Asymptotics and Confluence for a Singular Nonlinear -Difference-Differential Cauchy Problem

Abstract

We examine a family of nonlinear difference-differential Cauchy problems obtained as a coupling of linear Cauchy problems containing dilation difference operators, recently investigated by the author, and quasilinear Kowalevski type problems that involve contraction 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 Gevrey structure are exhibited. The other feature concerns the problem of confluence of these solutions as tends to 1.

1. Introduction

In this paper, we study a particular family of nonlinear difference-differential Cauchy problems displayed as follows for prescribed Cauchy data where are integers, is a real number and where the symbol from the leading term of the equation (1) stands for a nonconstant element of , and represents some well-chosen finite subset of . The right part of (1) is a polynomial of degree at most 2 in the variables for , holomorphic relatively to on some disc in centered at the origin and which depends polynomially on 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 difference-differential Cauchy problems outlined as for given Cauchy data where is a suitable natural number, the piece from the leading term of (3) is the polynomial appearing in (1), is the integer stemming from (1), stands for the dilation operator acting on functions through for arising in (1) and the right handside 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 , for some integer , of holomorphic solutions to (3) and (4) could be modeled on products , where stands for some small disc centered at 0 in and where is a suitable set of bounded sectors whose union covers some neighborhood of 0 in , see Definition 1. These solutions were expressed through Laplace transforms of order , along half lines in convenient directions where the Borel map is holomorphic relatively to on and is compelled to bear exponential growth rate (17) w.r.t on some unbounded sector . In [1], we addressed two important features of these solutions (i)Asymptotic behaviour w.r.t the time variable as (ii)Confluence property as tends to 1

Regarding the first point, a fine structure of mixed Gevrey and Gevrey type was disclosed. Namely, as expounded in Theorem 3 and 19, all the partial maps share a common formal power series , where is bounded holomorphic on , as so-called Gevrey asymptotic expansion of mixed order on , meaning that two constants can be pinpointed with whenever , for all integers .

Concerning the second item, discussed in Subsections 5.1 and 5.5, we have shown that for any prescribed sector from the covering , the corresponding solution (whose reliance on the parameter is flagged by an index ) to (3), (4) merges uniformly on , as tends to 1 for some fixed , to a holomorphic map on which itself solves some linear PDE Cauchy problem given by (177), (178). More precisely, some constant (unrelated to ) could be singled out with provided that .

The problem (1), (2) examined in this work is actually obtained by means of a procedure (described in Subsection 2.2) which consists in coupling the singular linear Cauchy problem (3), (4) with a quasilinear Kowalevski type problem which involves the contraction operator acting on functions by means of , framed as for assigned Cauchy data where is an appropriate integer, is some polynomial in with holomorphic coefficients on , the linear piece as well as the data (9) stand for properly chosen polynomials and where the forcing term is required to solve the linear problem (3), (4). Notice that the appearance of the difference operator is mandatory when some time derivative 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 time borders the origin(iii)Confluence aspects as

The first item is completed in Subsection 3.3 (Theorem 16 and its Corollary 17) where a finite set of holomorphic solutions to (1), (2) is built up on domains , provided that the radii of and are taken small enough. Furthermore, the solutions can be represented as Laplace transforms of order , along the same halflines as in (5) where the Borel map remains holomorphic w.r.t on but suffers now (at most) exponential growth rate (91) of order relatively to on (and not in general exponential increase as it was the case for ).

The second item is achieved in Subsection 4.5 (Theorem 30) where the existence of a formal power series , with holomorphic coefficients on , is established which stands for the common Gevrey asymptotic expansion of mixed order on of the partial maps , , satisfying therefore similar estimates to (6).

The last item is explained in Subsection 5.7 (Theorem 44). For any given sector from the covering , the related solution (whose dependence on is marked by the index ) to (1), (2) converges uniformly on , as , to a holomorphic map on 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 Theorems 16, 30, and 44 lean on statements established in [1]. In essence, the features of the solutions of (1), (2) reached in this paper are identical to those of the solutions of (3), (4) achieved in [1]. However, their proofs differ fundamentally. Indeed, the construction of the sectorial local holomorphic solutions 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 of (3), (4). The asymptotic features relatively to time of the solutions were obtained by dint of a version the so-called Ramis-Sibuya theorem which banks on sharp estimates of the difference . 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 difference operators on the Borel maps of the limit maps and (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 nonlinear difference differential equation (1) involves both dilations and contractions w.r.t the time variable by means of the presence of operators for both and . The same property arises in the framework of nonlinear difference equation in the study of the so-called Painlevé equations. Indeed, the discrete versions of the first and second Painlevé equations are expressed through the next two equations for with for some , , and some parameter, see for instance [2, 3]. For an excellent comprehensive and introductive book to -Painlevé equations and more generally to integrable discrete dynamical systems, we mention [4].

Regarding the existence of local holomorphic solutions to nonlinear difference equations, we may refer to some recent works. Indeed, for meromorphic or holomorphic solutions around the origin for special type of nonlinear difference equations such as the Painlevé equations, we can mention [5, 6]. Some category of nonlinear difference equations of the form where 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 difference equations with so-called irregular singularity at the origin, or . In the recent work [8], Gontsov et al. provide sufficient conditions for the convergence of so-called generalized power series with complex coefficients and complex exponents that are solutions of algebraic difference equations where stands for some polynomial, providing in particular local sectorial holomorphic solutions to these equations.

In the context of nonlinear 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 for , , , , and where is some well-prepared analytic function in its arguments.

On the subject of confluence for linear difference equations, some recent references have been pointed out on our latest contribution [1]. The confluence in the framework of nonlinear 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 Painlevé VI equation have been recently undertaken by Dreyfus and Heu in [10]. They construct some well-prepared analog Hamiltonian system where and show that its discreet solution given in the form of two sequences , for given and encodes the Taylor series coefficients of the holomorphic solution to the (formal limit as tends to 1) nonautonomous Hamiltonian system with initial condition and , which defines the sixth Painlevé equation for some prescribed rational map .

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.

Let be integers and be a real number. We set a polynomial with complex coefficients such that

Let be a finite subset of . For all and all , we fix polynomials and with complex coefficients.

We focus on the next singular linear Cauchy problem with polynomial coefficients in time, for given Cauchy data

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. Let be an integer. For all , we select open sectors centered at 0 (and do not contain 0) with given radius that fulfill the next three features: (i)The intersection of any two consecutive sectors of the family is nonempty, namely for all , with the convention that .(ii)The intersection of any three elements in is empty(iii)The union of the sectors covers some punctured neighborhood of the origin in The family is then named a good covering in .
The notion of admissible set of sectors is depicted in the next.

Definition 2. We set as an integer and set as a good covering in . We consider a set of unbounded sectors centered at 0 that endorse the next two properties: (1)Each sector does not contain any of the roots of the polynomial , for (2)For all , there exists a constant such that for all , one can single out a direction (that may depend on ) such that both conditions hold.We say that the set of sectors 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 Section 2.1 of [1] hold true. Fix a good covering in and a set of unbounded sectors chosen in a way that the data forms an admissible set of sectors.
Then, for all , one can construct a solution to the Cauchy problem (11), (12) that is bounded and holomorphic on and that can be expressed through a Laplace transform of order , for where stands for the disc centered at 0 with radius for some well-chosen constant . The Borel map represents a holomorphic function on the domain whose Taylor expansion is subjected to the next bounds for all , all , for suitably chosen constants and , .

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

2.2. Setup of the Main Nonlinear Cauchy Problem

Let be an integer and as defined in the previous subsection. Let be a finite subset of . For all , we fix a bounded and analytic function on a disc centered at 0 with some radius . Furthermore, we define a polynomial for some integer , where the coefficients are bounded and holomorphic on and for all , we denote polynomials with complex coefficients written in the form where stands for a finite subset of .

The finite set is compelled to fulfill to the next list of requirements:

(A1) There exists a real number for which for all .

(A2) The next inequalities hold provided that .

We consider the next nonlinear nonhomogeneous Cauchy problem for given Cauchy data where the forcing term 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 where the forcing term 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 order , namely, along the halfline appearing in the representation (15). So far, the so-called Borel map is supposed to be holomorphic with respect to on the unbounded sector and analytic w.r.t on some small disc centered at 0 with radius . For the Laplace transform to be well defined, we make the further assumption that has at most exponential growth of order w.r.t on , uniformly in on , meaning the existence of two constants with for all . Once we assume that such solutions exists, we will derive some functional equations that the Borel map 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 satisfying the above requirements.

2.3. An Auxiliary Cauchy Problem Satisfied by the Borel Map

We first need to remind the reader the next proposition which is a slightly modified version of Proposition 4 of [1].

Proposition 4. We set as a complex Banach space. Let be an integer and let be a holomorphic function on the open unbounded sector , continuous on . The existence of two constants and such that is assumed for all . Then, the Laplace transform of order of in the direction is defined by along a half-line , where depends on and is chosen in such a way that , for some fixed . The function is well defined, holomorphic, and bounded in any sector where and . (a)The action of the Laplace transform on entire functions is described as follows: if is an entire function on , with growth estimates (26) and with Taylor expansion , then defines an analytic function near the origin w.r.t with convergent Taylor expansion (b)The actions of the irregular operator and the monomial on the Laplace transform are expressed through the next formulas for every integers , and for all with . Here, stands for the convolution product (c)Let be holomorphic maps with the same feature (26) as above. Assume moreover, that is equipped with a product in a way that becomes a Banach algebra. Then, the next multiplicative formula holds for all with , where represents the convolution product (d)The action of the dilation commutes with the Laplace transform, for any integer , namely holds for all for .The point (A) allows the Cauchy data (23) to be expressed through Laplace transforms of order , of polynomials given by , for .
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 map is subjected to the next nonlinear and nonhomogeneous convolution Cauchy problem for prescribed Cauchy data

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)Let be a real number. We set for all integers . We fix a real number. We denote the vector space of all functions that are holomorphic on the unbounded sector (determined in Theorem 3) for which the norm is finite.(b)Let be a real number. We set as the vector space of all holomorphic functions near with holomorphic coefficients on such that the norm is finite. The normed space 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. Let be integers such that Then, one can find a constant (relying on ) such that for all , where stands for the -times iteration of the integration map .

Proof. Let with for all . We check that In the next lemma, we provide bounds for the coefficients of this last series.

Lemma 7. The next bounds hold for all .

Proof. We observe from (35) that where for all . Besides, we notice that for all , since the function is increasing on provided that . We deduce that where for all , all . In the next step, we supply bounds for the map . We check that for all and in a row with the classical estimates for any given integers , , we deduce that for all , all .
At last, collecting (40), (41), and (42) gives rise to the forecast bounds (39).
Owing to the above lemma, we deduce the next bounds Furthermore, we see that for some constant (relying on ), for all under the constraint (37). Finally, gathering (43) and (44) yields the expected bounds (38).

Proposition 8. There exists a constant (depending on ) such that for all .

Proof. Let where , for all . By construction, one can check that

In the next lemma, estimates for the coefficients of the latter series are disclosed.

Lemma 9. There exists a constant depending on such that for all , all such that .

Proof. Departing from the very definition of the norms, we can factorize the upper bounds as follows where for all with . Since is an increasing sequence, we observe that together with , and we get for all . We deduce that with From now on, we perform computations based on the ones already done in our previous work [13]. We first assume that .
We make the change of variable , in the above integral part Besides, using a partial fraction decomposition, we can split the next integral along as for all . Furthermore, the change of variable , for enables us to reach the bounds for some constant provided that with . In a similar way, one can find a constant for which as long as with .
Gathering (51), (52), (53), and (54) yields that provided that .
It remains to check the case . In that situation, the quantity can be computed explicitly and turns out to be a finite positive real number.
Finally, collecting the intermediate upper estimates (48), (50), (55), and (56) gives rise to Lemma 9.
From the expansion (46) along with the lemma 9, we obtain the next bounds from which Proposition 8 follows.

Proposition 10. Let and natural numbers such that Then, there exists a constant (depending upon ) such that for all .

Proof. We check that Proposition 10 is a direct consequence of Proposition 6 and Proposition 8. Indeed, we set for a given . According to Proposition 6, we observe that belongs to and that for some constant provided that (58) holds. Furthermore, we set . By construction, we notice that . Then, we get that Owing to Proposition 8, we get the next bounds for some constant (relying on ). At last, we set which yields the result.

Proposition 11. Let be a holomorphic function on a disc with radius . Then, the next bounds hold for all , where

Proof. Let with for all . We first expand the product which allows to control its norm Besides, since for , we notice that for . Hence, 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 Definition 5.

Proposition 12. We set where the polynomials are given by the Cauchy data (33). Then, (a)For all integers , , the maps belong to , for all .(b)For all integers , , , the maps appertain to , for all .

Proof. We first make the Taylor expansion explicit provided that .
We focus on (a). By the very definition of the norm, we get and we can find a constant (relying on ) for which since , for all , where and . As a consequence, which is a finite quantity.
We turn our attention to (b). The definition of the norm yields We need bounds for the coefficients of this latter polynomial in . Indeed, where is defined in (65). We make the change of variable , for in the above integral and observe that , . This gives rise to a constant (depending on ) such that Finally, which represents a positive real number.

In the following proposition, we claim that the forcing term of the Cauchy problem (22), (23) belong to the Banach space described in Definition 5.

Proposition 13. For all , the map belongs to the space , for all , provided that .

Proof. According to the Taylor expansion (16), the very definition of the norm yields Then, we need to control the coefficients of this latter series. Namely, one can find a constant (relying on given in (17) and on ) with since for all . Consequently, whenever 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 where the map is defined in (63), for some expression .

We check that solves the problem (32), (33) if the quantity fulfills the next fixed point equation

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

In the next proposition, we give sufficient conditions under which represents a shrinking map on some small ball centered at 0 in the space .

Proposition 14. Under the assumption (20), there exists some small real number such that if , one can select a radius such that satisfies the next two properties.
Let be the ball centered at 0 in with radius . (1) maps into , meaning that (2)For all , we have

Proof. We discuss the first point 1. Let belong to with .
Under the constraint (20) and owing to Propositions 6, 11, and 12, we get a constant such that Condition (30) and Propositions 10, 11, and 12 allow us to reach a constant for which According to Propositions 6, 8, 11, and 12, we obtain constants with Propositions 6, 8, 10, 11, and 12 grant the existence of constants such that We first choose the radius in a way that for some fixed . Then, we select sufficiently small with in a way that the next inequality holds, This last inequality can be achieved since the quantities and tend to zero as 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. Let with , for .
Under the condition (30), Propositions 6 and 11 yield a constant for which and Propositions 10 and 11 breed a constant such that In order to deal with the nonlinear terms, we use the next identity which helps to factorize the next difference which leads, by means of Proposition 6, 8, 11, and 12, to constants for which Furthermore, according to Proposition 10, we get a constant with Using the factorization (81), Propositions 6, 8, 11, and 12 grant the existence of constants such that From now on, we select the radius and small enough with such that in order that the next condition holds, 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 constants and 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 Subsection 3.1.

Proposition 15. We take for granted that assumption (20) holds. We fix the constants and as in proposition 14. Then, the convolution Cauchy problem (32), (33) possesses a solution which belongs to the space , for any given provided that . Furthermore, one can single out a constant (relying on ) such that

Proof. According to Proposition 14, we can apply the classical fixed point theorem for shrinking maps in complete metric spaces to the map . By construction, is a complete metric space for the distance since the vector space equipped with the norm is a Banach space. Furthermore, under the constraints imposed in Proposition 14, the map is shrinking of Lipschitz type . As a result, has a unique fixed point, denoted , meaning that This means in particular that we obtain a solution (unique in the ball ) for the equation (111). Moreover, owing to Proposition 6, we check that The decomposition (69) then confirms that the map belongs to , 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 of each bounded sector described in Theorem 3 fulfills for , where is introduced in (14) and where .
Then, the Cauchy problem (22), (23) possesses a holomorphic solution on the product , for some radius small enough. Furthermore, can be expressed by means of a Laplace transform of order , for along a halfline which appears in the representation (15). The Borel map represents a holomorphic function on the domain whose Taylor expansion is submitted to the following estimates for all , all , for well selected constants , any given , where is the sequence defined by (34). In particular, the Borel map suffers the next bounds estimates for .

Proof. The proof is a direct consequence of Proposition 15 and of the construction made in Subsection 2.3.

In the next corollary, we show that turns out to solve a nonlinear Cauchy problem with analytic coefficients in space near the origin and polynomial in time , which involves both differential operators and dilatations/contractions difference operators acting on time.

Corollary 17. The holomorphic map solves on the product a particular nonlinear Cauchy problem which is polynomial in time of the form for given Cauchy data of the form together with for well-chosen polynomials with complex coefficients for (which depend on ).
The set is a finite subset of which satisfies in particular the constraint for all and is a finite subset of with the property that whenever .
For all , the coefficients are polynomial in and holomorphic w.r.t the disc . The map is polynomial in the variable , a polynomial of degree at most 2 in the variables for and relies holomorphically on the variable in the disc .

Proof. We introduce the next two difference differential operators and where stands for the dilation operator acting on functions through . By construction, the holomorphic map satisfies and the map fulfills for . By coupling (95) and (96), we obtain that which can be expanded in the more precise shape and gives rise to the equation (136). Concerning the Cauchy data, the map is compelled to the conditions (33) which are rewritten in (93). Furthermore, the fact that is subjected to the constraints (12) at can be recast in the form for which can be rephrased through the Cauchy conditions (138) for suitably selected polynomials with complex coefficients for , since for all provided that .

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

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 for prescribed Cauchy data where is a finite subset of and where the coefficients are polynomial in and holomorphic relatively to on the disc given in Subsection 2.2.

According to the condition (31) imposed on the set in (22), the next feature holds for the set , for all . Indeed, it is straight to check that for any given integer , the decomposition holds for suitable polynomials with real coefficients that rely on .

4.1. Reduction to an Auxiliary Cauchy Problem

We first write down the convergent Taylor expansions of the coefficients of (98) at , for all , all . Furthermore, we expand the forcing term at

that converges provided that , for all according to Theorem 3. Finally, we recast the analytic solution to (98), (99) obtained in Theorem 16 as Taylor series at , which is convergent provided that and . The constant is in particular taken small enough in a way that and . By plugging the above expressions in the main equation (147), we check that solves (98), (99) if and only if the sequence of functions satisfies the next recursion for all , with prescribed conditions

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 Subsection 4.4.

At the onset, we apply the general differential operator for all integers to the latter recursion (104) and get for all , all .

We set as a proper subsector centered at 0. We introduce the next set of sequences where by definition, for , along with for all . Since the map leaves stable (i.e., ) for any integer , we deduce from the recursion (106) a sequence of inequalities for all .

We introduce a sequence denoted which fulfills the next recursion relation with given initial data

By construction, in comparing (110) and (111), (112), we observe the crucial fact that for all . Let us define the next formal generating series together with

A direct computation following from the recursion (111), (112) shows that the formal series solves the next nonlinear Cauchy problem for prescribed Cauchy data where for , whose coefficients are defined by (108). Since are polynomials in , we deduce that are also polynomials in the variable , for .

4.2. Asymptotic Expansions and Bounds for the 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 of the maps (described in Theorem 3).

Theorem 18. (1)There exists a formal power series with bounded holomorphic coefficients on some fixed disc for a well-chosen constant , which represent the common asymptotic expansion of all the functions on , for , uniformly relatively to on . It means that, for each , for each proper subsector , for each integer , one can single out a constant with for all .(2)The maps are infinitely often differentiable at the origin and for all , all , given that .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 Corollary 17 of Theorem 16 from [1], we derive important asymptotic bounds for the th derivative of the partial maps on bounded sectors. Indeed, the next result holds.

Theorem 19. For each , for each proper subsector , one can select two constants for which for all integers , all , where 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. Let be real numbers. We set and as real numbers. We define the space as the vector space of formal power series , such that the norm is finite. One checks that the vector space equipped with the norm represents a complex Banach space.

Remark 21. In the case , 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. Let . Then, the product belongs to and one can find a constant (depending on ) such that In other words, turns out to be a Banach algebra.

Proof. Let for two elements of . Their product writes By definition, its norm fufills the bounds Since , we first observe that for all integers with .
The next lemma is essential.

Lemma 23. There exists a constant (depending on ) such that for all integers with and .

Proof. We depart from the next inequality for all with and . These bounds straightly follow from the binomial expansion for each term of the identity , for .
In the next step, we need to prove the next.

Lemma 24. The next inequality holds for all Proof We recall the next identity defining the so-called Beta function (see [14], Appendix B) provided that are real numbers. As a result, observing that provided that , we get that from which Lemma 24 follows.
From the functional property for any , we can factorize Combining the bounds (125) and the expansions (127), we get that where where , with . Besides, a constant (relying on ) can be found such that for all . 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

The next proposition helps us in addressing the linear part of the Cauchy problem (116), (117).

Proposition 25. Let be integers under the requirement Then, the linear operator is bounded from the space into itself. In other words, there exists a constant (depending on ) with for all .

Proof. Let be in . The action of the difference differential operator is expressed through and enables us to rewrite the norm where for all , . In the continuing part of the proof, we show that the sequence is actually bounded by some constant.
Indeed, recall from [14], Appendix B, that for a given , one can find a constant (relying on ) such that provided that . Consequently, we get two constants and (depending on ) with for all , , provided that (131) holds. On the other hand, for all under the assumption (131). As a result, we obtain for all , . Finally, based on (133) and (136), we deduce 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 space for well-chosen parameters.

Proposition 26. Let chosen as in Section 2. (1)The series appertains to for small enough(2)The series and belong to for small enough(3)We set up For any nonnegative integers , the polynomials belong to for any given .

Proof. We focus on the first point (1). According to Theorem 3 stated in Subsection 4.2, with the help of the Cauchy formula, we deduce that the sequence defined in (109) is subjected to bounds of the form for some selected constants , whenever , . Besides, from the formula (182), we observe in particular that provided that . This last inequality combined with the functional relation for any begets for all , all . Thereby, we deduce constants with for all . Bestowing the last expansion of (115) implies the next norm bounds provided that are constrained to and .
We address the second point (2). According to the assumptions of Section 2, the functions and are polynomial in and bounded holomorphic on some disc relatively to , for all . From the Cauchy formula, we deduce that the first two sequences in (109) satisfy the bounds for some well-chosen constants and , for all . Owing to (139) and since the map is increasing on (see [14], Appendix B), we check that for all . Whence, constants and can be found with for all . Thanks to the first two expansions of (115), we deduce the control of the norms together with whenever are submitted to , and , .
Finally, the last point (3) is straightforward since for any given integers , the quantity is a polynomial in the variables and is therefore finite for any fixed .

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 where the polynomial is displayed in (137), for some formal series to be determined.

We observe that fulfills the problem (116), (117) if the expression solves the next fixed point equation

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 Section 4.3. In order to fulfill this objective, we set up the next nonlinear mapping defined as

In the next proposition, we discuss sufficient conditions under which represents a shrinking map acting on some small ball centered at the origin in the space .

Proposition 27. Taking for granted the condition (149), some small real number can be singled out such that if , one can choose a radius such that suffers the next features.
Let be the ball centered at 0 with radius in the space . (1)The next inclusion holds.(2)For any , we have

Proof. We behold the first property. Let belong to the ball . Under the constraint (100), Propositions 22, 25, and 26 allow the next inequality to hold true along with for some constants and . Eventually, due to Propositions 22 and 26, we arrive at for some constants and .

From now on, we take the radius in a way that for some fixed . Then, we pick up a tiny constant with aiming to the next constraint

This latter inequality can be reached since the quantities and are small as 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. Let be taken within the ball . The above inequality (153) prompts the next bounds for the linear piece

With an eye toward the nonlinear block, by means of the basic identity , we factorize the difference of squares as

Consequently, on account of Propositions 22, 25, and 26, the next estimates follow. Then, we pick out a small constant with intending to the next condition

Storing up the previous bounds (157) and (158) under the restriction (159) triggers the Lipschitz property (152).

At last, we compel the constants and 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 Section 4.3.

Proposition 28. Let us assume that the condition (149) holds. Let the constants be determined by Proposition 27. Then, the nonlinear Cauchy problem (116), (117) owns a formal power series solution that belongs to the space presuming that . Along with it, one can find a constant (resting on ) such that

Proof. On the basis of Proposition 27, the classical fixed point theorem for shrinking maps in complete metric spaces can be applied to the map from the plain observation that represents a complete metric space for the distance since the vector space endowed with the norm is a Banach space. Indeed, under the conditions asked in Proposition 27, the map appears to be of Lipschitz type and hence shrinking. Whence, has a unique fixed point, denoted , signifying that In particular, a unique solution for the equation (149) is found in the ball . Along with it, due to Proposition 25, norms estimates can be achieved, Consequently, the splitting (148) attests that the map appertains to 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. Let be a complex Banach space. We set an integer and a real number. Let be a holomorphic map, where stands for a bounded sector in centered at 0. Then, the map enjoys the property of having the formal series as Gevrey asymptotic expansion of mixed order on if for each closed proper subsector of centered at 0, two constants can be distinguished with for all integers and any .
We are ready to enunciate the second main outcome of the work.

Theorem 30. We consider the set of solutions to the main Cauchy problem (22), (23) established in Theorem 16. If one sets as the Banach space of bounded holomorphic functions on the disc equipped with the sup norm, then each partial map can be viewed as a holomorphic map from the sector into , for , as long as , where is set up in Theorem 16.
Hence, provided that is taken small enough, for all , the maps share a common formal power series as Gevrey asymptotic expansion of mixed order on . To rephrase it, for each , for each proper subsector , two constants can be singled out with for all , all integers .

Proof. According to the bounds (160) from Proposition 28 and Definition 20, the unique formal series solution of the Cauchy problem (116), (117) given by the expansion (114) is compelled to satisfy the next bounds for all . On the other hand, the next lemma will be required.

Lemma 31. There exist constants , such that for all real numbers .

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 for which for all . Therefrom, Besides, one can pinpoint two constants and such that for all and from the classical estimates for , we deduce that for all real numbers . 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 for all , together with for all , , for some constants , , and (relying and for ), where stands for a fixed subsector of .

From the Taylor formula with integral remainder, for each , , we can expand the function as follows where for all , all integers . The remainder can be estimated from above by means of the latter key bounds (169), for all .

For each , we define the formal power series

It turns out that can be rewritten in the form where the expressions are holomorphic on a disc provided that its radius is taken small enough. Indeed, the first term is expressed through the series which represents a holomorphic function on for since owing to the bounds (168), whenever . Besides, for all , the coefficients are given by the expansions which define holomorphic maps on , when since according to the bounds (169), given that .

Consequently, in view of both expansions (170) and (173) along with the remainder estimates (171), we deduce the next error bounds for all , all integers , given that , for chosen as above.

In the last part of the proof disclosed within the next lemma, we show that the coefficients do not depend on for all .

Lemma 32. The coefficients of the formal series given by (172) do not depend on , for all . As a result, the coefficients of the formal expansion (173) do not depend on for all and thereby each formal series turns out to be written as a single formal series with holomorphic coefficients on , for .

Proof. Paying regard to the relation (156), we get in particular the next recursion for the sequence , with prescribed set of data Now, we observe that the above prescribed quantities for and along with the forcing terms for all , all , do not depend on , 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 do not depend on for all . .
At last, the statement of Theorem 30 issues from the above lemma 32 and the error bounds (174).

5. Confluence as 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 solutions to the initial problem (22), (23) relatively to the parameter , constructed in Theorem 16. On that account, we denote the function . We also attach a second index to the Borel map by setting within the integral representation (88). From now on, the real parameter is chosen inside an interval for some fixed real number .

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 for given Cauchy data where all the data and the coefficients with along with the initial data for are already declared in Section 2.1.

In Section 8.1 of the paper [1], the next statement is outlined.

Proposition 33. Let be an admissible set of sectors as chosen in Definition 2. Let be one sector belonging to the family of unbounded sectors .
One can build up a solution to the Cauchy problem (177), (178) which is bounded holomorphic on a domain where the bounded sector belongs to the family of bounded sectors from and corresponds to under the requirement (2) of Definition 2 and where is some well-chosen constant. The map can be expressed by means of a Laplace transform of order , along a halfline described in Definition 2 (2), for all .
The Borel map represents a holomorphic function on . Its Taylor expansion suffers the next bounds for all , all , for fittingly chosen constants and , .

5.2. A Limit Nonlinear Cauchy Problem

In this subsection, a novel Cauchy problem is introduced that we call the limit problem as tends to 1. Its shape is displayed as follows for given Cauchy data where the forcing term is the holomorphic solution of the singular linear Cauchy problem (177), (178) unveiled in Proposition 33 of the former subsection. Besides, all the items along with the coefficients , for and the Cauchy data for are those already introduced in Section 2.2.

We aim for a solution to (182), (183) having the profile of a Laplace transform of order , along the halfline given in (179), where the Borel map is holomorphic with respect to on and analytic w.r.t on some small disc centered at 0 with radius .

The same computations as the ones of Section 2.3 by means of Proposition 4 shows that the Borel map solves that next auxiliary Cauchy problem for given Cauchy data

For later need, we make the change of variables and for in the integrals involved in (185). As a result, solves that next auxiliary Cauchy problem for given Cauchy data (186).

The next proposition comes along with the same steps as in the proofs of Proposition 15 and Theorem 16.

Proposition 34. Let be the admissible set of sectors distinguished in Proposition 33. Let be the infinite sector belonging to selected in Proposition 33.
A solution to the Cauchy problem (182), (183) can be reached, that is bounded and holomorphic on a domain for some small radius , where is the bounded sector from singled out in Proposition 33. Moreover, is described using a Laplace transform of order , for all , along the halfline that appears in the representation (179). The Borel map stands for a holomorphic map on the domain which solves the above auxiliary Cauchy problem (185), (187), and (186). Besides, the map belongs to the space and enjoys a decomposition of the form where is defined in (63) and belongs to and satisfies for some suitable constant .

In the ensuing corollary, we observe that actually solves a limit nonlinear Cauchy problem with analytic coefficients in space in the vicinity of the origin and polynomial in time .

Corollary 35. The analytic map solves on the product a particular Cauchy problem relying polynomially on time with the shape for given Cauchy data of the form together with for well selected polynomials , (that are independent from ). The set is a finite subset of for with the property that for any . The map is polynomial in time and in its arguments with and analytic w.r.t on .

Proof. We set up the differential operators and According to Proposition 34, satisfies and based on Proposition 33, the map solves provided that . By pairing (194) together with (195), we obtain that is represented by the equation (269). Regarding the Cauchy data, the map suffers the conditions (261) which are replicated in (192). Moreover, the constraints (178) put on can be reworded as for which begets the assumptions (193) for properly chosen polynomials with complex coefficients for .

5.3. Analytic Solutions to the Limit Singular Linear Cauchy Problem under the Action of a 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. Let be an integer and let . Two constants (that are unrelated to ) can be found such that for all , for all integers , where the constants , , , and the unbounded sector appear in Proposition 33.

5.4. Analytic Solutions to the Limit Nonlinear Cauchy Problem under the Action of a Difference Operator

This subsection is dedicated to the proof of the next technical proposition.

Proposition 37. Let be an integer and set . Then, one can find some constants and (unrelated to ) such that for any given , if is taken small enough, the next bounds 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 is compelled to solve. We first state a Cauchy problem satisfied by the quantity by substituting by in the equations (265), (186). On the way, we need to perform some practical computations. Namely, using the basic identity , we recast the next list of pieces in a suitable manner: and along with and 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 given in (199) owning the following shape: for given Cauchy data In the upcoming step, we plan to solve the problem (206), (207) within the Banach space described in Subsection 3.1. 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 , for . Indeed, we deduce that turns out to solve the next linear Cauchy problem for assigned Cauchy data 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 for some expression to be determined, where for , with .
We pinpoint the crucial fact that solves the problem (208), (209) if the quantity satisfies the next fixed point equation Our next duty will be to search for a solution and provide bounds relatively to for a solution to (212) in the Banach space , for any given , provided that is chosen small enough. With that in mind, we introduce the linear map In the next lemma, we discuss sufficient conditions under which acts as a shrinking map on a small ball centered at 0, whose radius depends on , in the space .

Lemma 38. We take for granted that the conditions (30) hold. Then, one can single out a small real number in a way that if , one can select a radius which is independent of (but relies on such that ), such that possesses the next two qualities: let be a ball centered at 0 with radius (1) maps into itself, meaning that (2)For any , we have

Proof. As a prefatory material, norm bounds estimates are required for some pieces of the map . (a)Indeed, at first we need estimates for the norm . According to the expansion (180) and the definition of the norm, we get and owing to the bounds (197) brought to mind in Proposition 36, we can upper bound the coefficients of this latter series by for some constant (unrelated to ), for all integers . Thereupon, it leads to provided that .(b)We focus on the quantities for integers under the constraint (20) and on According to the decomposition (189) with the bounds (190) and owing to Propositions 6 and 12, we obtain along with for some constant (independent of but relying on such that ) provided that . (c)We ask for sharp bounds for the norms , for integers under the constraint (20). Departing from the expansion (211), we observe that and its norm writes where since , for all . Furthermore, we can recast the difference as an integral by means of the parametrization with . Since is a polynomial with complex coefficients, its derivative can be written in the form for some finite subset of and complex coefficients . We set and since for all , all , we get in particular that whenever and . We deduce from (221) and (222) that for all . Thereupon, gathering (220) and (223) yields a constant for which Keeping in mind (219), we finally get a constant such that (d)Bounds for the quantity are also required, for integers under the constraint (20). Based on the decomposition (189) with the bounds (190), Propositions 10 and 12 allow to get a constant with presuming that . (e)We demanded also accurate estimates for the norms for integers under the constraint (20). From the expansion (211), we check that and therefore, On the basis of the upper bounds (223), by means of the change of variable with and keeping in mind the lower bounds , we deduce a constant for which At last, pairing (227) with (228) gives rise to a constant such that We are now ready to come to the core of the proof. We fix our attention on the first item 1.
Let belong to such that We provide explicit bounds for each block of the map .
Proposition 11 and (217) return Propositions 6 and 11 together with (225) trigger Proposition 11 and (226) afford Propositions 10 and 11 along with (229) prompt Propositions 6, 8, and 11 with the help of (217), (218) and (225) furnish Propositions 8, 10, and 11 together with (217) beget Proposition 6, 8, 10, and 11 coupled with (217), (218), and (225) breed From now on, we select a small sized quantity and suitable (taken independently of ) in a way that the next inequality holds Notice that the above constraint (238) can be achieved since all the quantities remain bounded for provided that , , and .
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 map . Let be elements of with for .
According to the computations made to treat the first property 1 of , we deduce forthrightly the next list of inequalities and along with and Hereafter, we choose small enough in a way that and gathering (239), (240), (241), and (242) gives rise to the shrinking property (215).
Eventually, we select the constants and 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 space .

Lemma 39. Let us presume that condition (30) holds. We single out the constants and (independently of in ) as in Lemma 38. Then, the linear Cauchy problem (208), (209) owns a solution that belong to the Banach space for any given . Along with it, a constant (unrelated to ) can be found with the next feature

Proof. Derived from Lemma 38, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map according to the fact stands for a complete metric space for the distance . Whence, carries a unique fixed point denoted inside the ball , meaning that As a result, a unique solution for the equation (292) is established in the ball . Furthermore, proposition 6 conjointly with the decomposition (210) and the bounds (225) certify that the map belongs to , solves the problem (208), (209), and suffers the next upper estimates for some constant which is unattached to in the range ..
According to Lemma 39, it turns out that the unique formal series in with holomorphic coefficients on solution of (208), (209) is subjected to the bounds (244). Since the difference , which represents in particular a formal power series in with holomorphic coefficients on , is shown to solve (208), (209) in the first step of Proposition 37, it must coincide with the solution 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 to the linear Cauchy problem (11), (12) relatively to the parameter set up in Theorem 3. This result will be applied in the ensuing subsection.

We consider the admissible set of sectors chosen in Proposition 33, where an unbounded sector and corresponding bounded sector are distinguished. We denote for , the solution of the problem (11), (12) displayed in Theorem 3, along the halfline taken in Proposition 33, where the Borel map has now an attached index in order to keep track of the dependence in . According to Theorem 3, represents a holomorphic function on the domain with a Taylor expansion of the form whenever , . We keep the notations of Section 5.1. In Section 8.3 of [1], the next result is stated.

Proposition 40. Let . Two constants (that are unconnected to ) can be singled out such that for all , all integers , where the constants , , 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 Theorem 44.

Proposition 41. Let . Then, one can select two constants and that are unrelated to , such that for any given , the next bounds hold, where stands for the solution to the Cauchy problem (32), (33) that belongs to the space for some (relying on but independent of ) built up in Proposition 15 and where 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 for which we seek upper bounds.
To that end, we need some prelusive computations. The basic identity 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 and along with the differences for the nonlinear terms that can be reshaped as and Owing to the fact that (resp. ) 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 as follows for prescribed vanishing Cauchy data In the ensuing part, we intend to solve the above problem (253), (254) by way of the Banach spaces introduced in Subsection 3.1. Namely, we search for a solution to (253), (254) shaped as for some expression to be specified. One checks that matches the problem (253), (254) if the map is subjected to the next fixed point equation In the sequel, we seek for a solution, for which sharp bounds relatively to are exhibited, to (256) inside the Banach space , for any prescribed in the condition that remains small enough. In order to meet this objective, we set up the next linear mapping In the following lemma, sufficient conditions are enunciated under which becomes a shrinking map on a tiny ball whose radius hinges on centered at the origin in the space .

Lemma 42. Assume that the condition (30) holds. Then, a small sized real number can be singled out in a way that if , a radius can be distinguished (in an unrelated manner to , but depending on for which ), such that acquires the next two hallmarks: let us denote the ball centered at 0 with radius in , (1) maps into itself, signifying that (2)The inequality holds as long as .

Proof. We first supply norm upper estimates for some parts of the map . (a)Upper bounds are established for the norm . Departing from the expressions (258) and (339), we arrive at and owing to the bounds (247) stirred up in Proposition 40, the coefficients of the above series can be upper bounded by for some constant (unrelated to ), for any integers . Thereby, it brings in on the condition that .(b)Bearing in mind (217), we already know that for some constant (independent of but relying on ), where are defined in Proposition 34, as long as .(c)We need accurate bounds for the norms , for nonnegative integers submitted to the condition (30). Calling to mind Lemma 39, the next splitting holds where belongs to the ball in for some constants and where is defined in (211). Owing to (225), we can set a constant with Besides, according to Proposition 6, we get a constant (independent of ) such that Thereupon, we deduce from (263) and (264) the awaited bounds provided that . (d)Keeping in mind (226), we observe that as long as that .(e)Precise bounds for the norms are required. Owing to the bounds (229), we can exhibit a constant with and according to Proposition 10, we get a constant (unrelated to ) with Thus, on grounds of (267) and (268), we reach the due bounds whenever . (f)We also need the bounds which is a particular case of (261) and according to Proposition 15, one can select constants (unrelated to but depending on ) with whenever .
We are now in position to reach the heart of the proof. We focus on the first attribute 1. We set in with We display explicit bounds for each piece of the map .
Proposition 11 and (261) beget Proposition 11 and (265) yield Propositions 6 and 11 trigger Proposition 11 and (266) prompt Proposition 11 and (269) return Propositions 10 and 11 furnish Propositions 6, 8, and 11 with the help of (270) and (271) promote Propositions 6, 8, 10, and 11 together with (270), (271) spark off From now onwards, we choose a small sized quantity and proper (taken freely from ) in a way that the next inequality holds Observe that the above constraint (281) is achievable since all the quantities remain bounded whenever , for given integers , .
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 map . Let be elements of such that for .
The estimates involved in the first property 1. of enable us to write down the next set of inequalities and along with and From this point forward, we pick up close enough to 0 in a way that The collection of bounds (282), (283), (284), and (285) contingent upon (286) sparks off the second feature (259).
At last, we single out the constants and 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 space .

Lemma 43. Take for granted that the condition (30) hold. We fix the constants and (independently of in ) as in Lemma 42. Then, the linear Cauchy problem (253), (254) carries a solution that belongs to the Banach space for any given . Along with it, a constant (unrelated to ) can be distinguished with the next property

Proof. Based on Lemma 42, the classical fixed point theorem for shrinking maps on metric spaces can be used for the map due to the fact that represents a complete metric space for the distance . Thus, gets a unique fixed point denoted inside the ball , meaning that Thereby, a unique solution for the equation (350) is confirmed in the ball . Furthermore, Proposition 6 conjointly with the decomposition (255) warrant that the map belongs to , solves the problem (253), (254) and is submitted to the next upper estimates for some constant which is unattached to in the range ..
According to Lemma 43, it turns out that the unique formal series in with holomorphic coefficients on solution of (253), (254) suffers the bounds (287). Since the difference , which represents in particular a formal power series in with holomorphic coefficients on , is shown to solve (253), (254) in the first step of Proposition 41, it must coincide with the solution 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) as

In this subsection, we unveil the third and last main result of the work.

Theorem 44. Let be the admissible set of sectors distinguished in Proposition 33. We denote the unbounded sector singled out in Proposition 33 and its corresponding bounded sector in accordance with the requirement (2) of Definition 2.
We denote the bounded holomorphic solution to (22), (23) on the product , given by a Laplace transform of order , see (88), constructed in Theorem 16. Besides, we consider the bounded holomorphic solution of the nonlinear limit problem (182), (183) on the domain expressed through a Laplace transform of order , see (188), built up in Proposition 34.
Then, a constant (unrelated to ) can be found such that for all . In other words, the solution of (22), (23) merges uniformly on to the solution of (182), (183) as .

Proof. We express both solutions and as Laplace transforms along a halfline assigned to the condition for some fixed constant , provided that , where the Borel map is outlined in (89) and is described in (189) whose Taylor expansion can be displayed as for .
The deviations bounds reached in Proposition 41 can be rephrased in the next explicit way for all , provided that .
Thereby, we can govern the difference in the following manner for all , all , bearing in mind that radius of fulfills according to (87). This achieves the expected bounds (289).

Data Availability

The work is self-contained and all data supporting the work are included in the manuscript.

Disclosure

The present work is registered as a preprint on http://preprints.org where it is quoted as [17].

Conflicts of Interest

The author declares that he has no conflicts of interest.