Abstract

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

1. Introduction

In this work, we focus on singularly perturbed linear partial q-difference differential equations which couple two categories of operators acting both on the time variable, so-called q-difference operators of irregular type and Fuchsian differential operators. As a seminal reference concerning analytic and algebraic aspects of q-difference equations with irregular type, refer [1], and for a far reaching investigation of Fuchsian ordinary and partial differential equations, refer [2].

Our equations are presented in the following mannerfor vanishing initial data , where are integers, represents the dilation map acting on time t for some real number , and stand for polynomials in . The main block is polynomial in the arguments , holomorphic in the perturbation parameter ϵ on a disc centered at 0 and in the space variable z on a horizontal strip of the form , for some . The forcing term is analytic relatively to and defines an entire function w.r.t t in with (at most) q-exponential growth (see (60), for precise bounds).

This paper is a natural continuation of the study [3] by Lastra and Malek and will share the same spine structure. Indeed, in [3], we aimed attention at the next problemfor vanishing initial data , where stand for polynomials in their arguments as above and is like the forcing term but with (at most) exponential growth in t. Under convenient conditions put on the shape of (2), we are able to construct a set of genuine bounded holomorphic solutions expressed as a Laplace transform of order k along a halfline and Fourier inverse integral in space z:where the Borel/Fourier map is itself set forth as a Laplace transform of order :where has (at most) exponential growth along and exponential decay in phase m on . The resulting maps are therefore expressed as iterated Laplace transforms following a so-called multisummability procedure introduced by Balser, see [4]. These functions define bounded holomorphic functions on domains for a well-selected bounded sector at 0 and where is a set of sectors which covers a full neighborhood of 0 and is called a good covering (cf. Definition 6). Additionally, the partial maps share on a common asymptotic expansion with bounded holomorphic coefficients on . This asymptotic expansion turns out to be (at most) of Gevrey order with , meaning that we can single out two constants such thatfor all , all .

We plan to obtain a similar statement for the problem under study (1). Namely, we will construct a set of genuine sectorial solutions to (1) and describe their asymptotic expansions as ϵ borders the origin. We first notice that our main problem (1) can be seen as a q-analog of (2), where the irregular differential operator is replaced by the discrete operator . This terminology originates from the basic observation that the expression approaches the derivative as q tends to 1. Here, as mentioned in the title, we qualify the q-analogy as partial since the Fuchsian operator is not discretized in the process. This suggests that, in the building procedure of the solutions (that will follow the same guideline as in [3]), the classical Laplace transform of order k shall be supplanted by a q-Laplace transform of order k as it was the case in the previous work [5] of the author where a similar problem was handled. However, due to presence of the Fuchsian operator , we will see that a single q-Laplace transform is not enough to construct true solutions and that a new mechanism of iterated q-Laplace and classical Laplace transforms is required. Furthermore, we witness that this enhanced multisummability procedure has a forthright effect on their asymptotic expression w.r.t ϵ. Namely, the expansions in the perturbation parameter are neither of classical Gevrey order as displayed in (3) nor of q-Gevrey order as in [5] (meaning that has to be replaced by in the control term of (3)). The asymptotic expansions we exhibit present a double scale structure which has a q-Gevrey leading part with order and a subdominant tail of Gevrey order , that we call Gevrey asymptotic expansion of mixed order (cf. Definition 8). Such a coupled asymptotic structure has already been observed in another setting by Lastra et al. in [6]. Indeed, we considered linear q-difference differential Cauchy problems with the shapefor suitably chosen analytic Cauchy dataand properly selected complex number with , where and are integers and B stands for a polynomial. When , the Fuchsian operator is responsible of the classical Gevrey part of the asymptotic expansion of the true solution which is shown to be of mixed order (in the sense of Definition 8) outside some q-spiral for some w.r.t t near 0, uniformly in z in the vicinity of the origin. Here, the solutions are expressed through a single q-Laplace transform and the contribution in the asymptotics emerges from a discrete set of singularities that accumulates at 0 in the Borel plane.

It is worthwhile mentioning that the approach which consists in building solutions by means of iterated q-Laplace and Laplace transforms stems from a new work by Yamazawa. In [7], he examines linear q-difference differential equations of the formfor the given holomorphic forcing term near the origin and where is a polynomial in with holomorphic coefficients w.r.t t near 0. Under special conditions on the structure of (4), he is able to construct a genuine solution obtained as a small perturbation of iterated truncated q-Laplace and Laplace of order 1 transforms of the iterated Borel and q-Borel transforms of a formal solution of (4). Furthermore, he gets in particular that has as asymptotic expansion of mixed order w.r.t t, uniformly in x near 0.

Notice that in our paper, the solutions are built up as complete iterated q-Laplace and classical Laplace transforms that are shown to be exact solutions of our problem (1). This is why the process we follow can actually be understood as an enhanced version of the multisummation mechanism introduced by Balser, see [4].

In a larger framework, this work is a contribution to the promising and fruitful realm of research in q-difference and q-difference-differential equations in the complex domain. For recent important advances in this area, we mention in particular the works by Tahara and Yamazawa [8–10]. Notice that the fields of applications of q-difference equations have also encountered a rapid growth in the last years. Some forefront studies in this respect are given, for instance, by [11–13] and references therein.

Now, we describe a little more precisely our main results obtained in Theorems 1 and 3. Namely, under convenient restrictions on the shape of (1) detailed in the statement of Theorem 1, we can manufacture a family of bounded holomorphic solutions on domains for a suitable bounded sector at 0, a strip of width , and belonging to a good covering in , which can be displayed as a q-Laplace transform of order k along a halfline and Fourier integral:

The q-Borel/Fourier map is itself shaped as a classical Laplace transform of order along :where has (at most) q-exponential growth of some order along (see (180)) and exponential decay in phase . In Theorem 3, we explain the reason for which all the partial functions share a common asymptotic expansion on with bounded holomorphic coefficients on , which turns out to be of mixed order . This last result leans on a new version of the classical Ramis—Sibuya theorem fitting the above asymptotics, which is fully expounded in Theorem 2.

Our paper is arranged as follows.

In Section 2, we recall the definition of the classical Laplace transform and its q-analog. We also put forward some classical identities for the Fourier transform acting on functions spaces with exponential decay.

In Section 3, we set forth our main problem (33) and we discuss the formal steps leading to its resolution. Namely, a first part is devoted to the inquiry of solutions among q-Laplace transforms of order k and Fourier inverse integrals of Borel maps W with q-exponential growth on unbounded sectors and exponential decay in phase leading to the first main integrodifferential q-difference equation (69) that W is asked to fulfill. A second undertaking suggests to seek for W, as a classical Laplace transform of suitable order of a second Borel map with again appropriate behaviour. The expression is then contrived to solve a second principal integro q-difference equation (81).

In Section 4, bounds for linear convolution and q-difference operators acting on Banach spaces of functions with q-exponential growth are displayed. The second key equation (81) is then solved within these spaces at the hand of a fixed point argument.

In Section 5, genuine holomorphic solutions W of the first principal auxiliary equation (69) are built up and sharp estimates for their growth are provided (cf. (146) and (147)).

In Section 6, we achieve our goal in finding a set of true holomorphic solutions (176) to our initial problem (33).

In Section 7, the existence of a common asymptotic expansion of the Gevrey type with mixed order is established for the solutions set up in Section 6. The decisive technical tool for its construction is detailed in Theorem 2.

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

Let be an integer. We remind the reader the definition of the Laplace transform of order as introduced in [14].

Definition 1. We set as some unbounded sector with bisecting direction and aperture and as a disc centered at 0 with radius . Consider a holomorphic function that vanishes at 0 and withstands the bounds. There exist and such thatfor all . We define the Laplace transform of of order in the direction d as the integral transformalong a halfline , where γ depends on T and is chosen in such a way that , for some fixed real number . The function is well defined, holomorphic, and bounded on any sector:where and .
If one sets , the Taylor expansion of , which converges on the disc , the Laplace transform has the formal seriesas Gevrey asymptotic expansion of order . This means that for all , two constants can be selected with the bounds:for all , all .
In particular, if represents an entire function w.r.t with bound (11), its Laplace transform does not depend on the direction d in and represents a bounded holomorphic function on , whose Taylor expansion is represented by the convergent series on .
Let be an integer and be a real number. At the next stage, we display the definition of the q-Laplace transform of order k which was used in a former work of Malek [5].
Let us first recall some essential properties of the Jacobi Theta function of order k defined as the Laurent series:for all . This analytic function can be factorized as a product known as Jacobi’s triple product formula:for all , from which we deduce that its zeros are the set of real numbers . We recall the next lower bound estimates on a domain, bypassing the set of zeroes of , from [5] Lemma 3, which are crucial in the sequel.

Lemma 1. Let . There exists a constant depending on and independent of such thatfor all satisfying , for all .

Definition 2. Let be a real number and be an unbounded sector centered at 0 with bisecting direction . Let be a holomorphic function, continuous on the adherence , such that there exist constants and withfor all . Let with . We put . We define the q-Laplace transform of order k of f in direction γ aswhere is a halfline in the direction γ.
The following lemma is a slightly modified version of Lemma 4 from [5].

Lemma 2. Let chosen as in Lemma 1 above. The integral transform defines a bounded holomorphic function on the domain for any radius , whereNotice that the value does not depend on such that due to the Cauchy formula.
The next lemma describes conditions under which the q-Laplace transform defines a convergent series near the origin.

Lemma 3. Let be an entire function with Taylor expansion fulfilling bound (19) for all . Then, its q-Laplace transform of order k, , does not depend on the direction and represents a bounded holomorphic function on with the restriction whose Taylor expansion is given by the convergent series .

Proof. The proof is a direct consequence of the next formulas:whenever and , where the last equality follows (for instance) from identity (4.7) from [15], for all .
We restate the definition of some family of Banach spaces mentioned in [14].

Definition 3. Let . We set as the vector space of continuous functions such thatis finite. The space endowed with the norm becomes a Banach space.
Finally, we remind the reader the definition of the inverse Fourier transform acting on the latter Banach spaces and some of its handy formulas relative to derivation and convolution product as stated in [14].

Definition 4. Let with and . The inverse Fourier transform f is given byfor all . The function extends to an analytic bounded function on the stripsfor all given .(a)Define the function which belongs to the space . Then, the next identity occurs.(b)Take and set as the convolution product of f and . Then, ψ belongs to , and moreover, for all .

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

We set as an integer. Let be integers. We set

We consider a finite set I of that fulfills the next feature,whenever and we set nonnegative integers withfor all .

Let , and be polynomials such thatfor all , all .

We consider a family of linear singularly perturbed initial value problemsfor vanishing initial data . Here, stands for a real number and the operator is defined as the dilation by q acting on the variable t through .

The coefficients are built in the following manner. For each , we consider a function that belongs to the Banach space for some , which depends holomorphically on the parameter ϵ on some disc with radius and for which one can find a constant with

We constructas the inverse Fourier transform of the map for all . As a result, is bounded holomorphically w.r.t ϵ on and w.r.t z on any strip for in view of Definition 4.

The presentation of the forcing term requires some preliminary groundwork. We consider a sequence of functions , for , that belongs to the Banach space with the parameters given above and which relies analytically and is bounded w.r.t ϵ on the disc . We assume that the next bounds,hold for all and given constants . We define the formal seriesfor some real number . We introduce the next Banach space.

Definition 5. Let and be real numbers. Let be an open unbounded sector with bisecting direction centered at 0 in . We denote the vector space of complex valued continuous functions on the adherence , which are holomorphic w.r.t u on and such that the normis finite. One can check that the normed space represents a Banach space.

Remark 1. The spaces above are faint modifications of the Banach spaces already introduced in the works of Dreyfus and Lastra [16–18].
The next lemma is a proper adjustment of Lemma 5 out of [5] to the new Banach spaces from Definition 5.

Lemma 4. Let be fixed as in (36). We take a number such thatLet be chosen as above. Then, the function belongs to the Banach space for any unbounded sector , any disc . Moreover, one can find a constant (depending on ) with

Proof. ound (36) implies thatAccording to the elementary fact that the polynomial admits its maximum value at , we deduce by means of the change of variable thatfor all . Therefore, we deduce thatwhich converges, provided that (39) holds, whenever .
We defineas the Laplace transform of w.r.t u of order in direction . Notice that two constants (depending on ) can be found such thatfor all . As a result, owing to bound (40) and the last part of Definition 1, we deduce that does not depend on the direction d and can be written as a convergent series:w.r.t τ near the origin. Now, we fix some real number such that . Then, one can sort a constant (depending on ) such thatfor all . This inequality is a consequence of the Stirling formula, which states thatas x tends to and from the existence of a constant (depending on ) withfor all . Consequently, it turns out that represents an entire function w.r.t τ such thatfor all . Furthermore, owing to bound (42), we know thatfor all , all . Henceforth, we get the next global boundsprovided thatfor all , , and .
Next, we setas the q-Laplace transform of w.r.t u of order k in direction d and Fourier inverse integral w.r.t m. We putfor all . We first provide bounds for this sequence of functions. Namely, we can get a constant (relying on ) withfor all , whenever and z belongs to the horizontal strip for some (see Definition 4). Owing to Lemma 3, we deduce that the function converges near the origin w.r.t T, where it carries the next Taylor expansion:for all and . In particular, the function is independent of the direction d chosen.
We now show that represents an entire function w.r.t T and supply explicit upper bounds. Namely, in accordance with (47), we obtainAgain, estimate (42) yieldsfor all , all , where is defined by . By gathering the two last above inequalities, the next global estimates can be figured out:for all , all and , provided thatLastly, we define the forcing term f as a time rescaled version of ,that represents a bounded holomorphic function w.r.t and and an entire function w.r.t t with q-exponential growth of order .
Throughout this paper, we are looking for time rescaled solutions of (33) of the formAs a consequence, the expression , through the change of variable , is asked to solve the next singular problem:At the onset, we seek for a solution that can be expressed as an integral representation via a q-Laplace transform of order k and Fourier inverse integral:where the inner integration is performed along a halfline in direction . Overall this section, we assume that the partial functions have at most q-exponential growth of order k on some unbounded sector centered 0 with bisecting direction d and belong to the Banach space mentioned in Definition 3, whenever . Precise bounds will be given later in Section 5. Here, we assume that .
Our aim is now the presentation of a related problem fulfilled by the expression . We first need to state two identities which concern the action of q-difference and Fuchsian operators on q-Laplace tranforms.

Lemma 5. The actions of the q-difference operators for integers and the Fuchsian differential operator are given by

Proof. The first identity is a direct consequence of the commutation formula (76) displayed in Proposition 6 from [5]. For the second, a derivation under the integral followed by an integration by parts implies the sequence of equalities:from which the forecast formula follows since the map is assumed to possess a growth of q-exponential order k and vanishes at .
The application of the above identities (66) and (67) in a row with (26) and (28) leads to the first integrodifferential q-difference equation fulfilled by the expression as long as solves (64):We turn now to the second stage of the procedure. Solutions of this latter equation are expected to be found in the class of Laplace transforms of order since by construction owns this structure after (44). Namely, we take for granted thatwhere stands for a halfline with direction , which belongs to where represents an unbounded sector centered at 0 with bisecting direction d. Within this step, we assume that the expression belongs to the Banach space introduced in Definition 5, for all , where the constants , and α are selected accordingly to the construction of the forcing term .
The next lemma has already been stated in our previous work [3].

Lemma 6. For all integers , positive integers and can be found such that

With the help of this last expansion, equation (69) can be recast in the form

This last prepared shape allows us to apply the next lemma that repharses formula (8.7) p. 3630 from [19], in order to express all differential operators appearing in (72) in terms of the most basic one .

Lemma 7. Let be integers. Then, there exist real numbers , , such thatBy convention, we take for granted that the above sum vanishes when .

Indeed, by construction of the finite set I, we can represent the next integers in a specific way:where for all and . As a consequence, we can further expand the next piece of (72) in its final convenient form:and we can remodel equation (72) in such a way that it contains only primitive building blocs:

Similarly to our previous technical Lemma 5, we disclose some useful commutation formulas dealing with the actions of the basic irregular operator , multiplication by monomials , and of the q-difference operator ,

Lemma 8.  (1)The action of the differential operators on is given by(2)Let be an integer. The action of the multiplication by on is described through the next formula:(3)Let be an integer. The action of the operator is represented through the following integral transform:

Proof. The first two formulas have already been given in our previous works [3, 20]. We focus on the third equality. By definition,and if one deforms the path of integration through which keeps the path invariant since , we get formula (79).
Departing from the arranged equation (76) with the help of Lemma 8, we can exhibit an ancillary problem satisfied by the expression ,

4. An Integral q-Difference Equation with Complex Parameter

The objective of this section is the construction of a unique solution of equation (81) just established overhead. This solution will be built among the Banach space displayed in Definition 5. Within the next three propositions, continuity of linear convolutions and q-difference operators acting on is discussed.

Proposition 1. Let be an integer and and be real numbers such that is an integer that are submitted to the next constraint:Let be a continuous function on , holomorphic w.r.t u on submitted to the boundfor all , for some constant . Then, the linear functionrepresents a continuous map from the Banach space into itself. In other words, some constant (depending on ) can be found withfor all .

Proof. Let f belong to . The map can be rewritten using the parametrization for , namely,for all , whenever . By definition, the next upper boundshold for all , . It follows thatfor all , all . Under condition (82), the expected bound (85) follows.

Proposition 2. Let be a complex number, be an integer, and be a real number withstanding the following condition:Then, we can sort a constant (depending on ) withfor all .

Proof. The proof is proximate to the one of Proposition 1 in [5] and similar to the one of Proposition 1 from [18]. We provide, however, a complete proof for the sake of a better readability.
Let belong to . By definition, we can perform the next factorizationSince the contractive map keeps the domain invariant, we deducewith , whereWe observe thatwhere is finite since is continuous on and is equal to .
In the remaining part of the proof, we show that is also finite. We first need to rearrange the pieces of . Namely, we expandSince as , we get two constants (depending on ) withfor all , . Gathering (95) and (96) with (97) gives rise to the boundwhich is finite owing to (89).

Proposition 3. We set polynomials such thatConsider a continuous function on , holomorphic w.r.t u on with the boundfor some constant . Then, there exists a constant (depending on Q, R, and μ) such thatwhenever f belongs to and belongs to .

Proof. The proof shares the same ingredients as the one of Proposition 2 of [3]. Again, we give a thorough explanation of the result. We take f inside and select belonging to . We first recast the norm of the convolution operator as follows:whereBy construction of the polynomials Q and R, one can sort two constants withfor all . As a consequence of (102), (104), and (100), with the help of the triangular inequality , we are led to the boundswhereis a finite constant under the first and last restriction of (99) according to the estimates of Lemma 2.2 from [21] or Lemma 4 of [22].
We disclose now additional assumptions on the leading polynomials and . These requirements will be essential in the transformation of our main problem (81) into a fixed point equation, as explained later in Proposition 4.
With this respect, the guideline is close to our previous study [3]. Namely, we assume the existence of an unbounded sectorial annulus:where direction and aperture for some given inner radius with the feature:We consider the next polynomial:In the following, we need lower bounds of the expression with respect to both variables m and u. In order to achieve this goal, we can factorize the polynomial w.r.t u, namely,where its roots can be displayed explicitely asfor all , for all .
We set an unbounded sector centered at 0, a small disc , and we adjust the sector in a way that the next condition holds. A constant can be chosen withfor all , all , provided that . Indeed, inclusion (108) implies in particular that all the roots , remain a part of some neighborhood of the origin, i.e., satisfy for an appropriate choice of . Furthermore, when the aperture is taken close enough to 0, all these roots stay inside a union of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in . We assign a sector withBy construction, the quotients live outside some small disc centered at 1 in for all , , . Then, (112) follows.
We are now ready to supply lower bounds for .

Lemma 9. A constant (depending on ) can be found withfor , all .

Proof. Departing from factorization (110), the lower bound (112) entailsfor all .
The next proposition discusses sufficient conditions under which a solution of the main integral q-difference equation (81) can be built up in the space .

Proposition 4. Let us assume the next extra requirements:for all . Furthermore, for each , we set an integer such thatand we take for granted thatholds. Then, for an appropriate choice of the constants (see (34)) that need to be taken close enough to 0 for all , a constant can be singled out in a manner that equation (81) gets a unique solution in the space with the condition:whenever , where ,r are chosen as above and are specified in Section 3 on the way to the construction of the forcing term .

Proof. The proof relies strongly on the next lemma which discusses contractive properties of a linear map.

Lemma 10. For all , we define the map asUnder the additional requirement (116)–(118), one can select the constants , for , and a real number in a way that this map acts on some neighborhood of the origin of the space in the following way:(i)The inclusion holds, where stands for the closed ball of radius ϖ centered at 0 in , for all .(ii)The map is contractive, namely, whenever , for all .

Proof. We first control the forcing term. Owing to bound (76) in Lemma 4, together with (114), we can exhibit a constant (relying on ) withwhere is a constant that is set in (36), whenever .
We deal with the first property (121). Let us take in under the constraint .
We fix some complex number such that , and we redraft the norm of the next integral expression as follows:for all , , where is an integer chosen as in (117) andWe observe that a constant (depending on ) can be picked up withfor all , . Indeed, from (114), we obtainfor all , where the right-hand side is finite owing to the suitable choices of and in (118).
Under requirements (32) and (116), an application of Proposition 3 yields a constant (depending on and μ) such thatwhereConditions (116) and (117) allow us to call back Proposition 2 in order to get a constant (depending on ) withwhereLastly, Proposition 1 gives rise to constants (depending on ) and (depending on ) withBy compiling (128)–(132), we obtainfor all , .
We now turn to the second principal pieces of . Following the same lines of arguments as above, we obtain thatwherefor all , and . In order to give bounds for , we make use of Proposition 1 which affords a constant (depending on ) withBy combining (134) and (136), we obtainfor all , and .
In the next step, we impose the constants , , to stay close enough to 0 in order that a constant can be singled out withThe collection of (123), 133 and (137) submitted to condition (138) yields the inclusion (121).
The next part of the proof is devoted to the explanation of the contractive property (122). Indeed, consider two functions and inside the ball . Then, an application of the two inequalities (133) and (137) overhead leads tofor all , andfor all , and .
This time, we require the constants , , to withstand the next inequalityOwing to (139) and (140), under demand (141), we obtain (122).
In conclusion, we choose the constants , in order that both (138) and (141) hold conjointly. This yield Lemma 10.
We go back to the core of Proposition 4. For , chosen as in the lemma above, we consider the closed ball that stands for a complete metric space for the distance . According to the same lemma, we observe that induces a contractive application from into itself. Then, according to the classical contractive mapping theorem, the map carries a unique fixed point that we set as ; meaning thatthat belongs to the ball , for all . Furthermore, the function depends holomorphically on ϵ in . Let the termbe taken from the right to the left-hand side of (81) and then divide by the polynomial defined in (109). These operations allows (81) to be exactly recast into equation (142) above. Consequently, the unique fixed point of obtained overhead in precisely solves equation (81).

5. An Integrodifferential q-Difference Equation with a Complex Parameter

In this section, we build up a solution to the integrodifferential q-difference equation (69) with the shape of a Laplace transform of order in direction d. Furthermore, we provide sharp bounds of this solution for large values of its q-Borel and Fourier variables τ and m.

Proposition 5. We depart from the solution of the integral equation (81) that has just been constructed in Proposition 4. We defineas the Laplace transform of of order in direction d, where the integration path belongs to the sector . Then, for all , the map is continuous on a domain and depends holomorphically on in , where represents an unbounded sector with bisecting direction d and opening that fulfillsfor defined as the aperture of the sector . Furthermore, the map withstands the next accurate bounds:(1)One can single out three constants (depending on ), (depending on ), and (relying on ) such that for all with , and .(2)One can find a constant (depending on ) withwhenever with , all , all .
In particular, one can sort two constants (depending on ) withfor all , and , where was introduced in Section 3 just above (47).
Finally, satisfies the first auxiliary integrodifferential q-difference equation (69) on the domain .

Proof. Bound (119) in Proposition 4 can be recast aswhich holds for all . The integral representation (144) yieldswhenever for a well-chosen direction γ (that may depend on τ) such that for some fixed constant that exists under the requirement (145).
In the second part of the proof, we are scaled down to provide bounds for the next associated function:when is chosen large enough. The next lemma holds.

Lemma 11. One can select two constants (depending on ) and (relying in ) such thatfor all .

Proof. We first make the change of variable in the integral above:On the other hand, we need the next expansions:We cut the integral expression in two pieces:whereprovided that .
We control the first piece . We observe that when . From (154), we deduce the inequalitiesfor all and . Therefore,whereBy construction, a constant (depending on ) can be found withfor all .
In a second step, we evaluate the part . Expansion (154) affords us to writewhereBesides, we can check that there exists a constant (depending on ) such thatprovided that , when . We deduce thatwherewhen . Furthermore, one can sort a constant (depending on ) such thatfor all . Hence,withwhen . We perform the linear change of variable in this latter integralin order to express it in terms of the Gamma function . Keeping in mind the Stirling formula (48), we get a constant (which depends on ) such thatfor all .
Finally, splitting (155) together with the collection of bounds (159)–(162) and (165)–(171) gives rise to the expected bound (152).
This last lemma combined with estimate (150) yield the announced upper bounds (146) and (147). In order to deduce the particular estimate (148) from (146) and (147), we observe that for any given (even close to 0), we can sort a constant (which relies on ) such thatfor all .
In the final part of the proof, the function is shown to fulfill the second main equation (69). In this respect, we tread rearwards the construction discussed in Section 3. Indeed, according to the fact that solves (81) and appertains to the space , for a well-chosen sector , the three identities of Lemma 8 can be applied in order to check that is a genuine solution of the integrodifferential-q-difference equation in prepared form (76). Ultimately, a successive play of Lemma 7 followed by Lemma 6 transforms equation (76) into the expected one (69).

6. Construction of a Finite Set of True Sectorial Solutions to the Main Initial Value Problem

We return to the first part of the formal constructions undertaken in Section 3 in view of the gain made in solving the two auxiliary problems (81) and (69) throughout Sections 4 and 5.

We need to state the definition of a good covering in , and we introduce a fitted version of a so-called associated sets of sectors to a good covering which is analog to the one proposed in our previous work [3].

Definition 6. Let be an integer. We consider a set of open sectors centered at 0, with radius for all for which the next three properties hold:(i)The intersection is not empty for all (with the convention that )(ii)The intersection of any three elements of is empty(iii)The union equals for some neighborhood of 0 in Then, the set of sectors is named a good covering of .

Definition 7. We consider(i)A good covering of whose radius satisfies (ii)A set of unbounded sectors , , centered at 0 with bisecting direction and small opening (iii)A set of unbounded sectors , , centered at 0 with bisecting direction and aperture , for some integer (iv)A fixed bounded sector centered at 0 with radius and a disc suitably selected in a way that the next features are conjointly satisfied:(a)Bound (112) holds, provided that , for all (b)The set fulfills the next properties:(1)The intersection is not empty for all (with the convention that )(2)The union covers (c)For all , all and all : where with any fixed real number close to 0.When the above features are verified, we say that the set of data is admissible.
We settle now the first principal result of the work. We construct a set of actual holomorphic solutions to the main initial value problem (33) defined on sectors , , of a good covering in . Besides, we are able to monitor the difference between consecutive solutions on the intersections .

Theorem 1. We ask the record of requirements (29)–(32), (34), (36), (39), (53), (61), (108), and (116)–(118) to hold. Let us distinguish an admissible set of dataas described in the definition above.
Then, for a suitable choice of the constants (c.f. (34)) close enough to 0 for all , a collection of true solutions of (33) can be singled out. More precisely, each function stands for a bounded holomorphic map on the product for any given and appropriate small radius . Additionally, is represented as a q-Laplace transform of order k and Fourier inverse integral:where . Furthermore, the map is itself fashioned as a Laplace transform of order :whose integration halfline is enclosed in and where belongs to the Banach space for the unbounded sector , provided that .
Finally, some constants can be found withfor all integers , all , whenever , where by convention, we set .

Proof. We first single out an admissible set of data . Under the requirements enounced in Theorem 1, Proposition 5 can be called in order to find a family of functions:expressed as a Laplace transform of order in direction such that of a Borel map which is holomorphic w.r.t u on , w.r.t ϵ on , and continuous relatively to , coming along with a constant such thatfor all , , and . Furthermore, the function solves the first auxiliary integrodifferential q-difference equation (69) on and suffers the bounds:for some constants , for , provided that , , and .
We now revisit the first stage of the formal construction from Section 3. Namely, we set the next q-Laplace transform of order k and Fourier inverse mapalong a halfline . Paying heed to the upper bound (181) and to Lemma 2 together with basic features about Fourier transforms discussed in Definition 4, we notice that stands for(a)A bounded holomorphic function w.r.t T on a domain for some small radius , where is described in (173)(b)A bounded holomorphic application relatively to the couple on , for any given Additionally, since solves (69), Lemma 5 leads to the claim that must fulfill the singular equation (64) on .
In conclusion, the function defined asrepresents a bounded holomorphic function w.r.t t on for some close enough to 0, , for any given , owing to assumption 3 of Definition 7. Moreover, solves the main initial value problem (33) on the domain , for any .
In the second half of the proof, we explain bound (178). Here, we follow a similar roadmap based on path deformation arguments as in our previous work [3]. Indeed, for , the partial functionis holomorphic on the sector . By the Cauchy theorem, we can bend each straight halfline , into the union of three curves with appropriate orientation depicted as follows:(1)A halfline for a given real number (2)An arc of circle with radius denoted joining the point which is taken inside the intersection (that is assumed to be nonempty, see Definition 7, 2.1) to the halfline (3)A segment As a result, the difference can be decomposed into a sum of five integrals along these curves:Bounds for the first piece,are now considered. The arguments followed are proximate to the ones displayed in the proof of Theorem 1 from [5]. Owing to Lemma 1 and bound (181), we obtainfor all , , and . We need the next two expansions:Hence,for all , , and . We now specify estimates for some pieces of these last upper bounds. Namely, since as x tends to 0, we get a constant (depending on ) such thatfor all . Since , we also notice thatfor all , , and . Furthermore,whenever and together withprovided that . Finally, there exists a constant (depending on ) withInequality (189) together with the collection of bounds (190)–(194) yield two constants and (which rely on ) such thatfor all , , and . We want to express these last bounds in terms of sequences now. The discussion hinges on the next lemma.

Lemma 12. The following inequalityholds for all , all integers .

Proof. By performing the change of variable with the help of the computation already undertaken in Lemma 4, we obtainfor all given and integer .
Consequently to (195) and (196), two constants (depending on ) can be picked up withwhenever , , and , for all integers .
With a similar discussion, we can exhibit comparable bounds for the next term:Namely, two constants and (depending on ) can be found withfor all , , and . Furthermore, we can single out two constants (resting on ) such thatprovided that , , and , for all integers .
In the next step, we turn to the first integral along an arc of circle:Making use of Lemma 1 and (181), gives rise to the inequalityfor all , , and . We require once more the expansion:We deduce thatOwing to the hypothesis , we check that (191) holds and bearing in mind (194), we arrive at the existence of two constants and (depending on the constants ) withfor all , , and . Calling back Lemma 12 gives rise to two additional constants (subjected to ) withfor all , , and , for all given integers .
The second integral along an arc of circlecan be managed in the same way. Indeed, one can single out two constants and (relying on ) such thatprovided that , , and . Moreover, we can find two constants (that hinge on ) such thatwhenever , , and , for all integers .
In the remaining part of the proof, we inspect the last integral along the segment:We require a lead-in lemma which supplies exponential flatness for the difference .

Lemma 13. For each , we can sort two constants such thatfor all , all and all , whenever it is assumed thatfor some fixed close enough to 0 and any given positive real number , under the convention that .

Proof. We first observe that all the maps , , are analytic continuations on the sector of a unique holomorphic function that we name on the disc which suffers the same bound (180). Furthermore, the application is holomorphic on when , and its integral is therefore vanishing along an oriented path described as the union of(a)A segment linking 0 to (b)An arc of circle with radius joining the points and (c)A segment attaching and the originAs a result, taking heed of the integral representation (179) of and , we can convert the difference into a sum of three integrals:where the integrations paths are two halflines and an arc of circle staying aside from the origin that are depicted as follows:We consider the first integral along a halfline in the above splitting:The direction (which might depend on τ) is properly chosen in order thatfor all , for some fixed . Besides, let be any given positive real number (even close to 0). Then, we can find a constant (depending on ) such thatfor all . According to estimate (180), we obtain thatfor all , all , provided that withfor a given .
In a similar manner, we disclose bounds for the next integral over a halfline:Indeed, the direction (that relies on τ) is properly chosen in order thatfor all , for some fixed . The use of (180) together with a record of bounds ressembling (219) allowsto hold whenever , , and restricted to (220) for given and .
In the remaining part of the lemma, we evaluate the third integral along an arc of circle:The circle satisfies the lower boundsfor some fixed , for all (if ) or (if ) granting that . Again, estimate (180) together with (218) for brings onfor all , and submitted to (220) for given and .
By collecting the above inequalities (219), (223), and (226) applied to the splitting of (214), we achieve the announced bounds (212).
Onwards, we take for granted that the real number selected in the above deformation (1), (2), and (3) suffers restriction (213) and . Bound (212) in a row with Lemma 1 yields:wherefor all , all and all . Bound control given below in (235) are now provided for this parameter depending on last integral. The ongoing reasoning leans on the next elementary lemma.

Lemma 14. (1)The next inequality holds for all integers and all positive real numbers , where and is a constant depending on .(2)For all , all , the estimate arises for all integers .

Proof. For the first item (1), using the change of variable we observe thatfor all integers . On the other hand, from the Stirling formula (48), we get a constant (depending on ) such thatfor all . Gathering (231) and (232) yields (229).
The second item (2) can be treated in a similar way through the successive changes of variables and by using the computation already carried out in Lemma 4,whenever .
A consecutive application of (229) and (230) gives rise to the boundsfor all , all , all integers . In other words, we get two constants (which rely on ) such thatwhenever , , for all integers .
Finally, blending (227) and (235) yields two constants (which rely on ) with the estimateprovided that , , and , for all integers .
Lastly, the collection of estimates (198), (201), (207), (210), and (236) applied to the of splitting (185) induces the next boundsfor all , all integers . Since the sequence grows faster than any geometric sequence with ratio , the last inequality (237) warrants the forecast bounds (178).
In the next proposition, we show that difference (178) of neighboring solutions of (33) turn out to be flat functions for which accurate bounds are displayed.

Proposition 6. Let be the set of actual solutions of (33) built up in Theorem 1. Then, we can find constants and (which rely on ) such thatwhereprovided that with , for all , where by convention .

Proof. Let be real numbers and be integers. We define the functionfor all . Keeping in mind the Stirling formula (48), we can find two constants (depending on ) withfor all integers . Hence,where and . In the next part of the proof, we exhibit explicit bounds for the function . We follow a similar strategy as in the recent work [7]. We select a real number small enough in order that for all given , there exists a positive real number such thatWe focus on the integer , where denotes the floor function. By construction, we have . Therefore,which implies thatand henceOn the other hand, we can express in term of the variable x by means of the Lambert function. Namely, we set as the principal branch of the Lambert function defined on and which solves the functional equationfor all . Since relation (243) can be recast in the formwe deduce thatFurthermore, owing to the paper [23], the next sharp lower boundshold for all . Finally, since , the above facts (246)–(250) give rise to the boundswhereprovided that , where . Finally, from (242) and (251), we deduce thatwhenever , which implies the forseen bounds (238) when looking back to estimate (178).

7. Asymptotic Expansions in the Perturbation Parameter

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

We first put forward the notion of asymptotic expansion with double Gevrey and q-Gevrey scales for formal power series introduced by Lastraet al. [6]. Here, we need a version that involves Banach valued functions which represents a straightforward adaptation of the original setting.

Definition 8. Let be a complex Banach space. We set as two integers and as a real number. Let be a bounded sector in centered at 0 and be a holomorphic function. Then, f is said to possess the formal seriesas Gevrey asymptotic expansion of mixed order on if for each closed proper subsector of centered at 0, one can choose two constants withfor all integers and any .
In the literature, the Ramis—Sibuya theorem is known as a cohomological criterion which ensures the existence of a common Gevrey asymptotic expansion of a given order for families of sectorial holomorphic functions (see [24], p.121 or [25], Lemma XI-2-6). Here, we propose a variant of this result which is adapted to the Gevrey asymptotic expansions of the mixed order disclosed in the above definition.

Theorem 2. Consider a complex Banach space and set a good covering in (described in Definition 6). Let be a set of holomorphic maps from into . We define the cocycle , , that stands for a holomorphic function from into , with the convention and .
Assume that the ensuing two requirements hold.(1)The functions are bounded on , for ,(2)The functions suffer the next sequential constraint on ; there exist two constants with provided that , for all integers , all . In other words, has the null formal series as Gevrey asymptotic expansion of mixed order on , for .Then, all the functions , , share a common formal power series as Gevrey asymptotic expansion of mixed order on .

Proof. The entire discussion leans on the following central lemma.

Lemma 15. For all , the cocycle splits, which means that bounded holomorphic functions can be singled out with the next feature:for all , where by convention . Furthermore, a sequence of elements in can be built up such that for each and any closed proper subsector with apex at 0, one can find withfor all , all integers .

Proof. The proof mimics the arguments of Lemma XI-2-6 from [25] with fitting adjustment in the asymptotic expansions of the functions constructed by means of the Cauchy—Heine transform.
For all , we choose a segment:These ς segments divide the open punctured disc into ς open sectors , wherewhere by convention . Letfor all , for , be defined as a sum of Cauchy—Heine transforms of the functions . By deformation of the paths and without moving their endpoints and letting the other paths , untouched (with the convention that ), one can continue analytically the function onto . Therefore, defines a holomorphic function on , for all .
Now, take . In order to compute , we writewhere the paths and are obtained by deforming the same path without moving its endpoints in such a way that(a) and (b) is a simple closed curve with positive orientation whose interior contains ϵTherefore, due to the residue formula, we can writefor all , for all (with the convention that ).
In a second step, we derive asymptotic properties of . We fix an and a proper closed sector contained in . Let (resp. ) be a path obtained by deforming (resp. ) without moving the endpoints in order that is contained in the interior of the simple closed curve (which is itself contained in ), where is a circular arc joining the two points and . We get the representationfor all . One assumes that the path is given as the union of a segment , where and and a curve such that , , and for all . We also assume that there exists a positive number with for all . By construction of the path , we get that the function defines an analytic function on the open disc .
It remains to give estimates for the integral . Let be an integer. From the usual geometric series expansion, one can writewherefor all .
Keeping in mind (256) for the special value and (266), we get some constants such thatfor all . In particular, we deduce the existence of two constants (depending on ) withfor all . Indeed, recall from [24], Appendix B, that for any given real number , as x tends to . Hence, a constant (depending on ) can be sort withfor all . Consequently, (268) follows from (267) and (269).
Moreover, one can choose a positive number (depending on ) such that for all and all . Bringing to mind (256) for the peculiar value and (266) give rise to two constants such thatFor that reason, we can find constants (relying on ) such thatfor all . Namely, from (269) we notice thatUsing comparable arguments, one can give estimates of the form (265), (266), (268), and (271) for the other integralsfor all .
As a consequence, for any , there exist coefficients , , and two constants such thatfor all integers , all .
Besides, identity (263) and sequential assumption (256) imply in particular that the difference has the null formal series as asymptotic expansion (in the Poincaré sense) on . Owing to the unicity of the asymptotic expansions on sectors, we deduce that all the formal series , , are equal to some formal series denoted . Lemma 15 follows.
We introduce the bounded holomorphic functionsfor all , all . By definition, for any , we observe thatfor all . Therefore, each stands for the restriction on of a global holomorphic function called on . Since remains bounded on , the origin turns out to be a removable singularity for which, as a result, defines a convergent power series on .
Finally, one can recastfor all , all . Furthermore, represents a convergent power series and has the formal series as Gevrey asymptotic expansion of mixed order on , for all . The conclusion of Theorem 2 follows.

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

Within this section, we explain the second principal result of our work.

Theorem 3. We set as the Banach space of complex value-bounded holomorphic functions on the product equipped with the supremum norm, where the sector , radius σ, and width are settled in Theorem 1.
Then, for all , the bounded holomorphic functions from into constructed in Theorem 1 share a common formal power seriesas Gevrey asymptotic expansion of mixed order . Videlicet, for all , two constants can be found withfor all integers , provided that .

Proof. We focus on the set of functions , , constructed in Theorem 1. For all , we setEach defines a bounded holomorphic map from into the Banach space described in the statement of Theorem 3. Furthermore, bound (178) implies that the cocycle fulfills the sequential bound (256) on for any . Then, Theorem 2 can be applied in order to get a formal power series which stands for the Gevrey asymptotic expansion of mixed order of each on , for all .

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Conflicts of Interest

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