Abstract

We study a singularly perturbed PDE with quadratic nonlinearity depending on a complex perturbation parameter . The problem involves an irregular singularity in time, as in a recent work of the author and A. Lastra, but possesses also, as a new feature, a turning point at the origin in . We construct a family of sectorial meromorphic solutions obtained as a small perturbation in of a slow curve of the equation in some time scale. We show that the nonsingular parts of these solutions share common formal power series (that generally diverge) in as Gevrey asymptotic expansion of some order depending on data arising both from the turning point and from the irregular singular point of the main problem.

1. Introduction

In this work, we consider a family of nonlinear singularly perturbed equations of the formwhere , , , and are polynomials with complex coefficients and is an analytic function in the vicinity of the origin with respect to and in and holomorphic with respect to on a horizontal strip in of the form for some .

Here we consider the case when vanishes identically near 0. The point is known to be called a turning point in that situation; see [1, 2] for a more detailed description of this terminology in the linear and nonlinear settings. Let us recall the definition of the valuation of an analytic function near as the smallest integer with the factorization for an analytic function near with . The most interesting case examined in this work is when the valuation of with respect to is larger than the valuation or since the problem cannot be reduced to the case by dividing (1) by a suitable power of and ; see Remark 13.

In our previous study [3], we already have considered a similar problem which corresponds to the situation when for our equation (1). Namely, we focused on the following problem:for given vanishing initial data , where , , , and are polynomials with complex coefficients and is a forcing term constructed as above. Under appropriate assumptions on the shape of (2), we established the existence of a family of actual bounded holomorphic solutions , , for some integer , defined on domains , for some fixed bounded sector with vertex at 0 and , a set of bounded sectors whose union covers a full neighborhood of 0 in . These solutions are obtained by means of Laplace and inverse Fourier transforms. On each sector , they share with respect to a common asymptotic expansion which defines a formal series with bounded holomorphic coefficients on . Moreover, this asymptotic expansion is shown to be of Gevrey order (at most) that appears in the highest order term of the operator which is of irregular type in the sense of [4] outlined as , for some integer and a polynomial with complex coefficients. Conjointly, since the aperture of the sectors can be chosen slightly larger than , the functions can be viewed as -sums of the formal series as defined in [5].

In this work, our goal is to achieve a similar statement, namely, the existence of sectorial holomorphic solutions and asymptotic expansions as tends to 0. However, the main contrast with problem (2) is that, due to the presence of the turning point, our solutions are no longer bounded in the vicinity of the origin, being meromorphic in both time variable and parameter . Namely, we build a set of actual meromorphic solutions to problem (1) of the form where , are some rational numbers, is an integer, and is a nonidentically vanishing root of a second-order algebraic equation with polynomial coefficients related to the polynomials , , see (70), and where is a bounded holomorphic function on products similar to the ones mentioned above, which can be expressed as a Laplace transform of some order and Fourier inverse transform along some half line , for some positive rational number , where represents a function with at most exponential growth of order on a sector containing with respect to , with exponential decay with respect to on and with analytic dependence on near 0 (see Theorem 19). Furthermore, we show that these functions own with respect to a common asymptotic expansion which represents a formal series with bounded holomorphic coefficients on . We specify also the nature of this asymptotic expansion which turns out to be of Gevrey order (at most) . Besides, since the aperture of the sectors may be selected slightly larger than , the functions can be identified as -sums of the formal series (Theorem 21). By construction, the integer shows up in the highest order term of the operator which is of irregular type of the form , with , for some integers , and a polynomial with complex coefficients. The rational number is built with the help of the integers , , , and and the rational numbers , ; see (86). According to the fact that , are mainly related to constraints assumed on the polynomials and (see (66), (67)), we observe that the Gevrey order of the asymptotic expansion involves information coming both from the highest irregular term and from the two polynomials and that shape the turning point at , whereas, in our previous contribution [3], the Gevrey order was exclusively stemming from the irregular singularity at .

The kind of equations with quadratic nonlinearity we investigate in this work is strongly related to singularly perturbed ODEs which are nonsingular at the origin of the form for some analytic functions , small complex parameter , and a complex additional parameter , described in the seminal joint paper by Canalis-Durand et al., see [6], where they study asymptotic properties of actual overstable solutions near a slow curve (meaning that ) in the case when the Jacobian is not invertible at . The main notable difference is that we assume the origin to be at the same time a turning point and an irregular singularity. More precisely, with the rescaling map the transformed equation (64) possesses a rational slow curve and remains a turning point and an irregular singularity for this new equation.

The construction of the distinguished solution performed in Section 4 and the parametric Borel/Laplace summable character of these solutions shown in Section 7 are also intimately linked to recent developments of exact WKB analysis of formal and analytic solutions to second-order linear ODEs of Schrödinger type. Namely, letbe a singularly perturbed ODE where is a small complex parameter and is some polynomial with complex coefficients. WKB solutions of (5) are known as special solutions that are described as an exponential where the expression satisfies a so-called Riccati equationThis last equation possesses formal power series solutions , where satisfies the quadratic equation . Once is fixed, we get two formal solutions , where for any . Notice that solves the first-order Riccati equation with turning points at the roots of . Our main PDE (1) resembles this last one provided that is a polynomial and with the significant distinction that our equation only involves differential operators with irregular singularity at . An essential feature of the theory is that the formal series are -summable in suitable directions with respect to (that are related to the function ) for any fixed . Different proofs of this fact can be found in [7–10]. Our second main statement, Theorem 21, can be considered as a similar contribution for some higher order PDEs of this latter result. Furthermore, in our study we are also able to describe the behaviour of our specific solutions near .

For more recent and advanced works related to WKB analysis and local/global studies of solutions to linear ODEs near turning points, we refer to contributions related to the 1D Schrödinger equation with simple poles [11], with merging pairs of simple poles and turning points [12], and with merging triplet of poles and turning points [13, 14] and for analytic continuation properties of the Borel transform (resurgence) of WKB expansions in the problem of confluence of two simple turning points we quote [15]. Concerning the structure of singular formal solutions to singularly perturbed linear systems of ODEs with turning points we point out [16] solving an old question of Wasow. We mention also preeminent studies on WKB analysis for higher order differential equations which reveal new Stokes phenomena giving rise to so-called virtual turning points [17, 18].

In the framework of linear PDEs, normal forms for completely integrable systems near a degenerate point where two turning points coalesce have been obtained in [19], which is a first step toward the so-called Dubrovin conjecture which concerns the question of universal behaviour of generic solutions near gradient catastrophe of singularly Hamiltonian perturbations of first-order hyperbolic equations; see [20]. We mention also that sectorial analytic transformations to normal forms have been obtained for systems of singularly perturbed ODEs near a turning point with multiplicity using the recent approach of composite asymptotic expansions developed in [2]; see [21].

The paper is organized as follows. In Section 2, we recall the definition introduced in the work [3] of some weighted Banach spaces of continuous functions with exponential growth on unbounded sectors in and with exponential decay on . We analyze the continuity of specific multiplication and linear/nonlinear convolution operators acting on these spaces.

In Section 3, we remind the reader of basic statements concerning -Borel-Laplace transforms, a version of the classical Borel-Laplace maps already used in previous works [3, 22, 23] and Fourier transforms acting on exponentially flat functions.

In Section 4, we display our main problems and explain the leading strategy in order to solve them. It consists in four operations. In a first step, we restrict our inquiry for the sets of solutions to time rescaled function spaces; see (63). Then, we consider candidates for solutions to the resulting auxiliary problem (64) that are small perturbations of a so-called slow curve which solves a second-order algebraic equation and which may be singular at the origin in . In a third step, we search again for time rescaled functions solutions for the associated problem (84) solved by the small perturbation of the slow curve; see (85). In the last step, we write down the convolution problem (95) solved by a suitable -Borel transform of a formal solution to the attached problem (87).

In Section 5, we solve the main convolution problem (95) within the Banach spaces described in Section 2 using some fixed point theorem argument.

In Section 6, we provide a set of actual meromorphic solutions to our initial equation (61) by executing backwards the operations described in Section 4. In particular, we show that our singular functions actually solve problem (164) which is a factorized part of (61) with a more restrictive forcing term. Furthermore, the difference of any two neighboring solutions tends to 0 as tends to 0 faster than a function with exponential decay of order .

In Section 7, we show the existence of a common asymptotic expansion of Gevrey order for the nonsingular parts of these solutions of (61) and (164) based on the flatness estimates obtained in Section 6 using a theorem by Ramis and Sibuya.

2. Banach Spaces with Exponential Growth and Exponential Decay

We denote by the open disc centered at with radius in and by its closure. Let be an open unbounded sector in direction and be an open sector with finite radius , both centered at in . By convention, these sectors do not contain the origin in .

We first give definitions of Banach spaces which already appear in our previous work [3].

Definition 1. Let and be real numbers. We denote by the vector space of functions such that is finite. The space endowed with the norm becomes a Banach space.

As a direct consequence of Proposition  5 from [3], we notice the following.

Proposition 2. The Banach space is a Banach algebra for the convolution product Namely, there exists a constant (depending on ) such that for all .

Definition 3. Let and , be real numbers. Let and be integers. Let . We denote by the vector space of continuous functions on , which are holomorphic with respect to on and such thatis finite. One can check that the normed space , is a Banach space.

Throughout the whole section, we keep the notations of Definitions 1 and 3.

In the next lemma, we check that some parameter depending functions with polynomial growth with respect to the variable and exponential decay with respect to the variable , which will appear later on in our study (Section 5), belong to the Banach spaces described above.

Lemma 4. Let , be integers. Let be a polynomial that belongs to such that for all . We take a function located in and we consider a continuous function on , holomorphic with respect to on , such that for all , all .
Then, the function belongs to . Moreover, there exists a constant (depending on and ) such thatfor all .

Proof. By definition of the norm and bearing in mind the constraint on the polynomial , we can writewhich yields the lemma since an exponential grows faster than any polynomial.

The next proposition provides norm estimates for some linear convolution operators acting on the Banach spaces introduced above. These bounds are more accurate than the one supplied in Proposition  2 from [3]. These new estimates will be essential in Section 5 in order to solve problem (95). The improvements are due to the use of thorough upper bounds estimates of a generalized Mittag-Leffler function described in the proofs of Propositions  1 and  5 from [23].

Proposition 5. Let , , be real numbers with . Let and be polynomials with complex coefficients such that and with for all . We consider a continuous function on , holomorphic with respect to on , such that for all , all . We make the following assumptions:(1) If , then there exists a constant (depending on and ) such thatfor all .
(2) If and , then there exists a constant (depending on , , , , and , ) such thatfor all .

Proof. By definition of the norm, we can writewhere Again by the definition of the norm of and by the constraints on the polynomials , , we deduce thatwhereWe perform the change of variable inside the integral which is a part of that yieldsAs a result, we obtain the boundswhereWe now proceed as in Proposition  1 of [23]. We split the function into two pieces and study them separately. Namely, we decompose , where We first provide estimates for .
(a) Assume that . We see that for all , for . Hence, from the first constraint of (16), we get for all . Subsequently, we obtainwhich is finite due to the second assumption of (16).
(b) Assume that . We notice that for all , for . Therefore, again from the first constraint of (16) we get for all . Consequently, we obtainwhich is finite due to the second assumption of (16).
In a second step, we study .
We see that for all . Hence,where for all . Taking account of the estimates in [23] which are deduced from the asymptotic behaviour for large of the generalized Mittag-Leffler function , for , we get a constant (that depends on , , , ) such thatfor all , provided the first and last constraints of (16) hold.
(1) We consider the first case when .
Bearing in mind (31) and (33), we deduce thatwhich is finite. On the other hand, when , we make the change of variable inside and, taking (31) into account, we getwhich is finite provided that the constraints (16) are fulfilled.
(2) We examine the second case when and .
We use this time the fact that for all and the bounds (33) in order to getOn the other hand, the bounds on the domain have already been treated above owing to (35).
Finally, gathering (21), (24), (28), (30), (34), (35), and (36) yields the statement of Proposition 5.

The forthcoming proposition presents norm estimates for some bilinear convolution operators acting on the aforementioned Banach spaces.

Proposition 6. There exists a constant (depending on and ) such thatfor all .

Proof. We follow the same guidelines as in the proof of Proposition  3 from [3]. By definition of the norm, we can writewhereBy definition of the norms of and and according to the triangular inequality for all , we deduce thatwhereWe provide upper bounds that can be split in two parts,whereis finite under the condition that according to Lemma  4 of [24] andWe carry out the change of variable inside the integral piece of which yields the boundswhere A change of variable in this last expression followed by a partial fraction decomposition allow us to writewhich acquaints us with the fact that is finite provided that and bounded on with respect to .
At last, collecting (38), (40), (42), (43), (45), and (47) leads to the statement of Proposition 6.

3. Borel-Laplace and Fourier Transforms

In this section, we review some basic statements concerning a -Borel summability method of formal power series which is a slightly modified version of the more classical procedure (see [5], Section  3.2). This novel version has already been used in works such as [3, 22] when studying Cauchy problems under the presence of a small perturbation parameter. We remind also the reader of the definition of Fourier inverse transform acting on functions with exponential decay.

Definition 7. Let be an integer. Let be the sequence Let be a complex Banach space. We say a formal power series is -summable with respect to in the direction if the following assertions hold: (1)There exists such that the -Borel transform of , , is absolutely convergent for , where (2)The series can be analytically continued in a sector for some . In addition to this, the extension is of exponential growth at most in , meaning that there exist such that Under these assumptions, the vector valued Laplace transform of along direction is defined by where is the path parametrized by , for some appropriate direction depending on , such that and for some .
The function is well defined and turns out to be a holomorphic and bounded function in any sector of the form , for some and . This function is known as the -sum of the formal power series in the direction .

The following are some elementary properties concerning the -sums of formal power series which will be crucial in our procedure.

(1) The function admits as its Gevrey asymptotic expansion of order with respect to in . More precisely, for every , there exist such that for every and . Watson’s lemma (see Proposition  11 p. 75 in [25]) allows us to affirm that is unique provided that the opening is larger than .

(2) Whenever is a Banach algebra, the set of holomorphic functions having Gevrey asymptotic expansion of order on a sector with values in turns out to be a differential algebra (see Theorems  18, 19, and 20 in [25]). This and the uniqueness provided by Watson’s lemma allow us to obtain some properties on -summable formal power series in direction .

By we denote the product in the Banach algebra and also the Cauchy product of formal power series with coefficients in . Let be -summable formal power series in direction . Let be integers. Then , , and , which are elements of , are -summable in direction . Moreover, one has for every .

The next proposition is written without proof for it can be found in [3], Proposition  6.

Proposition 8. Let and which belong to , where is a Banach algebra. Let be integers. The following formal identities hold.

In the last part of the section, we recall without proofs some properties of the inverse Fourier transform acting on continuous functions with exponential decay on ; see [3], Proposition  7 for more details.

Proposition 9. (1) Let be a continuous function with a constant such that for all , for some . The inverse Fourier transform of is defined by the integral representation for all . It turns out that the function extends to an analytic function on the horizontal stripLet . Then, we have the commuting relationfor all .
(2) Let and let , the convolution product of and , for all . From Proposition 2, we know that . Moreover, the following formulaholds for all .

Let , be integers. For all , let , be nonnegative integers and be complex numbers with such that for . For all , we consider nonnegative integers , and complex numbers with such that for . For all , we denote by and nonnegative integers such that for . For , we set nonnegative integers , , and such that for .

Let , , be polynomials which can be factorized as , , for some common integer , where and are polynomials that satisfyfor all and all .

We consider the following nonlinear singularly perturbed PDE:The coefficients are constructed as follows. For all , we consider functions that belong to the Banach space for some and . We define where is the integer introduced above, for . We setwhere denotes the Fourier inverse transform defined in Proposition 9. From (58), it turns out by construction that one can write , where is the inverse Fourier transform of .

Remark 10. The reason why we make these factorizations hypotheses on the polynomials , , and the functions will be explained later on in Remark 14 of next section and is related to the construction of the Banach spaces in Section 2 and their Fourier inverse transforms.
Within this work, we will search for time rescaled solutions of (61) of the formwhere are two rational numbers and . Then, the expression needs to formally solve the following nonlinear PDE:

4.1. Construction of a Distinguished Solution

We make the additional assumption that , set above can be chosen in such a way that the following inequalitiesfor all , andfor all and all , for some integer , together withfor all and all , for some integer , hold.

Remark 11. In the case , , the roots of the polynomial (in ) all have modulus equal to except the trivial root 0. The constraints (66) imply in particular that . As a result, all the nonvanishing roots of tend to as tends to 0 and 0 is therefore the only root (with order ) of in the vicinity of 0 as stays near the origin.
Let us assume that the expression is allowed to be written as a perturbation series with respect to :where the constant term is taken independently of and is not identically equal to 0. The coefficient is called the slow curve of (64) in the terminology of [6].
In the following, we make the assumption that solves the following second-order algebraic equation:As is not identically vanishing, it must be equal to . Bearing in mind that , we get its asymptotic behaviouras tends to 0.

Remark 12. Under the hypotheses (60) and (62), we observe, by factoring out the operator from (64), that must solve the related PDEwhere the forcing term is a polynomial in of degree less than . According to the assumptions (65), (66), and (67) and using the fact that , by taking into (72), we see that the constraint (70) is equivalent to the fact that . The precise shape of the term will be given later in Section 6; see (183).
In a first step, we express as a small perturbation of that can be expressed in the form where is a convergent series near ; namely,for some integer and some expression . By plugging this last expansion inside (64) and using the Leibniz rule, we getwhere we put by convention. At this level, we observe the important fact that the coefficient in front of contains the term that we want to set apart. As a result, we get the following equation satisfied by :We now introduce some additional constraints on the integers , , , , for , , and , , for . Namely, we impose that the following inequalities hold:for , together withfor all and finallyfor all .

Remark 13. (1) For the case , from (77), we need . As a consequence of (78), we get that , for . Let, for instance, , , , and . We set , , , and we choose the powers of and in the coefficients of (61) as follows:For these data, we can check that the constraints (65), (66), (67), (77), (78), and (79) above are fulfilled. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this special case, the main equation (61) writesWe can divide this last equation by , but not by , and the resulting equation still possesses a turning point and an irregular singularity at .
(2) For the case , we may take and hence one can choose some for some . Let, for instance, , , , and . We choose , , , and and we select the powers of and in the coefficients of (61) as follows:For these data, we can figure out that the constraints (65), (66), (67), (77), (78), and (79) above are satisfied. Moreover, all the forthcoming requirements (88), (105), (106), (107), and (184) stated in Theorem 19 are also verified. In this particular case, the main equation (61) writesWe can divide this latter equation by and by . The corresponding equation still suffers the presence of a turning point and an irregular singularity at .
In a second step, we divide the left- and right-hand sides of (76) by the monomial . We obtain the following equation:Notice that the additional constraints (65), (66), (67) and (77), (78), (79) ensure that the coefficients of the PDE (84) are analytic with respect to and on a neighborhood of the origin in . Moreover, the coefficient of is invertible at since . We will see later that this fact is essential in order to solve this equation within some function space of analytic functions.
We look for solutions which are rescaled in time of the formwhereAs a result, provided that holds, the expression is supposed to solve the following equation:We make further assumptions on the coefficients and for which are stronger than the constraint (79). Assume the existence of integers and such thatfor all . Then, for all , and all integers , with we deduce the existence of a nonnegative integer which is larger than 1 except such thatIndeed, if one puts , from (88), we can writeAccording to (88) and (89), with the help of formula (8.7) from [26], p. 3630, we can expand the following pieces appearing in (87) satisfied by :for all and all integers and such that , for some real constants , , and , .
In a third step, let us assume that the expression has a formal power series expansionwhere each coefficient is defined as an inverse Fourier transform for some function belonging to the Banach space and depending holomorphically on on some punctured disc centered at 0 with radius . We consider the formal power series obtained by formally applying -Borel transform with respect to and Fourier transform with respect to to the power series (92). The constraints (88) are introduced in such a way that satisfies some integral equation by making use of the properties of the -Borel transform of formal series and Fourier inverse transforms described in Propositions 8 and 9 with the help of the prepared expansions (91). Namely, after division by the power , which is by construction a common factor of the functions , , and for , , we get the new problemwith vanishing initial data , whereBy convention, the two sums appearing in (97) are vanishing provided that .

Remark 14. The hypotheses (60) and (62) ensure that (87) does not contain terms that involve isolated polynomials in which are not inverse Fourier transformable.

5. Analytic Solutions of a Convolution Problem with Complex Parameters

Our main goal in this section is the construction of a unique solution of problem (95) within the Banach spaces introduced in Section 2.

We make the following further assumptions. The conditions below are very similar to the ones proposed in Section  4 of [3]. Namely, we demand that there exists an unbounded sector with direction and aperture for some radius such thatfor all . The polynomial can be factorized in the formwherefor all and all .

We select an unbounded sector centered at 0 and a small closed disc and we require the sector to fulfill the next conditions.

(1) There exists a constant such thatfor all , all , and all . Indeed, from (99) and the explicit expression (101) of , we first observe that for every , all for an appropriate choice of and of . We also see that, for all and all , the roots remain in a union of unbounded sectors centered at 0 that do not cover a full neighborhood of the origin in provided that is small enough. Therefore, one can choose an adequate sector such that with the property that for all the quotients lay outside some small disc centered at 1 in for all and all . This yields (102) for some small constant .

(2) There exists a constant such thatfor some , all , and all . Indeed, for the sector and the disc chosen as above in , we notice that for any fixed , the quotient stays outside a small disc centered at 1 in for all and all . Hence (103) must hold for some small constant .

By construction of the roots (101) in the factorization (100) and using the lower bound estimates (102) and (103), we get a constant such thatfor all and all .

In the next proposition, we provide sufficient conditions under which the main convolution equation (95) possesses solutions in the Banach space described in Section 2.

Proposition 15. Under the additional assumptionsfor all , ,for all , such that , for andfor all , such that , there exist a radius , and a constant such that (95) has a unique solution in the Banach space which suffers the bounds for all , where the direction can be chosen for any sector that fulfills the constraints (102) and (103) above.

Proof. We undertake the proof with a lemma that studies some shrinking map on the Banach spaces mentioned above and reduces the main convolution problem (95) to the existence of a unique fixed point for this map.

Lemma 16. Taking for granted the fact that the assumptions (105), (106), and (107) hold, one can select the constant large enough and a constant small enough such that for all , the map defined aswherealong withsatisfies the next properties.
(i) The following inclusion holds:where is the closed ball of radius centered at 0 in , for all .
(ii) One hasfor all , for all .

Proof. We first deal with the property (113). Let and consider . We take such that .
We start providing norm estimates for each piece of the map .
From Lemma 4, we deduce the existence of a constant depending on , , , and for such thatAccording to Proposition 5(1), we obtain a constant depending on , , for , , for , , , and a constant depending on , , for , , and such thatMoreover, it appears from the proof of Proposition 5 that the next bounds hold for : there exist a constant depending on , , for , , and a constant depending on , , for such that for all . In the following, we will make use of the notations from the proof of Proposition 5. From the classical estimatesfor any real numbers , , we deduce that for all such that Furthermore, according to the Stirling formula as and bearing in mind the functional relation for all , we get two constants and independent of such thatOn the other hand, by direct inspection, we observe that there exists a constant (independent of and ) such thatFurthermore, there exists a constant depending on , , for and , such thatNow, after a thorough examination of the proof of Proposition 2 out of [23], one can check that there exists a constant independent of such thatfor all . Finally, gathering (120), (121), (122), and (123) yields the estimates (117).
Besides, we choose the radius large enough and in such a manner thatNotice that the infinite sums over the integers are convergent in the left-hand side of inequality (124), provided that is small enough, according to the fact that there exist two constants such that for all since is a convergent series near .
From the definition of given by (110), we deduce the following inequality:Hereafter, we focus on norm estimates for each part of the map .
We set Regarding Proposition 6, we get a constant (depending on , ) such thatOn the other hand, using Proposition 5(2), we grab a constant (depending on , , , , , for and , ) such thatTherefore, gathering (127) and (128) returnsBearing in mind Proposition 5(1), we get a constant (depending on , , , and , for ), such thatLikewise, we can apply Proposition 5(2) in order to exhibit a constant (depending on , , , , , and , for ) withfor all and with . Besides,provided that and , with .
Now, we choose and in such a way thatWith the help of the definition of given by (111), we deduce that Ultimately, we direct our attention to norm estimates for .
Taking notice of Proposition 5(1), we get a constant (depending on , , , , ), such thatMoreover, we can apply Proposition 5(2) in order to exhibit a constant (depending on , , , , and ) withfor all and with . Besides,provided that , , and with . Finally, we can select a constant (depending on , , and ) such thatfor all . We make the choice for the size of radius and in such a manner thatFrom the construction of the map , it is now clear thatEventually, gathering (125), (134), and (140) yields the first claim (113).
In the last part of the proof, we fix our attention to the affirmation (114). Let with We first prove that is a shrinking map. According to estimates (116) we obtain a constant (depending on , , , for , , for , , ) and a constant satisfying the estimates (117) such thatTherefore, we choose the radius large enough in order thatAs a result, we can set downWe turn to and show that it is a shrinking map as well. As a preparation, we may first rewriteFor , we set Regarding both the factorization (145) above and Proposition 6, we get a constant (depending on , ) such thatFrom (128) together with (147) we pick up a constant (depending on , , , , , for and , ) such thatBearing in mind (130), we get a constant (depending on , , , and , for ), such thatLikewise, we can apply (131) in order to exhibit a constant (depending on , , , , , and , for ) withfor all and with . Furthermore, from (132) we deduceprovided that and , with .
Now, we sort and in such a way thatSubsequently, we obtainThe last operation will be devoted to the proof that is a shrinking map.
Taking notice of (135), we get a constant (depending on , , , , and ), such thatMoreover, we may have a look at (136) in order to exhibit a constant (depending on , , , , and ) withfor all and with . Besides, from (137), we see thatprovided that , , and with . Finally, having a glance at (138), we can select a constant (depending on , , and ) such thatfor all . In the meanwhile, we select the size of radius in such a manner thatThe following inequality must then hold:Gathering (144), (153), and (159) legitimates estimates (114).
At the very end of the proof, we now take for granted that all conditions (124), (133), (139), (143), (152), and (158) hold for the radii and . Then both (113) and (114) hold at the same time and Lemma 16 is shown.

We consider the closed ball just built above in Lemma 16 which is actually a complete metric space with respect to the metric induced by the Banach space norm . From the lemma above, we get that is a contractive map from into itself. Due to the classical contractive mapping theorem, we deduce that the map has a unique fixed point denoted by in the ball , meaning thatfor a unique such that , for all . Moreover, the function depends holomorphically on in .

Now, if one sets apart the terms in the left-hand side and in the right-hand side of (95), we observe by dividing with the polynomial given in (100) that (95) can be exactly rewritten as (160). Therefore, the unique fixed point of in precisely solves problem (95) with vanishing initial data . This yields the proposition.

6. Singular Analytic Solutions on Sectors to the Main Problem

We go back to the sequence of formal constructions performed in Section 4 under the new light shed in Section 5 on problem (95).

We first recall the definitions of a good covering and associated sets of sectors as introduced in [3].

Definition 17. Let be an integer. For all , we consider open sectors centered at , with radius and opening with small enough such that , for all (with the convention that . Moreover, we assume that the intersection of any three different elements in is empty and that , where is some neighborhood of 0 in . Such a set of sectors is called a good covering in .

Definition 18. Let be a good covering in . Let be an open bounded sector centered at 0 with radius and consider a family of open sectors with aperture and where , for all , are directions which satisfy the following constraints: let be the roots of the polynomials (100) defined by (101) and and be unbounded sectors centered at 0 with directions and with small aperture. We assume the following.
(1) There exists a constant such thatfor all ; all ; all , for all .
(2) There exists a constant such thatfor some ; all ; all , for all .
(3) For all , all ; all ; we have that .
We say that the family is associated with the good covering .

In the next main first outcome, we construct a family of actual holomorphic solutions to the principal equation (61) which may be meromorphic at and defined on the sectors with respect to the complex parameter . Furthermore, we can also control the difference between any two neighboring solutions on the intersections and state that it is exponentially flat of order at most with respect to .

Theorem 19. One considers the nonlinear singularly perturbed PDE (61) and takes for granted that all the assumptions (60), (62), (65), (66), (67), (77), (78), (79), (88), (99), (105), (106), and (107) hold for some rational numbers , and integers , . Let a good covering in be given, for which a family of open sectors associated with this good covering can be singled out.
Then, there exist a radius large enough and small enough, for which a family of actual solutions of (61) can be built up. More exactly, the functions solve the following singularly perturbed PDE:with an additional part of forcing term where is given by expression (183) and defines a holomorphic bounded function provided that the additional constraints (184) are fulfilled. Each function can be decomposed aswhere is holomorphic on some disc , , and defines a bounded holomorphic function on for any given , with on . Furthermore, there exist constants and (independent of ) such thatfor all , for all (where by convention ).

Proof. We plan to construct actual solutions of the main equation (61) by performing backwards the sequence of constructions described in Section 4 starting from problem (95) solved in Section 5.
Let be a good covering in and let be a family of sectors associated with this good covering. From Proposition 15, we see that, for each direction , one can get a solution of the convolution equation (95) that belongs to the space and thus satisfies the following bounds:for all , all , and all , for some well chosen . Besides, these functions are analytic continuations with respect to of a common convergent series with coefficients in the Banach space solution of (95) for all . In particular, we see that the formal power series is -summable in direction as a series with coefficients in the Banach space for all in the sense of Definition 7. We denote byits -sum in direction , where , which defines an -valued analytic function with respect to on a sector for (where denotes the aperture of the sector ) and some (independent of ), for all .
Bearing in mind the identities of Proposition 8 and using the properties for the -sum with respect to derivatives and products (within the Banach algebra equipped with the convolution product as described in Proposition 2), we check that the functions must solve the following problem:whereWe examine now the functionwhich defines a bounded holomorphic function with respect to on , with respect to on for any , and for all on . Using the properties of the Fourier inverse transform described in Proposition 9 and watching out the expansions (91), we extract from equality (172) the next equation satisfied by ; namely,We now set and we focus on the functionwhich defines a bounded holomorphic function with respect to such that and with respect to on for any , for all . Having a quick look at (176), we observe that solves a related equation which after multiplication by yieldsIn the next step, we introduce the functionwhich defines a holomorphic function with respect to such that and with respect to on for any , for all . Notice that this function may be meromorphic at , provided that . Taking (178) into consideration, we see that the function solves the next PDE with forcing termwhich is exactly (72) announced in Remark 12 of Section 4.1, where is a contribution to the forcing term equal toUsing the fact that solves the second-order algebraic equation (70) and noticing the following identity: we can abridge the latter expression of asObserve that is bounded holomorphic with respect to and is analytic in near 0 provided that the following additional conditions hold:for all , , and .
Finally, we putwhich defines a holomorphic function with respect to on , with respect to for any , and with respect to , where and are sectors described in Definition 18. As a result, admits the decomposition (165) with which determines a bounded holomorphic function on for any given with the property for all . Again, the function may be meromorphic in both and in the vicinity of the origin. From (180) and (183) we deduce that solves the next main problemwith additional forcing term . As a spin-off, by applying the operator on the left- and right-hand side of this last equation, we see that is also an actual solution of problem (61) disclosed at the beginning of Section 4.
In the last part of the proof, we proceed to justify estimates (166). The steps of the verification are similar to the arguments displayed in Theorem  1 of [3], but we choose to present them for the sake of completeness. Let . By the sequence of constructions performed above, we see that the function can be written as a -Laplace and Fourier transformwhere . Using the fact that the function is holomorphic on for all , its integral along the union of a segment starting from 0 to , an arc of circle with radius which connects and , and a segment starting from to 0 are vanishing. Therefore, we can write the difference as a sum of three integrals:where , , and is an arc of circle with radius connecting and with a well chosen orientation.
We give estimates for the quantity By construction, the direction (which depends on ) is chosen in such a way that , for all , for all , and for some fixed . From the estimates (167), we get thatfor all and with , for some , and for all .
In the same way, we also give estimates for the integral Namely, the direction (which depends on ) is chosen in such a way that , for all , for all , and for some fixed . Again from the estimates (167) and following the same steps as in (190), we deduce thatfor all and with , for some , and for all .
Finally, we give upper bound estimates for the integral By construction, the arc of circle is chosen in such a way that , for all (if ) and (if ), for all , for all , and for some fixed . Bearing in mind (167) and (118), we get thatfor all and with , for some , and for all .
Finally, gathering the three above inequalities (190), (192), and (194), we deduce from decomposition (188) that for all and with , for some , and for all . Therefore, inequality (166) holds.

7. Parametric Gevrey Asymptotic Expansions of the Solutions

7.1. -Summable Formal Series and Ramis-Sibuya Theorem

We recall the definition of -Borel summability of formal series with coefficients in a Banach space as introduced in [5].

Definition 20. Let be an integer. A formal series with coefficients in a Banach space is said to be -summable with respect to in the direction if
(i) the existence of is ensured such that the following formal series, called formal Borel transform of of order , is absolutely convergent for ;
(ii) one can select a such that the series can be analytically continued with respect to in a sector . Moreover, there exist , and , such that for all .

If the definition above is fulfilled, the vector valued Laplace transform of order of in the direction is set as along a half line , where depends on and is chosen in such a way that , for some fixed , for all in a sector where and . The function is called the -sum of the formal series in the direction . It is bounded and holomorphic on the sector and has the formal series as Gevrey asymptotic expansion of order with respect to on . This means that, for all , there exist such that for all , all .

Now, we state a cohomological criterion for -summability of formal series with coefficients in Banach spaces (see [25], p. 121 or [27], Lemma  XI-2-6) which is known as the Ramis-Sibuya theorem in the literature.

Theorem (RS). Let be a Banach space over and be a good covering in . For all , let be a holomorphic function from into the Banach space and let the cocycle be a holomorphic function from the sector into (with the convention that and ). One makes the following assumptions.

(1) The functions are bounded as tends to the origin in , for all .

(2) The functions are exponentially flat of order on , for all . This means that there exist constants such thatfor all , all .

Then, for all , the functions are the -sums on of a common -summable formal series .

7.2. Parametric Gevrey Asymptotic Expansions of the Solutions and Construction of -Sums

In this subsection, we state the second main result of our work, namely, the existence of a formal power series in the parameter whose coefficients are bounded holomorphic functions on the product of a sector with small radius centered at 0 and a strip in which is the common Gevrey asymptotic expansion of order of the functions appearing in the expansion (165) of the solutions to the main equations (61) and (164) established in Theorem 19.

Theorem 21. Let one assume that the hypotheses of Theorem 19 hold. Then, there exists a formal power series whose coefficients belong to the Banach space of bounded holomorphic functions on equipped with supremum norm, where is defined in Theorem 19, and such that the functions from the decomposition (165) are its -sums on the sectors , for all , viewed as holomorphic functions from into . In other words, for all , there exist two constants such thatfor all ; all .

Proof. We consider the family of functions , , constructed in Theorem 19. For all , we define , which is by construction a holomorphic and bounded function from into the Banach space of bounded holomorphic functions on equipped with the supremum norm, where is introduced in Definition 18, is set in Theorem 19, and is the width of the strip on which the coefficients are defined with respect to (see (62)). Bearing in mind the estimates (166), we see that the cocycle is exponentially flat of order on , for any . Therefore, according to Theorem (RS) stated above, we obtain a formal power series such that the functions are the -sums on of as -valued functions, for all . The result follows.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.