Singular perturbation of nonlinear systems with regular singularity

We extend Balser-Kostov method of studying summability properties of a singularly perturbed inhomogeneous linear system with regular singularity at origin to nonlinear systems of the form \varepsilon zf^{\prime} = F(\varepsilon,z,f) with F a \mathbb{C}^{\nu}-valued function, holomorphic in a polydisc \bar{D}_{\rho}\times \bar{D}_{\rho}\times \bar{D}_{\rho}^{\nu}. We show that its unique formal solution in power series of \varepsilon, whose coefficients are holomorphic functions of z, is 1-summable under a Siegal-type condition on the eigenvalues of F_{f}(0,0,0). The estimates employed resemble the ones used in KAM theorem. A simple Lemma is developed to tame convolutions that appears in the power series expansion of nonlinear equations.


Introduction
Balser and Kostov [BK] have studied singularly perturbed linear system with regular singularity at z = 0 of the form f ′ means derivative of f w.r.t. z; A = A(ε, z) and b = b(ε, z) are, respectively, a ν × ν matrix and a ν-vector whose entries are holomorphic in the polydisc D R × D R , R > 0. 1 A is, in addition, such that A(0, 0) −1 exists. For such a system, there exists a unique formal solution in the ring O(r) [[ε]] 1 of formal power serieŝ in ε with coefficients a i (z) in the ring O(r) of holomorphic functions on D r , continuous in its closure, satisfying max |z−z 0 |≤r |a i (z)| ≤ Cµ i i! , i = 0, 1, 2, . . . for some positive constants C, µ and 0 < r < R. The authors have shown (see Theorem 1 and 2 of [BK]) thatf (ε, z) is the 1-Gevrey asymptotic expansion of a holomorphic function f (ε, z) in S(θ, γ; E) × D r , as ε tends to 0, if the closed sectorS(θ, γ; E) does not contain any ray on the direction of the eigenvalues λ j of A(0, 0): |arg λ j − θ| > γ/2 , j = 1, . . . , n . (1.4) The formal seriesf (ε, z) is thus 1-summable in the direction θ provided the eigenvalues of A(0, 0) satisfy a Siegel-type condition, i.e. the λ j satisfy (1.4) for some γ ≥ π. A nonlinear version of (1.1) appears as follows. Let f (ε, z) be the unique extension in S(0, γ; E) × D r , with ε = 2/N, of the meromorphic function where J κ (x) is the Bessel function of order κ. This function is related with the Fourier-Stieltjes transformσ N (x) of a uniform measure σ N on the N-dimensional sphere of radius √ N and we refer to [MC] and [MCG] for the motivations for its study. The N dependence in the argument is chosen in such way that φ ε (z) attains, as ε goes to 0, a limit function φ 0 (z) = −1 1 + √ 1 + 4z (1.6) (see Proposition 2.1 of [MCG]). φ ε satisfies an ordinary (Riccati) differential equation which, despite of being nonlinear, can be dealt by Balser-Kostov's method. It has been shown by the present authors (see Lemmas 3.2, 3.3 and 3.4 of [MC]) (a) existence of a unique formal solutionφ ε (z) in the form of (1.2), satisfying (1.3); (b)φ ε (z) is the 1-Gevrey asymptotic expansion of the holomorphic solution f (ε, z) of (1.7) in S(0, γ; E) × D r , as ε goes to 0 in S(0, γ; E); (c) choosing the sector S(θ, γ; E) of opening angle γ > π away from the negative real axis,φ ε (z) is, in addition, 1-summable in θ direction and its sum is equal to f (ε, z).
In the present article statements (a)-(c), together with the 1-summability, will be extended for more general ordinary differential equations of the form . . , f ν ) and F = (F 1 , . . . , F ν ) ν-vector functions, F i holomorphic in a polydisc, sayD ρ ×D ρ 1 ×D ν ρ , for some ρ 1 > ρ > 0. As in ( [BK]), the ν × ν matrix A 0,1 (0) = F f (0, 0, 0) is assumed to be invertible, a condition that makes (1.8) to possess a regular singularity at z = 0, and every eigenvalue of A 0,1 (0) satisfies condition (1.4). Equation (1.7) is of the form (1.8) with ν = 1 and 2 Balser-Kostov summability proof in [BK] of the formal seriesf solution does not follow the usual route: the (formal) Borel transformBf off is analytically continued along some sector of infinite radius (see e.g. [Ba]). Their proof establishes instead Grevrey asymptotic expansion directly from the equation (1.1), making resource of an auxiliary Lemma regarding an infinite system of linear equation of the same type. Although (1.8) is nonlinear, the system of infinitely many equations obtained by taking derivatives of (1.8) with respect to ε is linear, indeed of the type stated in Lemma 3 of [BK], and Balser-Kostov's method carries over to equation of the form (1.8).
The layout of this paper is as follows. In Section 2 we prove existence of a unique solution of (1.8) in power series of z. In Section 3 we show that the formal power series solution of (1.8) is Gevrey of order 1. In Section 4 Gevrey asymptotics are established. Our main result, the 1-summability of the formal solution of (1.8), is stated in Section 5 and proved using Propositions 2.2-4.1 of the previous sections. The main ingredient (Lemma 2.3), is employed to tame arbitrarily large number of convolutions arised in the expansion of F in power series of f .
A n,m (ε) regarded as a multilinear operator, endowed with an operator norm induced by the Euclidean space C ν : holomorphic in D ρ as a function of ε.
Proof Since (2.3) solves (1.8), its coefficients f k (ε) satisfy the formal relations (2.4) whose solution depends on the existence of inverse matrix (εkI − A 0,1 (ε)) −1 for every k ∈ N and ε ∈ S (0, γ, E). Assuming (1.4) holds for every eigenvalues of A 0,1 (0), let γ and E be such that (2.9), and consequently (2.10), holds. Hence, f k (ε) given by (2.8) is bounded uniformly in S (0, γ; E), uniquely defined for every k ∈ N and, in view of these, holomorphic in S (0, γ; E). Let φ l and α n,m be the supremum in S(0, γ; E) of f l (ε) and A n,m (ε) , respectively: (2.12) Now, we prove that the majorant series ∞ l=1 φ l σ l converges and is bounded by ρ for some 0 < σ < ρ. For this, the following lemma will be stated more generally than it is needed for this section.
From the uniform convergence of (2.3) we conclude that, for any fixed z ∈ D σ , the solution f (ε, z) tends to where f * (z) is the unique solution of equation for f , by the analytic implicit function theorem (see e.g. Section 2.3 of [Be] or the next section, for an alternative solution). Note that the solution f * (z) is regular at z = 0 since, by (2.3), it must satisfy f (0) = 0 and this concludes the proof of Proposition 2.2.
Remark 3.1 Regarding the radius of convergence of the power series of f * (z) one can estimate it a little better using the Cauchy majorant method as in Section 3.2 of [Be] (see also [BK], Section 1, for the linear equation). Multiplying (3.5) by z k , summing over k and replacing the inequality by equality, yields is holomorphic in a disc D σ 1 with σ 1 = ρ 1 ρ/(ρ + 4cC) < ρ 1 , proportional to ρ 1 . In Section 2, we have chosen ρ 1 so large that (2.12) can be written as (2.15) and the radius of convergence σ, obtained applying Lemma 2.3 to convolutions, is proportional to ρ instead (see expression after (2.19)). Despite of this loss, the method introduced there is undeniably practical, more adaptable to diverse situations and, for these reasons, we shall apply it here and in further sections.
Proposition 3.2 Suppose the formal power series (1.2) satisfies equation (1.8), formally, with F = F (ε, z, f ) obeying the hypotheses stated after (1.8). Then, the coefficients (a i (z)) i≥0 of (1.2) are analytic functions of z in the open disc D κ and there exist positive constants C and µ such that holds for all i ≥ 0 and z ∈D σ , with σ < κ < ρ. In other words, the formal power series is of Gevrey order 1, i.e.,f (ε, z) ∈ O(σ)[[ε]] 1 .
Proof Substituting the power series (1.2) into (3.1), we are thus led to a system of equations which has already been solved for a 0 (z), and for i ≥ 1 We observe that the sum over m has no limit as the sequence a(z) = (a k (z)) k≥0 starts from k = 0 and the convolution product of any two sequences α = (α k ) k≥0 and β = (β k ) k≥0 is now defined by To isolate a i , the largest index term in (3.7), we have to show that the matrix (recall B 0,1 (0) = A 0,1 (0)) is invertible for every z ∈ D κ for some κ ≤ ρ. For this, we take κ so small that c sup It follows from (3.7) and (3.9) that and this relation determines uniquely a i (z) in terms of earlier coefficients. Note that a i (z) is holomorphic in D κ and, by (3.5) and (2.19) sup z∈Dκ (0) |a 0 (z)| ≤ δA , (3.11) by letting κ small enough, for any δ > 0. Now, to obtain an estimate on the growth rate of |a i (z)|, let ϕ i denote the i-th Nagumo norm of a i (z) and let β n,m the supremum in D κ of B n,m (z) . The properties we shall use on Nagumo's norms is proved in ( [BK]) and references therein and are sumarized by for any two functions f and g holomorphic in D κ and nonnegative integers k, l.

Gevrey asymptotics
In order to set up an equation involving derivatives of f with respect to ε, we write and φ(ε, z) = (φ i (ε, z)) i≥0 for the sequence of those functions defined on S(0, γ; E) × D κ (0); analogously to (2.1) and (3.1), we write for the i-th derivative of h with respect to the first argument divided by i!. The i-th total derivative of F with respect to ε can thus be written as andG i (ε, z, φ 0 , . . . , φ i−1 ) depends only on derivatives of f with respect to ε of order lower than i. Differentiating equation (1.8) i times with respect to ε, dividing by i!, we have (4.4) may be think as inhomogeneous holomorphic function of (ε, z) in S(0, γ; E) × D σ (0), and for i = 0 simply (1.8): Proposition 4.1 Let f (ε, z) be the unique holomorphic solution of (1.8) on S(0, γ; E)× D σ (0) with σ, γ and E as in Proposition 2.2. There exist 0 < σ 1 ≤ σ, 0 < E 1 ≤ E and positive constants C and µ such that holds for all i ≥ 0 and every point (ε, z) in S(0, γ; E 1 ) ×D σ 1 (0).
Proof The case i = 0 follows straightforwardly from Proposition 2.2. (4.3) is a linear singular perturbation equation with regular singularity which can be dealt with the following auxiliary result due to Balser-Kostov [BK] (see Lemma 3 therein). For this, we drop temporarily all subindices i in (4.3). Let and consider a sequence (ψ k (ε, z)) k≥0 satisfying the system absolutely convergent for |z| ≤ σ, uniformly in S(0, γ; E). For H given by (4.1) and (4.4) this will actually be proven by induction when we resume the proof of Proposition 4.1. We write, in addition, , if |c k | ≤ C k holds for all k. If f is a ν-vector or a ν × ν matrix f (z) ≪ F (z) means majorized relation for each component.