;1τ−wn;qτn,σ,U=supτ∈U1+τ2k∣τ∣exp−σrbnτk∣wn;1τ−wn;qu∣≤q−1C11C12nn!supx≥01+x2kexp−σxkexpk1log2x+u0+αlogx+u0≤q−1C⌣11C12nn!,for some constant C⌣11>0 (unrelated to q), for any integers n≥0. Thereby, it brings in
(356)w;1τ,z−w;qτ,zσ,Z⌣0,U≤q−1C⌣11∑n≥0C12Z⌣0n≤2q−1C⌣11,on the condition that Z⌣0<1/2C12.(b)
Bearing in mind (217), we already know that(357)τkh1∂zh2Θ;1τ,zσ,Z⌣0,U≤M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U,for some constant M1>0 (independent of q but relying on q0), where Z0;1,υ;1>0 are defined in Proposition 34, as long as Z⌣0<Z0;1.(c)
We need accurate bounds for the normsτkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U, for nonnegative integers h1,h2,h3 submitted to the condition (30). Calling to mind Lemma 39, the next splitting
(358)Δh3Θ;1τ,z=∂z−κΔh3Ξ;1τ,z+Δh3Ψ⌣τ,z,holds where Δh3Ξ;1τ,z belongs to the ball Bυ~1−q−h3 in Gσ,Z~0,Uk for some constants Z~0,υ~>0 and where Δh3Ψ⌣τ,z is defined in (211). Owing to (225), we can set a constant Nk,h1,h2,κ,Z~0>0 with
(359)τkh1∂zh2Δh3Ψ⌣τ,zσ,Z~0,U≤1−q−h3Nk,h1,h2,κ,Z~0.
Besides, according to Proposition6, we get a constant M1>0 (independent of q) such that
(360)τkh1∂z−κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M1Z~0κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M1Z~0κ−h2υ~1−q−h3.
Thereupon, we deduce from (263) and (264) the awaited bounds(361)τkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U≤τkh1∂z−κ−h2Δh3Ξ;1τ,zσ,Z~0,U+τkh1∂zh2Δh3Ψ⌣τ,zσ,Z~0,U≤M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0,provided that Z⌣0<Z~0.
(d)
Keeping in mind (226), we observe that(362)τk∫0τkτk−sh0/k−1sh1∂zh2Θ;1s1/k,zdssσ,Z⌣0,U≤M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U,as long as that Z⌣0<Z~0.(e)
Precise bounds for the norms(363)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U,are required. Owing to the bounds (229), we can exhibit a constant N¯k,h0,h1,h2,κ,Z~0>0 with
(364)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Ψ⌣s1/k,zdssσ,Z~0,U≤1−q−h3N¯k,h0,h1,h2,κ,Z~0,and according to Proposition 10, we get a constant M3>0 (unrelated to q) with
(365)τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Δh3Ξ;1s1/k,zdssσ,Z~0,U≤M3Z~0κ−h2Δh3Ξ;1τ,zσ,Z~0,U≤M3Z~0κ−h2υ~1−q−h3.
Thus, on grounds of (267) and (268), we reach the due bounds(366)τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U≤τk∫0τkτk−sh0/k−1sh1∂z−κ−h2Δh3Ξ;1s1/k,zdssσ,Z~0,U+τk∫0τkτk−sh0/k−1sh1∂zh2Δh3Ψ⌣s1/k,zdssσ,Z~0,U≤M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0,whenever Z⌣0<Z~0.
(f)
We also need the bounds(367)Θ;1τ,zσ,Z⌣0,U≤M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U,which is a particular case of (261) and according to Proposition 15, one can select constants M1,Z0,υ>0 (unrelated to q but depending on q0) with
(368)Θ;qτ,zσ,Z⌣0,U≤Θ;qτ,zσ,Z0,U≤M1Z0κυ+Ψ⌣τ,zσ,Z0,U,whenever Z⌣0<Z0.
We are now in position to reach the heart of the proof. We focus on the first attribute 1. We setΔ1;qΞτ,z in Gσ,Z⌣0,Uk with
(369)Δ1;qΞτ,zσ,Z⌣0,U≤υ⌣q−1.
We display explicit bounds for each piece of the mapEΔ1;qΞ.
Proposition11 and (261) beget
(370)dh¯zkτkh1−kq−h3τkh1∂zh2Θ;1τ,zσ,Z⌣0,U≤kh11−q−h3kh1dh¯Z⌣0×M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U.
Proposition11 and (265) yield
(371)dh¯zkq−h3τkh1∂zh2Δh3Θ;1τ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1×M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0.
Propositions6 and 11 trigger
(372)dh¯zkq−h3τkh1∂z−κ−h2Δ1;qΞq−h3τ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1M1Z⌣0κ−h2Δ1;qΞτ,zσ,Z⌣0,U≤dh¯Z⌣0kh1q−h3kh1M1Z⌣0κ−h2υ⌣q−1.
Proposition11 and (266) prompt
(373)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1ksh1−kq−h3s1/kkh1∂zh2Θ;1s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh11−q−h3kh1M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U.
Proposition11 and (269) return
(374)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1∂zh2Δh3Θ;1s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0.
Propositions10 and 11 furnish
(375)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Δ1;qΞq−h3s1/k,zdssσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2Δ1;qΞτ,zσ,Z⌣0,U≤dh¯Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2υ⌣q−1.
Propositions6, 8, and 11 with the help of (270) and (271) promote
(376)a0zτk∫0τk∂z−κΔ1;qΞτk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞs1/k,zdsτk−ssσ,Z⌣0,U≤∣a0∣Z⌣0M2∂z−κΔ1;qΞτ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞτ,zσ,Z⌣0,U≤∣a0∣Z⌣0M2M1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,U≤∣a0∣Z⌣0M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κυ⌣q−1.
Propositions6, 8, 10, and 11 together with (270), (271) spark off
(377)ahz1Γh/kτk∫0τkτk−sh/k−1s∫0s∂z−κΔ1;qΞs−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞs11/k,zds1s−s1s1dssσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3τk∫0τk∂z−κΔ1;qΞτk−s11/k,zΘ;1s11/k,z+Θ;qτk−s11/k,z∂z−κΔ1;qΞs11/k,zds1τk−s1s1σ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2∂z−κΔ1;qΞτ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞτ,zσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞτ,zσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κυ⌣q−1.
From now onwards, we choose a small sized quantityZ⌣0>0 and proper υ⌣>0 (taken freely from q∈1,q0) in a way that the next inequality holds
(378)∑h¯=h0,h1,h2,h3∈C;h0=0kh11−q−h3kh1∣dh¯∣Z⌣0×M1Z0;1κ−h2υ;1+τkh1∂zh2Ψ⌣τ,zσ,Z0;1,U+∣dh¯∣Z⌣0kh1q−h3kh1×M1Z~0κ−h2υ~1−q−h3+1−q−h3Nk,h1,h2,κ,Z~0+∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2υ⌣q−1+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z⌣01Γh0/kkh11−q−h3kh1M3Z0;1κ−h2υ;1+τk∫0τkτk−sh0/k−1sh1∂zh2Ψ⌣s1/k,zdssσ,Z0;1,U+∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z~0κ−h2υ~1−q−h3+1−q−h3N¯k,h0,h1,h2,κ,Z~0+∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2υ⌣q−1+∣a0∣Z⌣0M2M1Z⌣0κυ⌣q−1M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+∑h=1A∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κυ⌣q−1×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+2q−1C⌣11≤υ⌣q−1.
Observe that the above constraint (281) is achievable since all the quantities(379)1−q−h3q−1=1q1−1/qh31−1/q=1q∑j=0h3−11/qj,1−q−h3kh1q−1=1q1−1/qh3kh11−1/q=1q∑j=0h3kh1−11qj,remain bounded whenever q∈1,q0, for given integers h3≥1, h3kh1≥1.
Eventually, piling up the inequalities (260), (273), (274), (275), (276), (277), (278), (279), and (280) reached above under the condition (378) produces the awaited item (258).
We focalize on the second aspect 2. of the mapE. Let Δ1;qΞ1,Δ1;qΞ2 be elements of Gσ,Z⌣0,Uk such that
(380)Δ1;qΞjτ,zσ,Z⌣0,U≤υ⌣q−1,for j=1,2.
The estimates involved in the first property 1. ofE enable us to write down the next set of inequalities
(381)dh¯zkq−h3τkh1∂z−κ−h2Δ1;qΞ1q−h3τ,z−Δ1;qΞ2q−h3τ,zσ,Z⌣0,U≤∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2Δ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,and
(382)dh¯z1Γh0/kτk∫0τkτk−sh0/k−1kq−h3s1/kkh1×∂z−κ−h2Δ1;qΞ1q−h3s1/k,z−Δ1;qΞ2q−h3s1/k,zdssσ,Z⌣0,U≤∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2Δ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,
along with(383)a0zτk∫0τk∂z−κΔ1;qΞ1τk−s1/k,z−Δ1;qΞ2τk−s1/k,zΘ;1s1/k,z+Θ;qτk−s1/k,z∂z−κΔ1;qΞ1s1/k,z−Δ1;qΞ2s1/k,zdsτk−ssσ,Z⌣0,U≤a0Z⌣0M2∂z−κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,UΘ;1τ,zσ,Z⌣0,U+Θ;qτ,zσ,Z⌣0,U∂z−κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U≤a0Z⌣0M2M1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,UM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U,and
(384)ahz1Γh/kτk∫0τkτk−sh/k−1×s∫0s∂z−κΔ1;qΞ1s−s11/k,z−Δ1;qΞ2s−s11/k,zΘ;1s11/k,z+Θ;qs−s11/k,z∂z−κΔ1;qΞ1s11/k,z−Δ1;qΞ2s11/k,zds1s−s1s1dssσ,Z⌣0,U≤∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U×M1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,UM1Z⌣0κΔ1;qΞ1τ,z−Δ1;qΞ2τ,zσ,Z⌣0,U.
From this point forward, we pick upZ⌣0>0 close enough to 0 in a way that
(385)∑h¯=h0,h1,h2,h3∈C;h0=0∣dh¯∣Z⌣0kh1q−h3kh1M1Z⌣0κ−h2+∑h¯=h0,h1,h2,h3∈C;h0≥1∣dh¯∣Z⌣01Γh0/kkh1q−h3kh1M3Z⌣0κ−h2+∣a0∣Z⌣0M2M1Z⌣0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U+∑h=1A∣ah∣Z⌣01Γh/kM3M2M1Z⌣0κM1Z0;1κυ;1+Ψ⌣τ,zσ,Z0;1,U+M1Z0κυ+Ψ⌣τ,zσ,Z0,U≤12.
The collection of bounds (282), (283), (284), and (285) contingent upon (286) sparks off the second feature (259).
At last, we single out the constantsZ⌣0 and υ⌣>0 in a way that both constraints (281), (286) occur unitedly. Lemma 42 follows.
In the lemma to come, the linear Cauchy problem (253), (254) is solved within the Banach spaceGσ,Z⌣0,Uk.Lemma 43.
Take for granted that the condition (30) hold. We fix the constants Z⌣0>0 and υ⌣>0 (independently of q in 1,q0) as in Lemma 42. Then, the linear Cauchy problem (253), (254) carries a solution Δ1;qΘu,z that belongs to the Banach space Gσ,Z⌣0,Uk for any given σ>0. Along with it, a constant M⌣1>0 (unrelated to q∈1,q0) can be distinguished with the next property
(386)Δ1;qΘu,zσ,Z⌣0,U≤M⌣1Z⌣0κq−1.Proof.
Based on Lemma42, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map E:Bυ⌣q−1⟶Bυ⌣q−1 due to the fact that Bυ⌣q−1,d represents a complete metric space for the distance dx,y=x−yσ,Z⌣0,U. Thus, E:Bυ⌣q−1⟶Bυ⌣q−1 gets a unique fixed point denoted Δ1;qΞu,z inside the ball Bυ⌣q−1, meaning that
(387)EΔ1;qΞ=Δ1;qΞ.
Thereby, a unique solutionΔ1;qΞ for the equation (350) is confirmed in the ball Bυ⌣q−1. Furthermore, Proposition 6 conjointly with the decomposition (255) warrant that the map
(388)Δ1;qΘu,z=∂z−κΔ1;qΞu,z,belongs to Gσ,Z⌣0,Uk, solves the problem (253), (254) and is submitted to the next upper estimates
(389)Δ1;qΘu,zσ,Z⌣0,U≤M1Z⌣0κΔ1;qΞu,zσ,Z⌣0,U≤M1Z⌣0κυ⌣q−1≤M⌣1Z⌣0κq−1,for some constant M⌣1>0 which is unattached to q in the range 1,q0.□.
According to Lemma43, it turns out that the unique formal series in z with holomorphic coefficients on U solution of (253), (254) suffers the bounds (287). Since the difference Θ;1u,z−Θ;qu,z, which represents in particular a formal power series in z with holomorphic coefficients on U, is shown to solve (253), (254) in the first step of Proposition 41, it must coincide with the solution Δ1;qΘu,z constructed above in the second step and we arrive at the conclusion that the for sought estimates (248) necessarily hold for it. The proof of the proposition is completed.


## 5.7. Confluence for the Analytic Solutions of the Problem (22), (23) asq⟶1

In this subsection, we unveil the third and last main result of the work.Theorem 44.
LetD¯=T¯,U¯ be the admissible set of sectors distinguished in Proposition 33. We denote U the unbounded sector singled out in Proposition 33 and T its corresponding bounded sector in accordance with the requirement (2) of Definition 2.
We denotev;qt,z the bounded holomorphic solution to (22), (23) on the product T×DZ0/2, given by a Laplace transform of order k, see (88), constructed in Theorem 16. Besides, we consider the bounded holomorphic solution v;1t,z of the nonlinear limit problem (182), (183) on the domain T×DZ0;1/2 expressed through a Laplace transform of order k, see (188), built up in Proposition 34.
Then, a constantC>0 (unrelated to q∈1,q0) can be found such that
(390)supt∈T,z∈DZ⌣0/2∣v;1t,z−v;qt,z∣≤Cq−1,for all q∈1,q0. In other words, the solution v;qt,z of (22), (23) merges uniformly on T×DZ⌣0/2 to the solution v;1t,z of (182), (183) as q⟶1.Proof.
We express both solutionsv;qt,z and v;1t,z as Laplace transforms
(391)v;qt,z=k∫LγΘ;qu,zexp−utkduu,v;1t,z=k∫LγΘ;1u,zexp−utkduu.along a halfline Lγ=0,+∞e−1γ⊂U∪0 assigned to the condition
(392)coskγ−argt>Δ,for some fixed constant Δ>0, provided that t∈T, where the Borel map Θ;qu,z is outlined in (89) and Θ;1u,z is described in (189) whose Taylor expansion can be displayed as
(393)Θ;1u,z=∑β≥0Θ;1,βuzββ!,for u,z∈U×DZ0;1/2.
The deviations bounds reached in Proposition41 can be rephrased in the next explicit way
(394)Θ;1u,z−Θ;qu,z≤∑β≥0Θ;1,βu−Θ;q,βuzββ!≤M⌣1Z⌣0κq−1∣u∣1+u2k∑β≥0expσrbβuk∣z∣Z⌣0β≤2M⌣1Z⌣0κq−1∣u∣expσζbuk,for all u∈U∪0, provided that z∈DZ⌣0/2.
Thereby, we can govern the differencev;1t,z−v;qt,z in the following manner
(395)v;1t,z−v;qt,z=k∫LγΘ;1u,z−Θ;qu,zexp−utkduu≤2M⌣1Z⌣0κq−1∫0+∞expσζbrkexp−rrTkΔdr,for all t∈T, all z∈DZ⌣0/2, bearing in mind that radius rT of T fulfills
(396)rT<Δσζb1/k,according to (87). This achieves the expected bounds (289).



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*Source: 9637628-2022-08-08.xml*