Abstract

We investigate a family of nonlinear partial differential equations which are singularly perturbed in a complex parameter and singular in a complex time variable at the origin. These equations combine differential operators of Fuchsian type in time and space derivatives on horizontal strips in the complex plane with a nonlocal operator acting on the parameter known as the formal monodromy around 0. Their coefficients and forcing terms comprise polynomial and logarithmic-type functions in time and are bounded holomorphic in space. A set of logarithmic-type solutions are shaped by means of Laplace transforms relatively to and and Fourier integrals in space. Furthermore, a formal logarithmic-type solution is modeled which represents the common asymptotic expansion of the Gevrey type of the genuine solutions with respect to on bounded sectors at the origin.

1. Introduction

In this paper, we examine a family of singularly perturbed nonlinear partial differential equations modeled as for vanishing initial data . The constants are natural numbers and , , and for and for , where and are two finite subsets of the positive integers , which stand for polynomials with complex coefficients. The linear differential operator depends analytically in a perturbation parameter on a disc with radius centered at 0 and relies polynomially in the complex time and holomorphically with respect to the space variable on a horizontal strip framed as in , for some given width . The forcing term is a map of the logarithmic type represented as a sum where and are polynomials in , with holomorphic coefficients in on and in on . The map is a specific polynomial of degree at most 2 in its arguments , , and , which relies holomorphically in on and in on . The precise shape of is framed in (36).

The main objective of the work (depicted in Theorem 24 of Subsection 8.2) is the construction of a set of logarithmic-type holomorphic solutions to the nonlinear Equation (1), and the analysis of their asymptotic power series expansions in the small parameter on sectors in centered at 0.

The nonlinear term of (1) involves not only powers of and , but also powers of derivatives of where is a nonlocal operator acting on which represents the so-called monodromy operator around 0 relatively to . In the literature, the concept of formal monodromy around a point in appears in the construction of formal fundamental solutions to linear systems of differential equations with a so-called irregular singularity at the given point , known as the Levelt-Turrittin theorem, see [1]. It asserts that a differential system of the form for analytic coefficient matrix near 0 with , for an integer , with an irregular singularity at 0, possesses a formal fundamental solution with the shape for some well-chosen integer , where is a formal meromorphic invertible matrix, is a diagonal matrix whose coefficients are polynomials in with complex coefficients and is related to the so-called formal monodromy matrix by the formula . It is worth remarking that this formal monodromy matrix extends in the formal settings the so-called monodromy matrix that appears in the representation of fundamental matrix solutions to systems (3) with a regular singularity of the form where is an invertible matrix with meromorphic coefficients near 0, for a matrix giving rise to the monodromy matrix by means of . The matrix is obtained as an analytic continuation of the fundamental matrix solution along a simple loop going counterclockwise around the origin 0 with base point by means of the identity where denotes the analytic continuation along , see [2]. In the same manner as the analytic continuation operator acting on analytic functions, a formal monodromy operator acting on various spaces and rings (such as the so-called Picard-Vessiot rings) through the formulas for complex numbers and where is the symbol for the Log function has been introduced and studied from an abstract and algebraic point of view in the textbook [1].

In our context, the action of the formal monodromy on can be reformulated as a shift mapping on angles in polar coordinates by means of the change of functions for , with radius and angle , through the formula

In this way, the main Equation (1) can be recast as some nonlinear mixed type partial difference-differential equation for the map . This class of equations has become the object of many investigations these last years and possesses numerous applications to engineering problems and biology, see for instance the introductive book [3].

In the context of singularly perturbed differential equations, most of the papers in the literature are devoted to advanced or delayed equations of the form for some vector-valued function , where stands for a small positive parameter and where is some fixed constant. Some abstract convergence results and historical background can be found in [4]. The construction of solutions having asymptotic expansions of the form with error-bound estimates for the remainder , , is extensively discussed for instance in the recent textbook [5] for the linear setting and in the paper [6] for some nonlinear cases.

In the particular situation of nonlinear difference equations in the complex domain with the shape for valued analytic maps in a neighborhood of for some , we notice that important results concerning asymptotic features of their solutions as tends to infinity have been obtained by several authors, see [7–9].

In the framework of singularly perturbed partial differential equations, we refer to some interesting works in the case involving delay operators such as [10–13] or entailing nonlinearities with nonlocal operators of integral type related to the famous Schrödinger equation in physics such as [14–17].

We highlight our premise that the main Equation (1) counts in powers of the basic differential operator which is labelled of Fuchsian type. We refer to [18] for many sharp results about Fuchsian ordinary and partial differential equations. However, under the sufficient conditions required on (1) listed in Subsection 2.2, it pans out that (1) will be reduced throughout the work to a coupling of two partial differential equations, stated in (81) and (82), that comprise only powers of the basic differential operator , for a well-chosen integer , of irregular type in a complex variable . The definition of irregular-type differential operators is given in the classical textbook [19] in the ordinary differential equation settings displayed in (3) and in the work [20] in the framework of partial differential equations.

In the present contribution, we cook up a set of holomorphic solutions to (1) shaped as logarithmic-type maps that involve Fourier/Laplace transforms. Namely, under the list of requirements which mould (1) and detailed in Subsection 2.3, one can outline (i)A set of properly selected bounded open sectors for some finite set and centered at 0(ii)A family of holomorphic functions , , which conforms to solutions of (1) on the domain . Each solution , , is expressed as a sumwhere each component , , is represented as a Fourier/Laplace transform where the commonly named Borel/Fourier map stands for a function (i)Which is analytic near (ii)Being (at most) of exponential growth of some order on an infinite sector containing the half line with respect to for suitable direction (iii)Continuous and subjected to an exponential decay with respect to on with analytic dependence in on the punctured disc .

Furthermore, owing to their Laplace integral structure, the components own asymptotic expansions of Gevrey type in the parameter . Indeed, for given , all the partial functions , , share a common asymptotic formal power series expansion on , with bounded holomorphic coefficients on . These asymptotic expansions turn out to be of Gevrey order on every sector , meaning that constants can be singled out for which the error bounds hold for all integers , all , uniformly in and . At last, we verify that the formal logarithmic-type expression itself obeys the main Equation (1).

Throughout the proof of our main result, we show that the components , of the built-up solutions , , to (1) turn out to be embedded in a larger family of maps , , for all integers for some integer . These maps are bounded holomorphic on products where stands for a set of bounded sectors, entailing for , which represents a good covering in (see Definition 18). Each map , , is modeled as a rescaled version of a bounded holomorphic map through on domains for any fixed , where are bounded sectors bisected by the direction , depicted in Definition 19 of the work. The set of maps is shown to solve a specific nonlinear partial differential equation with coefficients that are polynomial in , holomorphic with respect to on and relatively to on displayed in (66). The set of maps conforms a particular nonlinear partial differential equation stated in (67) whose coefficients and forcing term bring in not only polynomials in and holomorphic dependence relatively to on and to on but also polynomial reliance on the maps and their derivatives with respect to and . In this sense, the maps , , solve a coupling of nonlinear partial differential equations. The asymptotic property for the components , , of stems from sharp exponential bound estimates for the differences of neighboring maps reached in Proposition 21, for which a classical statement for the existence of asymptotic expansions of the Gevrey type can be applied, see Subsection 8.1.

In this work, as mentioned above, we restrict ourselves to quadratic nonlinearities. Besides, they are chosen in a way to respect the natural triangular structure of the systems of partial differential equations satisfied by the components , stated in (256) and (257), which stems from the linear part of (1). It means that its resolution is reduced to the study of a coupling of two equations which comprise one single equation satisfied by and a second equation for with coefficients and forcing term that involve .

From a computational or numerical viewpoint, two cogent merits of our approach consist in the facts that (i)The coefficients and of the formal expansions , satisfy simple and explicit recursion relations (stated in (262) and (270))(ii)The well-known least-term truncation method applies in our context for the analytic components of our solutions to (1) since the expansions involved (15) are of Gevrey type, leading to error bounds with exponential accuracy. Namely, one can find constants , , such that the piecewise holomorphic map satisfies for all , uniformly in and . For a reference about this truncation method, we quote the textbook [21], Subsection 4.5.

The major drawback of our proposal is that strong regularity (namely holomorphy) is assumed on the solutions we build up and on the coefficients and forcing term of our main Equation (1). Besides, we need to work with a complex parameter . Most of the recent studies in the domain of singularly perturbed advanced-delay differential or partial differential equations are obtained under weaker requirements and involve a positive real parameter . For an overview on recent numerical approaches developed for problems both singularly perturbed and advanced-delay and for statements on their error bounds, we mention [22, 23].

The approach developed in this work can be extended to the construction of both formal and genuine holomorphic solutions to comparable problems as (1) with higher-order logarithmic terms for , for suitable nonlinear terms and forcing terms chosen properly in a similar way as the ones in the present work. We focus on the complete description for the case for the sake of simplicity in order to give the readers a clear idea of the main purpose of the study and avoiding cumbersome notations and computations.

Logarithmic-type solutions have been extensively studied in the framework of nonlinear partial differential equations with so-called Fuchsian type and described in the Chapter 8 of the textbook by Gérard and Tahara [18]. Namely, these authors consider nonlinear partial differential equations with the shape where for some integers , for analytic maps near the origin in . Under conditions of nonresonance of the characteristic exponents at combined with some Poincaré condition on the characteristic polynomial associated to (21), they have described the holomorphic solutions to (21) with at most polynomial growth in on bounded sectors centered at 0, for near the origin in as the maps written in the form of a convergent logarithmic type expression for where (i) stands for convergent power series near the origin(ii) and are the characteristic exponents with positive real parts at (iii) are holomorphic coefficients near

In the case of so-called equations of irregular type or non-Fuchsian type, in which our present work falls, fewer results are known and represent a favourable breeding ground for upcoming research. Nonetheless, in that trend, we mention the remarkable recent general result [24] obtained by Tahara. This work extends a paper by Yamazawa which treats linear partial differential equations, see [25]. Therein, the author examines nonlinear partial differential equations with , for some integers , which possess a formal series (which is divergent in the generic situation) solution where each term , , is analytic with respect to on some appropriate bounded sector centered at 0 in and holomorphic near 0 relatively to on some disc in . In general, these expressions might involve combinations of functions of the form for holomorphic maps , powers of and , and analytic functions with respect to on . The author introduced a so-called Newton polygon associated to Equation (23) along the formal solution . In the case this Newton polygon possesses slopes and under some additional technical requirements, the author builds up a new formal solution to (23) which is subjected to the next two features (i)The formal series and are asymptotically equivalent in the sense that for any , there exists , such thatfor all , , some constant , any , where and denote the partial sums of the first terms of and .

The formal series is multisummable on with respect to , uniformly in on , in a sense that enhances the classical multisummability process described in [19] and gives rise to a genuine holomorphic solution of (23) on crafted as iterated analytic acceleration operators and Laplace integral of some Borel transform of .

Thereupon, it turns out that admits as an asymptotic expansion as tends to 0 on in the sense that for any , there exists such that for all , some constant , any .

At last, in the linear setting, some general results reaching beyond the structure of logarithmic-type solutions have been achieved. Namely, for Cauchy problems involving linear differential operators of order with holomorphic coefficients in in , existence and uniqueness results for so-called ramified solutions around certain characteristic hypersurfaces in , provided that is ramified around , have been obtained by several authors, see [26–28].

2. Layout of the Main Equation

2.1. Formal Monodromy around the Origin

In this subsection, we define the notion of a formal monodromy operator around the origin acting on different classes of objects. Following the description of abstract formal monodromy operator as stated in Subsection 3.2 of [1], we first provide a definition of formal monodromy acting on logarithmic type expressions involving formal power series with coefficients in Banach spaces.

Definition 1. Let be a bounded open sector centered at 0 in and let be a strip with width . We denote the Banach space of bounded holomorphic functions on equipped with the sup norm and we set as the vector space of all formal series with coefficients belonging to . Let and be two elements of ; we set the formal logarithmic type expression where stands for the principal value of the logarithm of a complex number .
We define the formal monodromy operator around 0 relatively to , denoted as acting on by means of

The next definition of formal monodromy extends the concept of a monodromy operator around 0 acting on analytic functions on a punctured neighborhood of 0 as analytic continuation along a simple loop around the origin as described in [2], Section 16.

Definition 2. Let be bounded open sectors centered at 0 in and be a strip defined by (29). We set as the Banach space of bounded holomorphic functions on endowed with the sup norm. Let be two elements of . We set that represents a holomorphic function for all with . The formal monodromy operator around 0 relatively to denoted acts on through the formula Notice that if and are holomorphic on a full-punctured disc centered at 0 relatively to , the formal monodromy given above coincides with the analytic continuation along a simple loop skirting counterclockwise the origin 0 with base point .

We observe that each components , of (31) (resp. , of (33)) can be expressed by means of and (resp. and ) through the formulas where represents the identity operator acting on in (35) and on in (36).

2.2. Outline of the Main Problem

The principal problem under study in this work is shaped as follows: for vanishing initial data . On the way in reaching our main result Theorem 24, we need to impose a list of constraints on the building blocks of (37). Namely, (i)The numbers , , and , are integers that are subjected to the next restrictions (1)We assume the existence of an integer with (2)The inequalities hold for all .(3)The bounds are asked for all .(4)The lower estimates are mandatory for all .(ii)The constants , , and are nonvanishing complex numbers that are chosen close enough to 0 (the precise constraints that these numbers are asked to obey are stated later on in the work, see Sections 5 and 6.(iii)The maps , , and along with , are polynomials with complex coefficients. We require that for and for . Furthermore, we require the existence of an unbounded sectorial annulus with bisecting direction , aperture , and inner radius (prescribed later in the work), for which the next inclusion occurs.

The forcing term is built up in the next manner. In its construction, we make use of Banach spaces that have been introduced in [29] and brought into play in several works by the author.

Definition 3. Let . We set as the vector space of continuous functions such that is finite. The space endowed with the norm becomes a Banach space.

The forcing term is written as a sum where the components are set up as follows. Let be finite subsets of the positive integers. For and , we denote maps that (i)Appertain to the Banach space for some and for all .(ii)Rely analytically on on some disc with radius for which constants exist such that

For , let us introduce the polynomials in the variable with coefficients in , and set the integral representations where is a half line in direction that relies on under the constraint . We observe that and are polynomials in and can be expanded in the form where stands for the Gamma function, for coefficients given by the inverse Fourier integral expressions that are bounded holomorphic on the product , for any given , where is the horizontal strip given by (29), for . Eventually, we set the components of (47) as a time-rescaled version of , for , that represent bounded holomorphic functions on .

The coefficients , , , and , are manufactured as follows. Let , , , and , , be maps that (i)Belong to the Banach space , for the real numbers and given above(ii)That depend analytically in on and for which positive constants , , , and , can be singled out with

We set for , . The maps , , , and , represent bounded holomorphic maps on the product , for any prescribed .

3.1. A Coupling of Associated Partial Differential Equations

We seek for solutions to our main Equation (37) in the form for some expression in the four independent variables . We furthermore assume that is an affine map relatively to meaning that is the polynomial of degree at most one in .

We first disclose an equation fulfilled by provided that solves (37) given by (62). According to the usual chain rule applied at a formal level at this stage of the work, we first observe that

Besides, owing to the assumption that is affine in , we can decompose in the form for some expressions , . If one sets for , through (57), one arrives at the next expansion of ,

As a result, in view of formulas (35) and (36) together with identity (58) and definitions (54) and (60), we check that formally solves Equation (37) if the expression is subjected to the next equation

In the next step, we derive some coupling of partial differential equations that the components and are asked to fulfill and displayed in (66) and (67).

Owing to the fact that the operators and commute to each other, the binomial formula helps us to rewrite (62) in the form

Besides, from decomposition (59), we observe that whenever . We reach the next equation

Finally, by dint of identification of the powers of in the above equality, it turns out that this last Equation (65) holds if the expressions and are asked to satisfy the next coupling of two partial differential equations

3.2. A Coupling of Auxiliary Convolution Equations

We search for solutions to the coupling of partial differential Equations (66) and (67) in the form of a Laplace transform of some order and inverse Fourier integral for , where stands for a half line in suitable directions which depend on in a way that remains strictly positive.

Here, we assume that for all , the so-called Borel-Fourier maps , , belong to the Banach space for well-chosen constants and for the prescribed constants in Subsection 2.2 that is described in the upcoming definition.

Definition 4. Let be positive real numbers and be an integer. Let . We set as an unbounded sector centered at 0 with bisecting direction . We denote the vector space of all continuous maps on , holomorphic w.r.t on , such that the norm is finite. The vector space equipped with the norm turns out to be a Banach space.

The main purpose of this subsection is to determine coupling convolution equations for the Borel-Fourier maps outlined in (83)–(85). We depart from some features of the Laplace transforms under the action of multiplication by a monomial and differential operators that were already stated and proved in our foregoing work [30], Lemma 6.

Lemma 5. The next identities hold. (1)The action of the differential operator on the integral representations is given by (2)Let be an integer. The multiplication by acting on is expressed through (3)Let be a map that belongs to . We set  The action of multiplication by on is expressed by means of (4)Let , , be polynomials. The action of the differential operators combined with the product of the resulting functions for , maps into a Fourier-Laplace transform for given .

The next useful lemma already stated in the previous work by Lastra and Malek [31] will show up in the process.

Lemma 6. For all integers , positive integers , for can be singled out such that with .

With the help of this lemma, Equations (66) and (67) can be remodeled in the form

The upcoming identity will also be called into play for the derivation of the coupling convolution equations. This technical formula was introduced in the work [32].

Lemma 7. Let be integers. Real numbers , for can be found such that holds, where we assume by convention that the sum vanishes for .

Owing to the assumption (38), the splitting holds for suitable integers , provided that . Furthermore, under the constraint (39), the decomposition occurs for well-chosen integers , as long as and .

Ultimately, by means of the above two relations (79) and (80), Lemma 7 can be applied in order to rewrite both Equations (76) and (77), only with the help of the basic irregular differential operator . Namely, together with

On the ground of the identities disclosed in Lemma 5, this hindmost coupling of Equations (81) and (82) allows us to reach the next statement.

The maps , , displayed in (68) solve the closing coupling (81) and (82) if the Borel maps , , fulfill the next coupling of convolution equations along with where

4. Linear and Bilinear Convolution Operators Acting on Banach Spaces

In this section, we examine continuity properties of several linear and bilinear convolutions operators that are applied on the Banach spaces given in Definition 4 and that unfold in the above coupled Equations (83)–(85),.

Proposition 8. Let , be integers and set . Let be an unbounded sector centered at 0 with bisecting direction and fix as some positive real number. Let be a continuous map on the closure subjected to the upper bounds provided that , all , for some constant . We take for granted that Then, we can single out a constant (relying on , , , and ) for which holds as long as belongs to the Banach space .

Proof. Let . By definition, the bounds ensue provided that and . According to the assumption (86), the latter bounds warrant the next estimates for all , all .
We further perform the change of variable in the above integral and get as long as and .
We introduce the function for all . In the next lemma, we uncover upper bounds for for large values of .

Lemma 4. The function is well defined and continuous for all . Furthermore, there exists a constant for which for all .

Proof. We first explain why is well defined and continuous for . Indeed, by means of the change of variable for , we can recast in the form which is a finite quantity for all and represents a continuous map w.r.t , according to the last inequality of (87).
In order to reach bounds for large , we apply a strategy stemming from Proposition 8 in our joint work [33]. Namely, we split into two pieces, where We first focus on bounds for . Two cases arise. (i)Assume that . In that situation, we observe that provided that , for . Therefore, bearing in mind the constraints (75), for all .(ii)Suppose that . We check that for any . Hence, paying regard to (75), whenever .In the second step, we provide upper estimates for . We notice that , for . Hence, where for all . From the sharp bounds established in Proposition 8 of [30], we can pinpoint a constant (depending on ) with for all , under the conditions (87). As a result, we get that provided that .
At last, gathering the bounds (97), (98), and (102), we deduce the awaited bounds (93) from the splitting (95).

We turn to the bounds for the map . We identify two alternatives. (i)Assume that is chosen such that  Owing to Lemma 4 and the first constraint of (87), we get from the upper bounds (91) some constant with for all chosen under (103).(ii)Suppose that fulfills

Based on (91), we arrive at some constant such that whenever is restricted to (105).

Eventually, the combination of the above bounds (104) and (106) yields the expected result (88).

Proposition 10. Let be polynomials and be a real number subjected to the constraints for all . Then, a constant (depending on ,, and ) can be selected such that holds provided that and .

Proof. The proof mirrors the one of Proposition 10 in our recent work [34]. Indeed, let us choose in . The very definition of the norms displayed in Definitions 3 and 4 allows the bounds provided that and together with for all . These two bounds (109) and (110) yield the next estimates where According to the triangular inequality, we observe that for all real numbers and by the construction of the polynomials asked to fulfill (107), two constants can be pinpointed such that whenever . Thereby, the next upper bounds are reached whose right hand side is a finite quantity under the restrictions (107), owing to Lemma 2.2 from [29] or Lemma 4 of [35].
Eventually, gathering (111) and (115) yields the foretold bounds (108).

Proposition 11. Let be an integer. Let , and be polynomials with complex coefficients such that for all . We require the positive real number to satisfy Let be a continuous function on such that for all . Then, one can find a constant (relying on , , , , , and ) such that for all .

Proof. Take in the space . According to the definition of the norm, the next two bounds hold provided that and . These bounds together with the assumption (118) prompt for all and .
By construction, we check that some positive constants ,, and can be picked out in a way that for all . As a result and keeping in mind the inequality (113), we deduce the next bounds for the first piece of the right handside of (121), namely, where is a finite quantity under conditions (116) and (117), as explained in Lemma 6.2 from [29] or Lemma 4 of [35]. Besides, according to Lemma 7 of our recent work [36], there exists a constant (relying on ) such that for all , all .
Counting up the above two bounds (123) and (125), it results from (121) that whenever and . The estimates (119) follow.

5. Solving the First Convolution Equation (83)

In this section, we uniquely solve the auxiliary convolution Equation (83) stated in Subsection 3.2 within the Banach spaces displayed in Definition 4. Our approach consists in rearranging (83) into a fixed point equation (disclosed later on in (173)). In the first stage, we ask to perform a division by the next parameter depending polynomial provided that . Decisive lower bounds concerning are displayed in the next lemma.

Lemma 12. For a convenient choice of the inner radius and aperture of the sector (introduced in (44)), unbounded sectors centered at 0 with bisecting direction and a small radius can be distinguished in a way that the next lower estimates hold for some well-chosen constant , provided that , for all .

Proof. Owing to the fact that the complex roots , of can be explicitely computed, we factorize the polynomial as follows: with for all , for any and .
We pinpoint an unbounded sector centered at 0, a small disc and we position the sector given in (44) in a way that the next two properties hold: (1)A constant can be found such that for all , all , whenever .(2)There exists a constant with for some , all , all .We now explain how the above two bounds can be established. (i)We deem the first inequality (131) in observing that under hypothesis (45); the roots are bounded from below and obey for all , all for a suitable choice of the radii . Furthermore, for all , all ; these roots are penned inside an union of unbounded sectors centered at 0 that do not cover a full neighborhood of 0 in whenever the aperture of is taken small enough. Hence, a sector may be chosen such that Such a sector satisfies in particular that for all , the quotients lay outside some small disc centered at 1 in for all , all . Eventually, (131) follows.
The sector and disc are selected as above. The second lower bound (132) ensues from the fact that for any fixed , the quotient stays apart a small disc centered at 1 in for all , all .
Departing from factorization (129) and paying regard to the two lower bounds (131) and (132) reached overhead, we arrive at as long as , for all .

We introduce the next nonlinear map

In the next proposition, we establish that represents a shrinking map on some suitable ball of the Banach space mentioned in Definition 4.

Proposition 13. Let us select a well-chosen inner radius and aperture of the sector jointly with an unbounded sector and radius that heed the requirements of Lemma 12 and obey the additional condition Then, one can single out a radius small enough, constants and close enough to 0 and a fitting radius in a way that for all , the map enjoys the next two features (i)The inclusion holds, where we denote , the closed ball of radius centered at 0 in the space .(ii)The Lipschitz condition occurs for all .

Proof. We take aim at the first item stating the inclusion (137). We prescribe some real number and take in , for given , such that We provide explicit bounds for each term of the map applied to .
According to Proposition 8 and Lemma 12, we observe that for along with for with and as long as . In order to handle the next piece, under the constraint (136), we can recast for all , with and . Based on Lemma 12, we check that provided that , . Owing to assumptions (42) and (48), Proposition 10 together with (144) yields Besides, a constant can be picked up such that for all , assuming condition (136). Condition (40) together with (146) enables us to apply Proposition 8 and prompt Eventually, bearing in mind (55), we deduce from (145) complemented by (147) that The ensuing block is remodeled as for all , with , and , under (115).
Assumptions (42) and (48) and the upper bounds (144) warrant the application of Proposition 10 which triggers Condition (40) coupled with (146) grants the use of Proposition 8 and beget At last, not forgetting (55), we deduce from the joint bounds (150) and (151) that We control now the piece . In accordance with Lemma 12, we notice that provided that and , whose right handside is a finite quantity since holds from (45) for all . Besides, owing to the definition of given in Subsection 2.2 and bounds (49), we deduce for all , . The combination of the bounds (153) and (154) grants which represents a finite quantity bearing in mind that contains only positive integers.
We address the ensuing linear part of . Paying regard to (153) and the bounds (55), Proposition 10 prompts At last, we manage the nonlinear tail piece of . We first factorize where for all and , according to (128). This latter decomposition together with the assumption (43) enable the application of Proposition 11 which yields We select , , and close enough to 0 and take suitably in a proper way that the next inequality holds. Observe that the first six blocks of the left handside of (160) can be made small since they contain positive powers of , owing in particular to the constraint (41) imposed on (37) and its last two terms can be dwindled provided that the positive constants and are chosen nearby the origin.
Eventually, the collection of all the bounds overhead (140)–(142), (148), (152), (155), (156), and (159) restricted by (160) gives rise to inclusion (137).
We mind the second item addressing the Lipschitz feature. Take inside the ball of the space whose radius has been prescribed in the first item discussed above. We display norm estimates for each block of the difference . Based on the bounds reached formerly in the proof of the first item, we check the next list of six estimates. Namely, for along with for with and as long as . Furthermore, holds for and together with for , , and in a row with Upper estimates for the rear part of ask some groundwork. Indeed, according to the classical identity , we reshape Keeping in mind the factorization (157) with (158), Proposition 11 sparks a constant with The remodeling (167) of together with (168) and (169) leads to We enclose the constants , , and in the vicinity of the origin allowing the next inequality The merging of the above bounds (161)–(166) and (170) subjected to (171) triggers the Lipschitz attribute of . Notice that the foremost five blocks of the left handside of (171) can be taken small scaled since they contain positive powers of due to the constraint (41) imposed on (37) and its two tail terms can be downsized provided that the positive constants and are chosen close to the origin.
In the closing part of the proof, we fix the radius and select the quantities , together with close enough to 0 that conforms both (160) and (171). For these values, the map is endowed with both inclusion and shrinking properties (137) and (138) for all . Proposition 13 follows.

The forthcoming proposition displays a solution to the first convolution Equation (83) shaped in the Banach spaces described in Definition 4.

Proposition 14. Let us choose an appropriate inner radius and aperture of the sector together with an unbounded sector and radius that conforms the requirements of Lemma 12. Then, a radius and constants , can be pinpointed sufficiently close to 0 together with a proper radius in a manner that for all , a unique solution to (83) exists such that (i)The map appertains to under the constraint The partial map stands for an analytic map from into , for any prescribed and .

Proof. We take the constants , , together with reached in Proposition 13. We observe that the closed ball represents a complete metric space for the distance . Proposition 4 claims that induces a contractive map from into itself. It follows from the classical Banach fixed-point theorem that possesses a unique fixed point inside the ball , for all , meaning that holds. Furthermore, the map relies analytically on since does on the domain . On the other hand, we check that the convolution Equation (83) can be rearranged as the Equation (173) by shifting the term from the right to the left handside of (83) and dividing by the resulting equation by the map given by (127). As a result, the unique fixed point of enclosed in precisely solves (83). The result follows.

6. Building Up a Solution to the Second Convolution Equation (84) with (85)

In this section, we cook up a unique solution to the auxiliary convolution equation reached in (84) with (85) inside the Banach spaces described in Definition 4.

The roadmap follows the one of the previous section and consists in recasting (84) with (85) into a fixed-point equation for a certain nonlinear map , stated in Proposition 16.

The map is set up as follows. We mind the map stemming from Proposition 14 and the polynomial displayed in (127). Let where

In the next proposition, we discuss the Lipschitz feature of on some well-chosen ball in the Banach spaces depicted in Definition 4.

Proposition 15. Let a timely inner radius and aperture of the sector in a row with an unbounded sector and radius chosen to fulfill the specifications of Lemma 12. We also take for granted the additional condition (136) required for the sector and the disc .
Then, one can target a small radius along with constants , , for and proximate to 0, coupled to a fitted radius in a way that for all , the map boasts the next two properties (i) maps into itself, where stands for the closed ball of radius centered at 0 in the space .(ii)The norm downsizing condition holds whenever .

Proof. We heed the first item asserting the inclusion. We fix some real number and pick up an element in , for , with Concrete bounds are presented for each piece of the map applied to .
The estimates for the first three blocks of are merely the same as the ones obtained in (140)–(142). Namely, owing to Proposition 8 and Lemma 12, we observe that for along with for with and as long as .
The next two pieces of follow from Proposition 8 and Lemma 12 together with the estimates (172) reached in Proposition 14. Indeed, we arrive at for in a row with for with .
The estimates for the following two components of simply recast the ones obtained in (148) and (152). Indeed, for and in parallel with for and with . Furthermore, the two ensuing constituents of mirror the one reached in (148) and (152) and draw on the estimates (172) from Proposition 14. Namely, for and in tandem with provided that and with .
The next element of we pay regard is and is displayed in (176). Its bounds are obtained in a similar way as the ones reached in (155). Indeed, which can be subsided close to 0 provided that is tiny enough since .
We handle the second and third pieces of . Paying heed to (153) and the bounds (55), Proposition 10 kindles and bearing in mind the estimates (172) from Proposition 14, ensues.
Thanks to factorization (157) with (158) and the bounds (172) from Proposition 14, we can apply Proposition 11 in order to address the last three terms of . Namely, together with as well as We pin down the constants and , , for and proximate to 0 together with a suitable radius in a way that the next inequality holds where We check that all the terms on the left handside of (194) except can be tapered off since they contain positive powers of in particular due to the constraint (41). Besides, the constant can be lessen provided that the constants and , , for and are taken in the vicinity of 0.
At last, stacking up all the above bounds ((179)–(193), under the contingency (194) yield that maps into itself.
In the second part of the proof, we address the second item of Proposition 15. Let be elements of the ball of the space with radius chosen as in the first part of the proof.
We provide norm estimates for each part of the difference . The bounds for the foremost five blocks of the difference are barely the ones found in (161)–(165). Namely, for along with for with and as long as . Furthermore, holds for and together with for , and . Besides, bounds for the sixth piece of result from (189) and are written The treatment of the seventh piece of springs from (191). Indeed, The hindmost term of the difference can be processed in a similar way as for the difference (167) given by (170). Namely, We skirt the constants , , and , nearby the origin in a manner that the next inequality holds where We notice that all the terms appearing in the left handside of (204) excluding can be dwindled since they involve positive powers of according to the constraints (41). Furthermore, the term can be depleted whenever the constants and , are taken close to 0.
In the end, the coupling of all the above bounds (196)–(203)under condition (204) triggers the shrinking feature (177) for the map .
In conclusion, we select the radius and pinpoint the constants , , for , along with , for nearby the origin, in a way they obey both (194) and (204). These values taken for granted, the map fulfills both inclusion and shrinking properties described in the items of Proposition 15.

The oncoming proposition provides a solution to the second convolution equation (84) with (85) crafted in the Banach spaces displayed in Definition 4.

Proposition 16. Consider an appropriate inner radius and aperture of the sector together with an unbounded sector and radius that respect the requirements of Lemma 12. Then, a radius along with constants , for and , for can be pinned down nearby 0 together with an appropriate radius in a way that for all , a unique solution to (84), (85) exists that is favoured with the next features (i)the map belongs to under the restriction (ii)the partial map stands for an analytic map from into , for any prescribed and .

Proof. Let the constants , , for and , for together with be fixed as in Proposition 15. The proposition 6 asserts that induces a contractive map from the closed ball and complete space into itself for the distance inherited from the norm on the Banach space .
The classical Banach fixed point theorem then claims that boasts a unique fixed point inside the ball , for all . In other words, holds. Furthermore, the map depends analytically on since itself does on the domain . On the other hand, we observe that the convolution equation (84) can be reorganized as the equation (207) by moving the term from the right to the left handside of (84) and dividing by the resulting equation by the map given by (127). As a result, the unique fixed point of penned in precisely solves (84) and (85). The result ensues.

7. Building up a Finite Set of Holomorphic Solutions to the Coupling of Partial Differential Equations (66) and (67)

7.1. Fourier-Laplace Transforms Solutions to the Pairing (66), (67)

In this section, we exhibit genuine analytic solutions expressed by means of Fourier-Laplace transforms to the coupling (66) and (67) reached at the end of Subsection 3.1.

Proposition 17. For all unbounded sectors with bisecting direction and disc that obey the demands of Lemma 12, we introduce the two partial maps for , for all where the Borel map is manufactured in Proposition 14 and solves (83), the Borel map is crafted in Proposition 16 and fulfills (84) and (85), and the radius is taken in agreement with Propositions 14 and 16 and stands for a halfline in a direction suitably chosen and described below.
The maps , , are endowed with the next two properties.
They define holomorphic functions that are bounded by a constant not relying on on a product where represents a bounded open sector centered at 0 with bisecting direction , for any given . (i)The map solves the Equation (66) for prescribed initial data . The map is subjected to the Equation (67) for vanishing data The sector is submitted to the next technical constraints: (1)A positive real number can be singled out with the next property: for all , a direction (that might rely on ) can be favoured with (2)The radius of withstands the next upper bounds for some positive real number , where is defined in the above item.

Proof. We discuss the first item of the proposition. We mind the maps and constructed in Propositions 14 and 16, and we select a bounded sector that matches the above prerequisite (210) and (211). We set and take for given real number where is the direction chosen above. Then, then next two inequalities for the Borel map hold. (i)A constant can be found for which the next bounds hold for all , all .(ii)Similarly, a constant can be singled out with the bounds provided that and .As a result, we reach the next two upper bounds for the maps , . Namely, by means of the change of variable in the integral together with for all , and all . We observe that the right handside of both (215) and (216) are unconstrained constants relatively to on . The first item ensues.
Concerning the second item, we remind from Proposition 14 (resp., Proposition 16) that the Borel map (resp., ) is shown to solve the associated convolution Equation (83) (resp., (84) and (85)). By tracking reversedly the computations made in Subsection 3.2, we deduce that for all , the next properties hold.
The holomorphic map given by the expression (209) for obeys the Equation (81), then fulfills (76) and finally solves (66) on the domain , for prescribed initial data .
The holomorphic map expressed in the form (209) for conforms to the Equation (82), then satisfies (77) and finally is subjected (67) on the domain , for vanishing initial data .
The second item of Proposition 17 follows.

7.2. Construction of a Finite Family of Genuine Solutions to the Coupling (66) and (67) and Sharp Bounds for the Neighboring Differences of Related Maps

We need to refer to the usual definition of a good covering in given in the textbook [37].

Definition 18. Let be an integer. We consider a set of open bounded sectors centered at 0 endowed with the next three properties (1)The intersection of two neighboring sectors and is not empty for any , where the convention is chosen(2)The intersection of any three sectors , , and for distinct integers is empty(3)The union of all the sectors is subjected to for some neighborhood of 0 in .Such a set is designated as a good covering in .

The next definition displays some domains in which are crucially involved in the set up of genuine solutions.

Definition 19. We consider two finite sets of bounded open sectors centered at 0, and a bounded sector centered at 0, for which the next list of constraints is required. (1)For each and fixed , for some given radius , the sector has bisecting direction and obeys the next three rules (i)For each , one can single out an unbounded sector centered at 0 with bisecting direction that is subjected to the requirements of Lemma 12 (namely, for which the lower bounds (131) and (132) hold).(ii)For each , a positive real number can be selected in a way that for all , a direction (that might depend on ) can be found with (iii)The radius of is constrained to the next upper bounds for some positive real number , where is determined in the above item.(2)The radius of the sector satisfies the restriction where , are specified in 1. for . Besides, the sectors share a common radius given by , for .(3)For all , the sectors and stick to the feature provided that and .(4)The set stands for a good covering in . Furthermore, the aperture of the sector is taken nearby 0 in a way that the set is not empty.These sets and and the sector form a so-called fitting collection of sectors.

In the next proposition, we shape a finite family of analytic solutions to the coupling of auxiliary problems (66) and (67).

Proposition 20. We consider a fitting collection of sectors , , and in the sense of Definition 19. The solutions to (66) and (67) are cooked up as follows.
Equation (66) possesses a finite set of holomorphic solutions , for , on the domain , for all , where is proximate to 0, for any , that fulfills the initial condition . These maps enjoy the next two qualities: for each , (1)The map is bounded by a constant unconstrained to in , on the product .(2)The map is represented as Fourier inverse and Laplace transforms where the Borel maps appertain to the Banach space. are subjected to for suitable constants and radius , for all .
Equation (67) (where the expression needs to be replaced by ) owns a finite set of holomorphic solutions , for , on the domain , for all , where is closed to 0, for any , with the initial condition . These maps are endowed with the next two features: for each , (1)The map is bounded on the product by a constant not relying to in .(2)The map is expressed by means of a Fourier inverse and Laplace transforms where the Borel maps are crafted in the Banach space. with bounds for appropriate constants and radius , for all .

Proof. Proposition 20 is a downright consequence of Proposition 17 and of the very definition of fitting collections of sectors depicted in Definition 19.

In the next proposition, we examine a finite set of maps related to the analytic solutions of the coupling (66) and (67). In particular, we obtain a control on their consecutive differences which turns out to be an essential information in the study of their parametric asymptotic expansions.

Proposition 21. Let us prescribe a fitting collection of sectors , , and in accordance with Definition 19. For each , we set up the maps for , where are described in Proposition 20. The next attributes hold: for all , (i)The maps , , are bounded holomorphic on the product and satisfy ,(ii)One can exhibit constants and such that for all , all , all , for , where we adopt the convention .

Proof. The first item is a direct outcome of the properties of the maps , , stated in Proposition 20 and from the characteristics 2 and 3 of the sectors and listed in Definition 19.
The second item follows from a path deformation argument. Indeed, let us take and . For any given and fixed , the partial maps , , represent analytic continuation on the sector of a common analytic map we denote on the disc .
For any prescribed and , we deform the oriented path into the union of three oriented curves (i)Two halflines (ii)An arc of circle centered at 0 with radius that connects the above two halflines.Then, the classical Cauchy’s theorem enables us to reshape the next difference into a sum of three contributions We provide upper bounds for the first piece of (232) Based on the bounds (213)–(225) and (227) together with the requirements asked in Definition 19, we arrive at provided that , and .
In the same vein, we can get upper bounds for the second piece of (232). Namely, for all , and .
At last, we handle the integral along the arc of circle closing (232), Owing to the bounds (225) and (227), we observe that as long as , and . Besides, in view of the restrictions discussed in Definition 6.1, it follows that for all , , granted that the angle belongs to or . By virtue of (238) and (239), we come up with some constant with contingent upon , , and . Hence, we deduce that holds, where as long as , and .
In summary, the splitting (232) along with the bounds (234), (236), and (241) beget the awaited estimates (229).

8. Main Statement of the Paper: Construction of a Finite Set of Holomorphic Solutions to the Leading Problem (37)—Description of their Parametric Asymptotic Expansion

8.1. Parametric Gevrey Asymptotic Expansions of the Associated Maps (228)

We first call to mind a result known as the Ramis-Sibuya theorem, see Lemma XI-2-6 in [37].

Theorem 22. Let be a good covering in be fixed as described in Definition 18. We denote a Banach space over . For all , we set as holomorphic functions that obey the next requirements (1)The maps are bounded on for all (2)The difference defines a holomorphic map on the intersection which is exponentially flat of order , for some integer , meaning that one can select two constants for which holds provided that , for all . By convention, we set and .Then, one can find a formal power series with coefficients belonging to , which is the common Gevrey asymptotic expansion of order relatively to on for all the maps , for . It means that two constants can be singled out with the error bounds for all integers , all , all .

In the next proposition, we exhibit asymptotic expansions of Gevrey type for the two sets of related maps introduced in Proposition 21, , , relatively to the variable .

Proposition 23. We denote the Banach space of bounded holomorphic functions on the product which are valued, equipped with the sup norm. Then, for , a formal power series with coefficients , , in can be shaped that obey the next error bounds. For all , two constants and can be chosen with for all integers , all .

Proof. Let . For all , let us define the maps by the expression . For , these functions share the next two features: (i)The maps are bounded holomorphic on the sector , according to the first item of Proposition 21(ii)The differences are submitted to the bounds for the constants and obtained in Proposition 21, whenever , where the convention and is in use.As a result, requirements 1 and 2 of Theorem (22) are matched for the sets of maps , . We deduce the existence of formal series , , which are the Gevrey asymptotic expansion of order relatively to on shared by all the maps for . This is tantamount to the statement of Proposition 23 and the awaited bounds (246).

8.2. Statement of the Main Result

The next statement stands for the pinnacle of our work.

Theorem 24. Let us prescribe a fitting collection of sectors , , and accordingly to Definition 19. We take for granted that all the conditions (38)–(45), (47)–(49), (54), and (55) enumerated in Subsection 2.2 are fulfilled.
Then, provided that the constants and , , , along with and , are nonvanishing and taken proximate to 0, the main equation with vanishing initial data possesses a finite set of bounded holomorphic solutions , for all , where is the subset of introduced in item 4 of Definition 19, on the domain . In Equation (248), the formal monodromy operator around 0, acts on the analytic map through Definition 2 by use of (34). The next additional features hold. (i)For each , the solution can be expressed by means of a Fourier/Laplace transform where for Borel maps , , that belong to the Banach space under restrictions (194) and (196).(ii)The two components , , of are endowed with Gevrey asymptotic expansions given by (245) of order relatively to on displayed in (246).(iii)If one sets the formal expression then, conforms to the next equation where the formal monodromy operator around 0, acts on the formal expression by means of the formula (32) from Definition 1.

Proof. For all , where is the set described in item 4 of Definition 19, we define where the maps are introduced in (228) of Proposition 21.
As a result of the definition of together with the first item of Proposition 21 and the classical limit , for any natural number , we check that the map represents a bounded holomorphic function on the product that vanishes at , meaning that for all and .
According to Proposition 20, we know that for each ,the map stands for a solution of Equation (66) on the domain .
The map embodies a solution of (67) where the expression is asked to be replaced by on the domain .
Then, on the basis of the computations (65), (63), and (62) performed reversedly from Subsection 3.1, we deduce that solves the main Equation (37), rephrased as (248), on the domain , for all .
The first item of Theorem 24 follows from the Fourier/Laplace representation of the maps , , displayed in Proposition 20 that are used to define the components in (197).
The second item of Theorem 24 merely restates the result obtained in Proposition 23.
We focus on the third item. We first need to disclose partial differential equations that the maps , turn out to fulfill. Indeed, the usual chain rule enables the next computation for all , , provided that , and . According to the statement discussed in Proposition 20, the partial map matches Equation (66) on the domain , whenever ; we observe that the map satisfies the next equation as long as , , and . On the other hand, since the partial map obeys Equation (67) on the domain , for , it follows that the map fulfills the next equation coupled to (215), provided that , and .
The next classical result (stated in Proposition 17 p. 66 from [19]) will be essential to deduce recursion relations for the coefficients , of from the partial differential equations that govern the components , .

Proposition 25. Let be a holomorphic map from a bounded open sector centered at 0 in into a complex Banach space equipped with a norm . The next statements are equivalent (i)There exists a formal power series in which is the asymptotic expansion of on , meaning that for all closed sector of centered at 0, one can associate a sequence of positive real numbers such that for all , all integers .(ii)All th derivatives of denoted are continuous at 0 and satisfy for all integers .

We first derive some recursion relations for the coefficients , . To that aim, we take the derivative of order on the left and right handsides of (256) relatively to for any integer . Indeed, owing to the Leibniz rule, we deduce for all , all , and . Owing to the asymptotic expansion (246) for , the application of Proposition 25 yields the next limits for all integers and any given . We let tend to 0 on the sector in the above equality (260) and with the help of (261) combined with the observation that both maps and are holomorphic with respect to on the product , we reach the next relation for the coefficients , , for all , provided that , .

This enables us to display some partial differential equations fulfilled by the formal expansion . Namely, we know that the maps , , together with are analytic on the disc . Their convergent Taylor series are expressed as for all . Then, departing from (245), we get the formal Taylor expansion of the next pieces that involve . Namely, along with

As a result, relation (262) and the above formal expansions prompt the next partial differential equation satisfied by ,

In the next part of the proof, we exhibit recursion relations for the coefficients , . We proceed by taking the th derivative of both handsides of (257) with respect to for any given integer . Indeed, the Leibniz rule yields for all , all , and . Besides, the asymptotic expansion (246) for warrants the application of Proposition 25 in order to reach the limits for all integers and any prescribed . We allow the parameter to get close to 0 in relation (268). Based on the above limits, (269) combined with (261) and the fact that the maps and , rely holomorphically in the variable on the product , we obtain the next relation for the coefficients , , for all , whenever and .

This latter recursion relation leads to some partial differential equation governing the formal expression given by (245). In the process, we use the convergent Taylor expansions (263) together with

for which are valid for all and from which the next list of computations are deduced along with

for . Futhermore, the next identities hold with

As a consequence of the above computations, relation (270) triggers the next partial differential equation fulfilled by and coupled with (267),

In conclusion, we have checked by means of (267) that the power series formally solves the same partial differential equations as the function stated in (256). In addition, through (276) and (257), we observe that the formal power series and the map obey identical coupled partial differential equations. Then, drew on the computations (65), (63) and (62) performed reversedly from Subsection 3.1, we deduce that the formal expression stated in (252) conforms the same equation as the analytic map given in (248) and recast as (253) where the formal monodromy operator around 0 given by acts on the formal expression by dint of the formula (32) in Definition 1. This completes the proof of the third item of Theorem 24.

9. Conclusion and Perspectives

In this work, we have considered an initial value problem that is singularly perturbed in a complex parameter and possesses an irregular singularity in a complex time at the origin. This problem involves specific nonlocal nonlinearities, where the so-called formal monodromy operator plays a central role and can be viewed as a mixed-type partial differential and difference nonlinear equation when aiming at potential applications.

The presence of these nonlocal monodromy operators, which is the main novelty of our approach, enables the construction of holomorphic solutions with logarithmic expansions of finite type given by (12) or (20) in a more general setting. It is worth noting that in the case of polynomial nonlinearities involving only powers of the solution and its derivatives, logarithmic expansions of infinite type might appear (i.e., infinitely many powers of might be encountered), as shown in the studies [18, 24] quoted in the introduction of this work.

The terms comprising the nonlinear part of our problem are suitably selected in a way that the construction of logarithmic-type solutions can be reduced to the study of analytic solutions to a coupling of two singularly perturbed initial value problems with quadratic local nonlinearities with a particularly simple shape of triangular form. The treatment of more general nonlinearities leading to nontriangular structures looks far more challenging and is postponed to a future investigation.

At last, we expect that our approach can be adapted to other related problems, for instance in the context of difference equations which is another field of research of the author and his colleagues.

Data Availability

There a no underlying data supporting the results of the study.

Disclosure

The present work is registered as a preprint on http://preprints.org where it is quoted as [38].

Conflicts of Interest

The author declares that they have no conflicts of interest.