On parametric multisummable formal solutions to some nonlinear initial value Cauchy problems

We study a nonlinear initial value Cauchy problem depending upon a complex perturbation parameter $\epsilon$ whose coefficients depend holomorphically on $(\epsilon,t)$ near the origin in $\mathbb{C}^{2}$ and are bounded holomorphic on some horizontal strip in $\mathbb{C}$ w.r.t the space variable. We consider a family of forcing terms that are holomorphic on a common sector in time $t$ and on sectors w.r.t the parameter $\epsilon$ whose union form a covering of some neighborhood of 0 in $\mathbb{C}^{\ast}$, which are asked to share a common formal power series asymptotic expansion of some Gevrey order as $\epsilon$ tends to 0. The proof leans on a version of the so-called Ramis-Sibuya theorem which entails two distinct Gevrey orders. Finally, we give an application to the study of parametric multi-level Gevrey solutions for some nonlinear initial value Cauchy problems with holomorphic coefficients and forcing term in $(\epsilon,t)$ near 0 and bounded holomorphic on a strip in the complex space variable.

This work is a continuation of the study initiated in [14] where the authors have studied initial value problems with quadratic nonlinearity of the form (2) Q(∂ z )(∂ t u(t, z, ǫ)) = (Q 1 (∂ z )u(t, z, ǫ))(Q 2 (∂ z )u(t, z, ǫ)) for given vanishing initial data u(0, z, ǫ) ≡ 0, where D, ∆ l , d l , δ l are positive integers and Q(X), Q 1 (X), Q 2 (X), R l (X), 0 ≤ l ≤ D, are polynomials satisfying similar constraints to the ones envisaged for the problem (1). Under the assumption that the coefficients c 0 (t, z, ǫ) and the forcing term f (t, z, ǫ) are bounded holomorphic functions on D(0, r) × H β × D(0, ǫ 0 ), one can build, using some Borel-Laplace procedure and Fourier inverse transform, a family of holomorphic bounded functions u p (t, z, ǫ), 0 ≤ p ≤ ς − 1, solutions of (2), defined on the products T × H β × E p , where E p has an aperture slightly larger than π/k. Moreover, the functions ǫ → u p (t, z, ǫ) share a common formal power seriesû(t, z, ǫ) = m≥0 h m (t, z)ǫ m /m! as asymptotic expansion of Gevrey order 1/k on E p . In other words, u p (t, z, ǫ) is the k−sum ofû(t, z, ǫ) on E p , see Definition 9.
In this paper, we observe that the asymptotic expansion of the solutions u dp (t, z, ǫ) of (1) w.r.t ǫ on E p , defined asû(t, z, ǫ) = m≥0 h m (t, z)ǫ m /m! ∈ F[[ǫ]], inherites a finer structure which involves the two Gevrey orders 1/k 1 and 1/k 2 . Namely, the order 1/k 2 originates from the equation (1) itself and its highest order term ǫ (δ D −1)(k 2 +1)−δ D +1 t (δ D −1)(k 2 +1) ∂ δ D t R D (∂ z ) as it was the case in the work [14] mentioned above and the scale 1/k 1 arises, as a new feature, from the asymptotic expansionf of the forcing terms f dp (t, z, ǫ). We can also describe conditions for which u dp (t, z, ǫ) is the (k 2 , k 1 )−sum ofû(t, z, ǫ) on E p for some 0 ≤ p ≤ ς − 1, see Definition 10. More specifically, we can present our two main statements and its application as follows.
Main results Let k 2 > k 1 ≥ 1 be integers. We choose a family {E p } 0≤p≤ς−1 of bounded sectors with aperture slightly larger than π/k 2 which defines a good covering in C * (see Definition 7) and a set of adequate directions d p ∈ R, 0 ≤ p ≤ ς − 1 for which the constraints (173) and (174) hold. We also take an open bounded sector T centered at 0 such that for every 0 ≤ p ≤ ς − 1, the product ǫt belongs to a sector with direction d p and aperture slightly larger than π/k 2 , for all ǫ ∈ E p , all t ∈ T . We make the assumption that the coefficient c 0 (t, z, ǫ) can be written as a convergent series of the special form c 0 (t, z, ǫ) = c 0 (ǫ) n≥0 c 0,n (z, ǫ)(ǫt) n on a domain D(0, r) × H β ′ × D(0, ǫ 0 ), where H β ′ is a strip of width β ′ , such that T ⊂ D(0, r), ∪ 0≤p≤ς−1 E p ⊂ D(0, ǫ 0 ) and 0 < β ′ < β are given positive real numbers. The coefficients c 0,n (z, ǫ), n ≥ 0, are supposed to be inverse Fourier transform of functions m → C 0,n (m, ǫ) that belong to the Banach space E (β,µ) (see Definition 2) for some µ > max(deg(Q 1 ) + 1, deg(Q 2 ) + 1) and depend holomorphically on ǫ in D(0, ǫ 0 ) and c 0 (ǫ) is a holomorphic function on D(0, ǫ 0 ) vanishing at 0. Since we have in view our principal application (Theorem 3), we choose the forcing term f dp (t, z, ǫ) as a m k 2 −Fourier-Laplace transform f dp (t, z, ǫ) = k 2 (2π) 1/2 +∞ −∞ Lγ p ψ dp k 2 (u, m, ǫ)e −( u ǫt ) k 2 e izm du u dm, where the inner integration is made along some halfline L γp ⊂ S dp and S dp is an unbounded sector with bisecting direction d p , with small aperture and where ψ dp k 2 (u, m, ǫ) is a holomorphic function w.r.t u on S dp , defined as an integral transform called acceleration operator with indices m k 2 and m k 1 , ψ dp k 2 (u, m, ǫ) = L γ 1 p ψ dp k 1 (h, m, ǫ)G(u, h) dh h where G(u, h) is a kernel function with exponential decay of order κ = ( 1 k 1 − 1 k 2 ) −1 , see (116). The integration path L γ 1 p is a halfline in an unbounded sector U dp with bisecting direction d p and ψ dp k 1 (h, m, ǫ) is a function with exponential growth of order k 1 w.r.t h on U dp ∪ D(0, ρ) and exponential decay w.r.t m on R, satisfying the bounds (177). The function f dp (t, z, ǫ) represents a bounded holomorphic function on T × H β ′ × E p . Actually, it turns out that f dp (t, z, ǫ) can be simply written as a m k 1 −Fourier-Laplace transform of ψ dp k 1 (h, m, ǫ), f dp (t, z, ǫ) = k 1 (2π) 1/2 +∞ −∞ Lγ p ψ dp k 1 (u, m, ǫ)e −( u ǫt ) k 1 e izm du u dm, see Lemma 13.
Our first result stated in Theorem 1 claims that if the sup norms of the coefficients c 1,2 (ǫ)/ǫ, c 0 (ǫ)/ǫ and c F (ǫ)/ǫ on D(0, ǫ 0 ) are chosen small enough, then we can construct a family of holomorphic bounded functions u dp (t, z, ǫ), 0 ≤ p ≤ ς − 1, defined on the products T × H β ′ × E p , which solves the problem (1) with initial data u dp (0, z, ǫ) ≡ 0. Similarly to the forcing term, u dp (t, z, ǫ) can be written as a m k 2 −Fourier-Laplace transform u dp (t, z, ǫ) = k 2 (2π) 1/2 +∞ −∞ Lγ p ω dp k 2 (u, m, ǫ)e −( u ǫt ) k 2 e izm du u dm where ω dp k 2 (u, m, ǫ) denotes a function with at most exponential growth of order k 2 in u on S dp and exponential decay in m ∈ R, satisfying (187). The function ω dp k 2 (u, m, ǫ) is shown to be the analytic continuation of a function Acc dp k 2 ,k 1 (ω dp k 1 )(u, m, ǫ) defined only on a bounded sector S b dp with aperture slightly larger than π/κ w.r.t u, for all m ∈ R, with the help of an acceleration operator with indices m k 2 and m k 1 , Acc dp k 2 ,k 1 (ω dp k 1 )(u, m, ǫ) = L γ 1 p ω dp k 1 (h, m, ǫ)G(u, h) dh h .
We show that, in general, ω dp k 1 (h, m, ǫ) suffers an exponential growth of order larger than k 1 (and actually less than κ) w.r.t h on U dp ∪ D(0, ρ), and obeys the estimates (189). At this point u dp (t, z, ǫ) cannot be merely expressed as a m k 1 −Fourier-Laplace transform of ω dp k 1 and is obtained by a version of the so-called accelero-summation procedure, as described in [1], Chapter 5.
We stress the fact that our application (Theorem 3) relies on the factorization of some nonlinear differential operator which is an approach that belongs to an active domain of research in symbolic computation these last years, see for instance [6], [7], [12], [28], [29], [33].
We mention that a similar result has been recently obtained by H. Tahara and H. Yamazawa, see [31], for the multisummability of formal seriesû(t, x) = n≥0 u n (x)t n ∈ O(C N ) [[t]] with entire coefficients on C N , N ≥ 1, solutions of general non-homogeneous time depending linear PDEs of the form for given initial data (∂ j t u)(0, x) = ϕ j (x), 0 ≤ j ≤ m − 1 (where 1 ≤ m ≤ L), provided that the coefficients a j,α (t) together with t → f (t, x) are analytic near 0 and that ϕ j (x) with the forcing term x → f (t, x) belong to a suitable class of entire functions of finite exponential order on C N . The different levels of multisummability are related to the slopes of a Newton polygon attached to the main equation and analytic acceleration procedures as described above are heavily used in their proof.
It is worthwhile noticing that the multisummable structure of formal solutions to linear and nonlinear meromorphic ODEs has been discovered two decades ago, see for instance [2], [5], [8], [18], [21], [27], but in the framework of PDEs very few results are known. In the linear case in two complex variables with constant coefficients, we mention the important contributions of W. Balser, [4] and S. Michalik, [22], [23]. Their strategy consists in the construction of a multisummable formal solution written as a sum of formal series, each of them associated to a root of the symbol attached to the PDE using the so-called Puiseux expansion for the roots of polynomial with holomorphic coefficients. In the linear and nonlinear context of PDEs that come from a perturbation of ordinary differential equations, we refer to the works of S. Ouchi, [25], [26], which are based on a Newton polygon approach and accelero-summation technics as in [31]. Our result concerns more peculiarly multisummability and multiple scale analysis in the complex parameter ǫ. Also from this point of view, only few advances have been performed. Among them, we must mention two recent works by K. Suzuki and Y. Takei, [30] and Y. Takei, [32], for WKB solutions of the Schrödinger equation ǫ 2 ψ ′′ (z) = (z − ǫ 2 z 2 )ψ(z) which possesses 0 as fixed turning point and z ǫ = ǫ −2 as movable turning point tending to infinity as ǫ tends to 0.
In the sequel, we describe our main intermediate results and the sketch of the arguments needed in their proofs. In a first part, we depart from an auxiliary parameter depending initial value differential and convolution equation which is regularly perturbed in its parameter ǫ, see (72). This equation is formally constructed by making the change of variable T = ǫt in the equation (1) and by taking the Fourier transform w.r.t the variable z (as done in our previous contribution [14]). We construct a formal power seriesÛ (T, m, ǫ) = n≥1 U n (m, ǫ)T n solution of (72) whose coefficients m → U n (m, ǫ) depend holomorphically on ǫ near 0 and belong to a Banach space E (β,µ) of continuous functions with exponential decay on R introduced by O. Costin and S. Tanveer in [10].
As a first step, we follow the strategy recently developped by H. Tahara and H. Yamazawa in [31], namely we multiply each hand side of (72) by the power T k 1 +1 which transforms it into an equation (77) which involves only differential operators in T of irregular type at T = 0 of the form T β ∂ T with β ≥ k 1 + 1 due to the assumption (74) on the shape of the equation (72). Then, we apply a formal Borel transform of order k 1 , that we call m k 1 −Borel transform in Definition 4, to the formal seriesÛ with respect to T , denoted ω k 1 (τ, m, ǫ) = n≥1 U n (m, ǫ) τ n Γ(n/k 1 ) .
Then, we show that ω k 1 (τ, m, ǫ) formally solves a convolution equation in both variables τ and m, see (85). Under some size constraints on the sup norm of the coefficients c 1,2 (ǫ)/ǫ, c 0 (ǫ)/ǫ and c F (ǫ)/ǫ near 0, we show that ω k 1 (τ, m, ǫ) is actually convergent for τ on some fixed neighborhood of 0 and can be extended to a holomorphic function ω d k 1 (τ, m, ǫ) on unbounded sectors U d centered at 0 with bisecting direction d and tiny aperture, provided that the m k 1 −Borel transform of the formal forcing term F (T, m, ǫ), denoted ψ k 1 (τ, m, ǫ) is convergent near τ = 0 and can be extended on U d w.r.t τ as a holomorphic function ψ d k 1 (τ, m, ǫ) with exponential growth of order less than k 1 . Besides, the function ω d k 1 (τ, m, ǫ) satisfies estimates of the form: there exist constants ν > 0 and ̟ d > 0 with |ω d k 1 (τ, m, ǫ)| ≤ ̟ d (1 + |m|) −µ e −β|m| |τ | 1 + |τ | 2k 1 e ν|τ | κ for all τ ∈ U d , all m ∈ R, all ǫ ∈ D(0, ǫ 0 ), see Proposition 11. The proof leans on a fixed point argument in a Banach space of holomorphic functions F d (ν,β,µ,k 1 ,κ) studied in Section 2.1. Since the exponential growth order κ of ω d k 1 is larger than k 1 , we cannot take a m k 1 −Laplace transform of it in direction d. We need to use a version of what is called an accelero-summation procedure as described in [1], Chapter 5, which is explained in Section 4.3.
In a second step, we go back to our seminal convolution equation (72) and we multiply each handside by the power T k 2 +1 which transforms it into the equation (123). Then, we apply a m k 2 −Borel transform to the formal seriesÛ w.r.t T , denotedω k 2 (τ, m, ǫ). We show that ω k 2 (τ, m, ǫ) formally solves a convolution equation in both variables τ and m, see (125), where the formal m k 2 −Borel transform of the forcing term is set asψ k 2 (τ, m, ǫ). Now, we observe that a version of the analytic acceleration transform with indices k 2 and k 1 constructed in Proposition 13 applied to ψ d k 1 (τ, m, ǫ), standing for ψ d k 2 (τ, m, ǫ), is the κ−sum ofψ k 2 (τ, m, ǫ) w.r.t τ on some bounded sector S b d,κ with aperture slightly larger than π/κ, viewed as a function with values in E (β,µ) . Furthermore, ψ d k 2 (τ, m, ǫ) can be extended as an analytic function on an unbounded sector S d,κ with aperture slightly larger than π/κ where it possesses an exponential growth of order less than k 2 , see Lemma 4. In the sequel, we focus on the solution ω d k 2 (τ, m, ǫ) of the convolution problem (134) which is similar to (125) but with the formal forcing termψ k 2 (τ, m, ǫ) replaced by ψ d k 2 (τ, m, ǫ). Under some size restriction on the sup norm of the coefficients c 1,2 (ǫ)/ǫ, c 0 (ǫ)/ǫ and c F (ǫ)/ǫ near 0, we show that ω d k 2 (τ, m, ǫ) defines a bounded holomorphic function for τ on the bounded sector S b d,κ and can be extended to a holomorphic function on unbounded sectors S d with direction d and tiny aperture, provided that S d stays away from the roots of some polynomial P m (τ ) constructed with the help of Q(X) and R D (X) in (1), see (136). Moreover, the function ω d k 2 (τ, m, ǫ) satisfies estimates of the form: there exist constants ν ′ > 0 and υ d > 0 with for all τ ∈ S d , all m ∈ R, all ǫ ∈ D(0, ǫ 0 ), see Proposition 14. Again, the proof rests on a fixed point argument in a Banach space of holomorphic functions F d (ν ′ ,β,µ,k 2 ) outlined in Section 2.2. In Proposition 15, we show that ω d k 2 (τ, m, ǫ) actually coincides with the analytic acceleration transform with indices m k 2 and m k 1 applied to ω d k 1 (τ, m, ǫ), denoted Acc d k 2 ,k 1 (ω d k 1 )(τ, m, ǫ), as long as τ lies in the bounded sector S b d,κ . As a result, some m k 2 −Laplace transform of the analytic continuation of Acc d k 2 ,k 1 (ω d k 1 )(τ, m, ǫ), set as U d (T, m, ǫ), can be considered for all T belonging to a sector S d,θ k 2 ,h with bisecting direction d, aperture θ k 2 slightly larger than π/k 2 and radius h > 0. Following the terminology of [1], Section 6.1, U d (T, m, ǫ) can be called the (m k 2 , m k 1 )−sum of the formal seriesÛ (T, m, ǫ) in direction d. Additionally, U d (T, m, ǫ) solves our primary convolution equation (72), where the formal forcing termF (T, m, ǫ) is interchanged with F d (T, m, ǫ) which denotes the (m k 2 , m k 1 )−sum ofF in direction d.
In Theorem 1, we construct a family of actual bounded holomorphic solutions u dp (t, z, ǫ), 0 ≤ p ≤ ς − 1, of our original problem (1) on domains of the form T × H β ′ × E p described in the main results above. Namely, the functions u dp (t, z, ǫ) (resp. f dp (t, z, ǫ)) are set as Fourier inverse transforms of U dp , u dp (t, z, ǫ) = F −1 (m → U dp (ǫt, m, ǫ))(z) , f dp (t, z, ǫ) = F −1 (m → F dp (ǫt, m, ǫ))(z) where the definition of F −1 is pointed out in Proposition 9. One proves the crucial property that the difference of any two neighboring functions u d p+1 (t, z, ǫ) − u dp (t, z, ǫ) tends to zero as ǫ → 0 on E p+1 ∩ E p faster than a function with exponential decay of order k, uniformly w.r.t t ∈ T , z ∈ H β ′ , with k = k 2 when the intersection U d p+1 ∩ U dp is not empty and with k = k 1 , when this intersection is empty. The same estimates hold for the difference f d p+1 (t, z, ǫ) − f dp (t, z, ǫ).
The whole section 6 is devoted to the study of the asymptotic behaviour of u dp (t, z, ǫ) as ǫ tends to zero. Using the decay estimates on the differences of the functions u dp and f dp , we show the existence of a common asymptotic expansionû of Gevrey order 1/k 1 for all functions u dp (t, z, ǫ) (resp. f dp (t, z, ǫ)) as ǫ tends to 0 on E p . We obtain also a double scale asymptotics for u dp as explained in the main results above. The key tool in proving the result is a version of the Ramis-Sibuya theorem which entails two Gevrey asymptotics orders, described in Section 6.1. It is worthwhile noting that a similar version was recently brought into play by Y. Takei and K. Suzuki in [30], [32], in order to study parametric multisummability for the complex Schrödinger equation.
In the last section, we study the particular situation when the formal forcing term F (T, m, ǫ) solves a linear differential and convolution initial value problem, see (225). We multiply each handside of this equation by the power T k 1 +1 which transforms it into the equation (229). Then, we show that the m k 1 −Borel transform ψ k 1 (τ, m, ǫ) formally solves a convolution equation in both variables τ and m, see (233). Under a size control of the sup norm of the coefficients c 0 (ǫ)/ǫ and c F (ǫ)/ǫ near 0, we show that ψ k 1 (τ, m, ǫ) is actually convergent near 0 w.r.t τ and can be holomorphically extended as a function ψ dp k 1 (τ, m, ǫ) on any unbounded sectors U dp with direction d p and small aperture, provided that U dp stays away from the roots of some polynomial P m (τ ) constructed with the help of Q(X) and R D (X) in (225). Additionally, the function ψ dp k 1 (τ, m, ǫ) satisfies estimates of the form: there exists a constant υ > 0 with |ψ dp for all τ ∈ U dp , all m ∈ R, all ǫ ∈ D(0, ǫ 0 ), see Proposition 18. The proof is once more based upon a fixed point argument in a Banach space of holomorphic functions F d (ν,β,µ,k 1 ,k 1 ) defined in Section 2.1. These latter properties on ψ dp k 1 (τ, m, ǫ) legitimize all the assumptions made above on the forcing term F (T, m, ǫ). Now, we can take the m k 1 −Laplace transform L dp m k 1 (ψ dp k 1 )(T ) of ψ dp k 1 (τ, m, ǫ) w.r.t τ in direction d p , which yields an analytic solution of the initial linear equation (225) on some bounded sector S dp,θ k 1 ,h with aperture θ k 1 slightly larger than π/k 1 . It comes to light in Lemma 13, that L dp m k 1 (ψ dp k 1 )(T ) coincides with the analytic (m k 2 , m k 1 )−sum F dp (T, m, ǫ) ofF in direction d p on the smaller sector S dp,θ k 2 ,h with aperture slightly larger than π/k 2 . We deduce consequently that the analytic forcing term f dp (t, z, ǫ) solves the linear PDE (260) with analytic coefficients on D(0, r) × H β ′ × D(0, ǫ 0 ), for all t ∈ T , z ∈ H β ′ , ǫ ∈ E p . In our last main result (Theorem 3), we see that this latter issue implies that u dp (t, z, ǫ) itself solves a nonlinear PDE (264) with analytic coefficients and forcing term on The paper is organized as follows. In Section 2, we define some weighted Banach spaces of continuous functions on (D(0, ρ)∪U )×R with exponential growths of different orders on unbounded sectors U w.r.t the first variable and exponential decay on R w.r.t the second one. We study the continuity properties of several kind of linear and nonlinear operators acting on these spaces that will be useful in Sections 4.2, 4.4 and 7.2. In Section 3, we recall the definition and the main analytic and algebraic properties of the m k −summability. In Section 4.1, we introduce an auxiliary differential and convolution problem (72) for which we construct a formal solution. In Section 4.2, we show that the m k 1 −Borel transform of this formal solution satisfies a convolution problem (85) that we can uniquely solve within the Banach spaces described in Section 2. In Section 4.3, we describe the properties of a variant of the formal and analytic acceleration operators associated to the m k −Borel and m k −Laplace transforms. In Section 4.4, we see that the m k 2 −Borel transform of the formal solution of (72) satisfies a convolution problem (125). We show that its formal forcing term is κ−summable and that its κ−sum is an acceleration of the m k 1 −Borel transform of the above formal forcing term. Then, we construct an actual solution to the corresponding problem with the analytic continuation of this κ−sum as nonhomogeneous term, within the Banach spaces defined in Section 2. We recognize that this actual solution is the analytic continuation of the acceleration of the m k 1 −Borel transform of the formal solution of (72). Finally, we take its m k 2 −Laplace transform in order to get an actual solution of (167). In Section 5, with the help of Section 4, we build a family of actual holomorphic solutions to our initial Cauchy problem (1). We show that the difference of any two neighboring solutions is exponentially flat for some integer order in ǫ (Theorem 1). In Section 6, we show that the actual solutions constructed in Section 5 share a common formal series as Gevrey asymptotic expansion as ǫ tends to 0 on sectors (Theorem 2). The result is buit on a version of the Ramis-Sibuya theorem with two Gevrey orders stated in Section 6.1. In Section 7, we inspect the special case when the forcing term itself solves a linear PDE. Then, we notice that the solutions of (1) constructed in Section 5 actually solve a nonlinear PDE with holomorphic coefficients and forcing term near the origin (Theorem 3).

Banach spaces of functions with exponential growth and decay
The Banach spaces introduced in the next subsection 2.1 (resp. subsection 2.2) will be crucial in the construction of analytic solutions of a convolution problem investigated in the forthcoming subsection 4.2 (resp. subsection 4.4).

2.1
Banach spaces of functions with exponential growth κ and decay of exponential order 1 We denote D(0, r) the open disc centered at 0 with radius r > 0 in C andD(0, r) its closure. Let U d be an open unbounded sector in direction d ∈ R centered at 0 in C. By convention, the sectors we consider do not contain the origin in C.
Throughout the whole subsection, we assume µ, β, ν, ρ > 0, k, κ ≥ 1 and d ∈ R are fixed. In the next lemma, we check the continuity property under multiplication operation with bounded functions.
for all h(τ, m) ∈ F d (ν,β,µ,k,κ) . In the next proposition, we study the continuity property of some convolution operators acting on the latter Banach spaces.
It remains to consider the case k = 1. In that case, we know from Corollary 4.9 of [9] that there exists a constant j 1 > 0 such that (29) and (35), we deduce that From L'Hospital rule, we know that which is finite whenever κ ≥ 1. Therefore, we get a constantC 3.3.1 > 0 such that Taking into account the estimates for (24), (27), (28), (29), (36) and (37), we obtain the result (21) for k = 1. ✷ Definition 2 Let β, µ ∈ R. We denote E (β,µ) the vector space of continuous functions h : for all m ∈ R. Assume that µ > deg(Q) + 1. Let m → b(m) be a continuous function such that for all m ∈ R. Then, there exists a constant C 4 > 0 (depending on Q, R, µ, k, κ, ν) such that Proof The proof follows the same lines of arguments as those of Propositions 1 and 2. Let Again, we know that there exist constants Q, R > 0 such that for all m, m 1 ∈ R. By means of the triangular inequality |m| ≤ |m 1 | + |m − m 1 |, we get that where Under the hypothesis κ ≥ k and from the estimates (7), (11) and (19) in the special case χ 2 = 1/k and ν 2 = −1, we know that C 4.1 is finite.

2.2
Banach spaces of functions with exponential growth k and decay of exponential order 1 In this subsection, we mainly recall some functional properties of the Banach spaces already introduced in the work [14], Section 2. The Banach spaces we consider here coincide with those introduced in [14] except the fact that they are not depending on a complex parameter ǫ and that the functions living in these spaces are not holomorphic on a disc centered at 0 but only on a bounded sector centered at 0. For this reason, all the propositions are given without proof except Proposition 5 which is an improved version of Propositions 1 and 2 of [14]. We denote S b d an open bounded sector centered at 0 in direction d ∈ R andS b d its closure. Let S d be an open unbounded sector in direction d. By convention, we recall that the sectors we consider throughout the paper do not contain the origin in C.
Definition 3 Let ν, β, µ > 0 be positive real numbers. Let k ≥ 1 be an integer and let d ∈ R.
Throughout the whole subsection, we assume that µ, β, ν > 0 and k ≥ 1, d ∈ R are fixed. In the next lemma, we check the continuity property by multiplication operation with bounded functions.
Lemma 2 Let (τ, m) → a(τ, m) be a bounded continuous function on (S b d ∪ S d ) × R, which is holomorphic with respect to τ on S b d ∪ S d . Then, we have for all h(τ, m) ∈ F d (ν,β,µ,k) .
In the next proposition, we study the continuity property of some convolution operators acting on the latter Banach spaces.
Proof The proof follows similar arguments to those in Proposition 1. Indeed, let f (τ, m) ∈ F d (ν,β,µ,k) . By definition, we have Therefore, Now, we study the function F 1 (x). We first assume that −1 < χ 2 < 0. In that case, we have for all x > 0. Bearing in mind that 1 + χ 2 + ν 2 ≥ 0 and since 1 + x ≥ 1 for all x ≥ 0, we deduce that there exists a constant K 1 > 0 with We assume now that χ 2 ≥ 0. In this situation, we know that (x− h) χ 2 ≤ x χ 2 for all 0 ≤ h ≤ x/2, with x ≥ 0. Hence, for all x ≥ 0. Again, we deduce that there exists a constant K 1.
In the next step, we focus on the function F 2 (x). First, we observe that 1 for all x > 0. Now, from the estimates (18), we know that there exists a constant K 2.3 > 0 such that On the other hand, we also have that 1 + x ≥ x for all x ≥ 1. Since γ 1 ≥ ν 2 and due to (52) with (53), we get a constantF 2 > 0 with Gathering the estimates (47), (49), (51), (54) and (55), we finally obtain (45). ✷ The next two propositions are already stated as Propositions 3 and 4 in [14].
Proposition 7 Let k ≥ 1 be an integer. Let Q(X), R(X) ∈ C[X] be polynomials such that

Laplace transform, asymptotic expansions and Fourier transform
We recall a definition of k−Borel summability of formal series with coefficients in a Banach space which is a slightly modified version of the one given in [1], Section 3.2, that was introduced in [14]. All the properties stated in this section are already contained in our previous work [14].
Definition 4 Let k ≥ 1 be an integer. Let m k (n) be the sequence defined by is absolutely convergent for |τ | < ρ.
ii) there exists δ > 0 such that the series B m k (X)(τ ) can be analytically continued with respect to τ in a sector S d,δ = {τ ∈ C * : |d − arg(τ )| < δ}. Moreover, there exist C > 0 and If this is so, the vector valued m k −Laplace transform of B m k (X)(τ ) in the direction d is defined by where γ depends on T and is chosen in such a way that cos(k(γ − arg(T ))) ≥ δ 1 > 0, for some fixed δ 1 . The function L d m k (B m k (X))(T ) is well defined, holomorphic and bounded in any sector where π k < θ < π k + 2δ and 0 < R < δ 1 /K. This function is called the m k −sum of the formal seriesX(T ) in the direction d.
We now state some elementary properties concerning the m k −sums of formal power series.
1) The function L d m k (B m k (X))(T ) has the formal seriesX(T ) as Gevrey asymptotic expansion of order 1/k with respect to t on S d,θ,R 1/k . This means that for all π k < θ 1 < θ, there exist C, M > 0 such that is the unique holomorphic function that satisfies the estimates (60) on the sectors S d,θ 1 ,R 1/k with large aperture θ 1 > π k . 2) Let us assume that (E, ||.|| E ) also has the structure of a Banach algebra for a product ⋆. Let ] be m k −summable formal power series in direction d. Let q 1 ≥ q 2 ≥ 1 be integers. We assume thatX 1 (T ) +X 2 (T ),X 1 (T ) ⋆X 2 (T ) and T q 1 ∂ q 2 TX 1 (T ), which are elements of T E[[T ]], are m k −summable in direction d. Then, the following equalities hold for all T ∈ S d,θ,R 1/k . These equalities are consequence of the unicity of the function having a given Gevrey expansion of order 1/k in large sectors as stated above in 1) and from the fact that the set of holomorphic functions having Gevrey asymptotic expansion of order 1/k on a sector with values in the Banach algebra E form a differential algebra (meaning that this set is stable with respect to the sum and product of functions and derivation in the variable T ) (see Theorems 18,19 and 20 in [3]).
In the next proposition, we give some identities for the m k −Borel transform that will be useful in the sequel.
Proposition 8 Letf (t) = n≥1 f n t n ,ĝ(t) = n≥1 g n t n be formal series whose coefficients f n , g n belong to some Banach space (E, ||.|| E ). We assume that (E, ||.|| E ) is a Banach algebra for some product ⋆. Let k, m ≥ 1 be integers. The following formal identities hold.
In the following proposition, we recall some properties of the inverse Fourier transform for all x ∈ R. The function F −1 (f ) extends to an analytic function on the strip , the convolution product of f and g, for all m ∈ R. From Proposition 4, we know that ψ ∈ E (β,µ) . Moreover, we have 4 Formal and analytic solutions of convolution initial value problems with complex parameters 4.1 Formal solutions of the main convolution initial value problem for all m ∈ R, all 0 ≤ l ≤ D − 1. We consider sequences of functions m → C 0,n (m, ǫ), for all n ≥ 0 and m → F n (m, ǫ), for all n ≥ 1, that belong to the Banach space E (β,µ) for some β > 0 and µ > max(deg(Q 1 ) + 1, deg(Q 2 ) + 1) and which depend holomorphically on ǫ ∈ D(0, ǫ 0 ) for some ǫ 0 > 0. We assume that there exist constants K 0 , T 0 > 0 such that is a formal series with coefficients in E (β,µ) . Let c 1,2 (ǫ), c 0 (ǫ), c 0,0 (ǫ) and c F (ǫ) be bounded holomorphic functions on D(0, ǫ 0 ) which vanish at the origin ǫ = 0. We consider the following initial value problem for given initial data U (0, m, ǫ) ≡ 0.
Proof From Proposition 4 and the conditions stated above, we get that the coefficients U n (m, ǫ) ofÛ (T, m, ǫ) are well defined, belong to E (β,µ) for all ǫ ∈ D(0, ǫ 0 ), all n ≥ 1 and satisfy the following recursion relation Analytic solutions for an auxiliary convolution problem resulting from a m k 1 −Borel transform applied to the main convolution initial value problem We make the additional assumption that Using the formula (8.7) from [31], p. 3630, we can expand the operators Multiplying the equation (72) by T k 1 +1 and using (75), we can rewrite the equation (72) in the form Using (71) we get that for any κ ≥ k 1 , the function ϕ k 1 (τ, m, ǫ) belongs to F d (ν,β,µ,k 1 ,κ) for all ǫ ∈ D(0, ǫ 0 ), any unbounded sector U d centered at 0 with bisecting direction d ∈ R, for some ν > 0. Indeed, we have that ) By using the classical estimates for any real numbers m 1 ≥ 0, m 2 > 0 and Stirling formula Γ(n/k 1 ) for all n ≥ 1. Therefore, if the inequality A 2 < T 0 holds, we get the estimates On the other hand, we make the assumption that ψ k 1 (τ, m, ǫ) ∈ F d (ν,β,µ,k 1 ,k 1 ) , for all ǫ ∈ D(0, ǫ 0 ), for some unbounded sector U d with bisecting direction d ∈ R, where ν is chosen above. We will make the convention to denote ψ d k 1 the analytic continuation of the convergent power series ψ k 1 on the domain U d ∪ D(0, ρ). In particular, we get that ψ d k 1 (τ, m, ǫ) ∈ F d (ν,β,µ,k 1 ,κ) for any κ ≥ k 1 . We also assume that there exists a constant ζ ψ k 1 > 0 such that for all ǫ ∈ D(0, ǫ 0 ). In particular, we notice that for any κ ≥ k 1 . We require that there exists a constant r Q,R l > 0 such that Using the computation rules for the formal m k 1 −Borel transform in Proposition 8, we deduce the following equation satisfied by ω k 1 (τ, m, ǫ), In the next proposition, we give sufficient conditions under which the equation (85) has a solution where β, µ are defined above and for well chosen κ > k 1 .

Proposition 11
Under the assumption that where β, µ > 0 are defined in Proposition 10 which verifies ||ω d Proof We start the proof with a lemma which provides appropriate conditions in order to apply a fixed point theorem.
ds s satisfies the next properties.

Formal and analytic acceleration operators
In this section, we give a definition of the formal and analytic acceleration operator which is a slightly modified version of the one given in [1], Chapter 5, adapted to our definitions of m k −Laplace and m k −Borel transforms. First we give a definition for the formal transform.
] be a formal series. We define the formal acceleration operator with indices mk, m k bŷ Notice that if one defines the formal m k −Laplace transformL m k (f ) and the formal mk−Borel then the formal acceleration operatorÂ mk,m k can also be defined aŝ In the next definition, we define the analytic transforms.
Proposition 12 Letk > k > 0 be real numbers. Let S(d, π k + δ, ρ) be a bounded sector of radius ρ with aperture π k + δ, for some δ > 0 and with direction d. Let F : S(d, π k + δ, ρ) → C be a bounded analytic function such that there exist a formal seriesF (z) = n≥1 F n z n ∈ C[[z]] and two constants C, K > 0 with where γk is the closed Hankel path starting from the origin along the segment following the arc of circle 2 ) ] and going back to the origin along , 0] where 0 < δ ′ < δ that can be chosen as close to δ as needed. Then, the function (B d mk F )(Z) is analytic on the unbounded sector S(d, δ ′′ ) with direction d and aperture δ ′′ where 0 < δ ′′ < δ ′ which can be chosen as close to δ ′ as needed. Moreover, if (B mkF )(Z) = n≥1 F n Z n /Γ(n/k) denotes the formal mk−Borel transform ofF , then for any given ρ ′ > 0, there exists two constants C, K > 0 with where κ is defined as 1/κ = 1/k − 1/k. Finally, the mk−Borel operator B d mk is the right inverse operator of the mk−Laplace transform, namely we have that for all T ∈ S(d, π k + δ ′ , ρ/2).
Proof The proof follows the same lines of arguments as Theorem 2, Section 2.3 in [1]. Namely, one can check that if F (z) = z n , for an integer n ≥ 0, then for all Z ∈ S(d, δ ′′ ) by using the change of variable u = z/w 1/k in the integral (110) and a path deformation, bearing in mind the Hankel formula where γ is the path of integration from infinity along the the ray arg(w) = −π to the unit disc, then around the circle and back to infinity along the ray arg(w) = π. From the asymptotic expansion (109) and using the same integrals estimates as in Theorem 2, Section 2.3 in [1], together with the Stirling formula, for any given ρ ′ > 0, we get two constantsČ,Ǩ > 0 such that . Therefore (111) follows.
In the last part of the proof, we show the identity (112). We follow the same lines of arguments as Theorem 3 in Section 2.4 from [1]. Using Fubini's theorem, we can write Therefore, by direct integration, we deduce that u Tk Tk−uk has in the interior of γk exactly one singularity at u = T (since T is assumed to belong to S(d, π k + δ ′ , ρ/2)), this being a pole of order one, with residue −F (T )/k. The residue theorem completes the proof of (112). ✷ Proposition 13 Let S(d, α) be an unbounded sector with direction d ∈ R and aperture α. Let k > k > 0 be given real numbers and let κ > 0 be the real number defined by 1/κ = 1/k − 1/k. For all 0 < δ ′ < π/κ (which can be chosen close to π/κ), we define the kernel function where V d,k,δ ′ is the path starting from 0 along the halfline R + e i(d+ π 2k + δ ′ 2 ) and back to the origin . The function G(ξ, h) is well defined and satisfies the following estimates : there exist c 1 , c 2 > 0 such that for all h ∈ L d = R + e id and all ξ ∈ S(d, δ ′′ ) for 0 < δ ′′ < δ ′ (that can be chosen close to δ ′ ).
Proof We first show the estimates (116). We follow the idea of proof of Lemma 1, Section 5.1 in [1]. We make the change of variable u = hũ in the integral G(ξ, h) and we deform the path of integration in order to get the expression where γk is the closed Hankel path defined in Proposition 12 with the direction d = 0. Hence, we recognize that G(ξ, h) can be written as an analytic Borel transform G(ξ, h) = k(B 0 mk e k )(ξ/h) where e k (u) = e −(1/u) k . From Exercise 1 in Section 2.2 from [1], we know that e k (u) has0 as formal power series expansion of Gevrey order k on any sector S 0, π k +δ with direction 0 for any 0 < δ < π/κ. From Proposition 12, we deduce that (B 0 mk e k )(Z) has0 as formal series expansion of Gevrey order κ on any unbounded sector S 0,δ ′′ where 0 < δ ′′ < δ ′ < δ < π/κ (where δ ′′ can be chosen close to π/κ. Finally, using Exercice 3 in Section 2.2 from [1], we get two constants c 1 , c 2 > 0 such that |(B 0 mk e k )(Z)| ≤ c 1 e −c 2 |Z| −κ for all Z ∈ S 0,δ ′′ . The estimates (116) follow.
In order to show the asymptotic expansion with bound estimates (117), we first check that if f (h) = h n , for an integer n ≥ 0, then on S d,κ,δ,ρ . Indeed using Fubini's theorem, we can write From the definition of the Gamma function we know that and bearing in mind (113), after a path deformation, we recognize that Since the Taylor expansion of f at h = 0 is convergent, there exist two constants C f , K f > 0 such that Taking the expansion (119) and the exponential growth estimates (116), using the same integrals estimates as in Exercice 3 in Section 2.1 of [1], we get two constants C, K > 0 such that for all 1 ≤ l ≤ D − 1. In order the ensure the positivity of the integers d 2 l,k 2 , we impose the following assumption on the integers d 1 l,k 1 , for all 1 ≤ l ≤ D − 1. Indeed, by Definition of d 1 l,k 1 in (76), the constraint (120) rewrites Using the formula (8.7) from [31], p. 3630, we can expand the operators T δ l (k 2 +1) ∂ δ l T in the form where A δ l ,p , p = 1, . . . , δ l − 1 are real numbers, for all 1 ≤ l ≤ D.

Lemma 4 The function
is analytic on an unbounded sector S d,κ,δ with aperture π κ + δ in direction d, for any 0 < δ < ap(U d ) where ap(U d ) denotes the aperture of the sector U d , and has estimates of the form : there exist constants C ψ k 2 > 0 and ν ′ > 0 such that for any unbounded sector S d and bounded sector S b d with aperture π κ + δ, with δ as above, and we carry a constant ζ ψ k 2 > 0 with for all ǫ ∈ D(0, ǫ 0 ).
From the assumption that the function ψ d k 1 (τ, m, ǫ) belongs to the space F d (ν,β,µ,k 1 ,k 1 ) , see (82), we know that the m k 1 −Laplace transform defines a holomorphic and bounded function (by a constant that does not depend on ǫ ∈ D(0, ǫ 0 )) on a sector S d,θ,σ ′ in direction d, with radius σ ′ and aperture θ which satisfies π k 2 + π κ < θ < π k 2 + π κ + ap(U d ), where ap(U d ) is the aperture of U d , for some σ ′ > 0. Hence, by using a path deformation and the Fubini theorem, we can rewrite the function ψ d k 2 (τ, m, ǫ) in the form where V d,k 2 ,δ ′ ,σ ′ /2 is the closed Hankel path starting from the origin along the segment following the arc of circle [(σ ′ /2)e i(d+ π 2k 2 2 ) ] and going back to the origin , 0], where 0 < δ ′ < π κ + ap(U d ) that can be chosen close to π κ + ap(U d ). Therefore, from Proposition 12, we know that τ → ψ d k 2 (τ, m, ǫ) defines a holomorphic function on the unbounded sector S(d, δ ′′ ) where 0 < δ ′′ < δ ′ , which can be chosen close to δ ′ , for all m ∈ R, all ǫ ∈ D(0, ǫ 0 ). Now, we turn to the estimates (129). From the representation (131), we get the following estimates : there exist constants E 1 , E 2 , E 3 > 0 such that for all τ ∈ S(d, δ ′′ ), all m ∈ R, all ǫ ∈ D(0, ǫ 0 ). Besides, from the asymptotic expansion (111), we get in particular the existence of a constant E 0 > 0 such that for all τ ∈ S(d, δ ′′ ) ∩ D(0, ρ ′ ) and some ρ ′ > 0. Finally, combining the estimates (132) and (133) yields (129). ✷ We consider now the following problem ds s for vanishing initial data ω k 2 (0, m, ǫ) ≡ 0, where ψ d k 2 (τ, m, ǫ) has been constructed in Lemma 4. We make the additional assumption that there exists an unbounded sector We choose an unbounded sector S d centered at 0, a small closed discD(0, ρ) and we prescribe the sector S Q,R D in such a way that the following conditions hold.
1) There exists a constant M 1 > 0 such that for all 0 ≤ l ≤ (δ D − 1)k 2 − 1, all m ∈ R, all τ ∈ S d ∪D(0, ρ). Indeed, from (135) and the explicit expression (137) of q l (m), we first observe that |q l (m)| > 2ρ for every m ∈ R, all 0 ≤ l ≤ (δ D − 1)k 2 − 1 for an appropriate choice of r Q,R D and of ρ > 0. We also see that for all m ∈ R, all 0 ≤ l ≤ (δ D − 1)k 2 − 1, the roots q l (m) remain in a union U of unbounded sectors centered at 0 that do not cover a full neighborhood of the origin in C * provided that η Q,R D is small enough. Therefore, one can choose an adequate sector S d such that S d ∩ U = ∅ with the property that for all 0 ≤ l ≤ (δ D − 1)k 2 − 1 the quotients q l (m)/τ lay outside some small disc centered at 1 in C for all τ ∈ S d , all m ∈ R. This yields (138) for some small constant M 1 > 0.
2) There exists a constant M 2 > 0 such that Indeed, for the sector S d and the discD(0, ρ) chosen as above in 1), we notice that for any fixed 0 ≤ l 0 ≤ (δ D − 1)k 2 − 1, the quotient τ /q l 0 (m) stays outside a small disc centered at 1 in C for all τ ∈ S d ∪D(0, ρ), all m ∈ R. Hence (139) must hold for some small constant M 2 > 0.
By construction of the roots (137) in the factorization (136) and using the lower bound estimates (138), (139), we get a constant C P > 0 such that In the next proposition, we give sufficient conditions under which the equation (134) has a solution ω d k 2 (τ, m, ǫ) in the Banach space F d (ν ′ ,β,µ,k 2 ) where ν ′ , β, µ are defined above.
Proof We start the proof with a lemma which provides appropriate conditions in order to apply a fixed point theorem.

Remark:
The analytic functions F d (T, m, ǫ) (resp. U d (T, m, ǫ)) can be called the (m k 2 , m k 1 )−sums in the direction d of the formal series F (T, m, ǫ) (resp. U (T, m, ǫ)) introduced in the Section 4.1, following the terminology of [1], Section 6.1.
In the next proposition, we construct analytic solutions to the problem (72) with analytic forcing term and for vanishing initial data.
Proof Since the function ω d k 2 (u, m, ǫ) solves the integral equation (134), one can check by direct computations similar to those described in Proposition 8, using the integral representations (166) that U d (T, m, ǫ) solves the equation (123) where the formal series F (T, m, ǫ) is replaced by F d (T, m, ǫ) and hence solves the equation (72) where F d (T, m, ǫ) must be put in place of F (T, m, ǫ). ✷

Analytic solutions of a nonlinear initial value Cauchy problem with analytic forcing term on sectors and with complex parameter
Let k 1 , k 2 ≥ 1, D ≥ 2 be integers such that k 2 > k 1 . Let δ l ≥ 1 be integers such that We require that there exists a constant r Q,R l > 0 such that for all m ∈ R, all 1 ≤ l ≤ D. We make the additional assumption that there exists an unbounded sector with direction d Q,R D ∈ R, aperture η Q,R D > 0 for the radius r Q,R D > 0 given above, such that Definition 7 Let ς ≥ 2 be an integer. For all 0 ≤ p ≤ ς −1, we consider open sectors E p centered at 0, with radius ǫ 0 and opening π k 2 + κ p , with κ p > 0 small enough such that E p ∩ E p+1 = ∅, for all 0 ≤ p ≤ ς − 1 (with the convention that E ς = E 0 ). Moreover, we assume that the intersection of any three different elements in {E p } 0≤p≤ς−1 is empty and that ∪ ς−1 p=0 E p = U \ {0}, where U is some neighborhood of 0 in C. Such a set of sectors {E p } 0≤p≤ς−1 is called a good covering in C * .
Definition 8 Let {E p } 0≤p≤ς−1 be a good covering in C * . Let T be an open bounded sector centered at 0 with radius r T and consider a family of open sectors S dp,θ,ǫ 0 r T = {T ∈ C * /|T | < ǫ 0 r T , |d p − arg(T )| < θ/2} with aperture θ > π/k 2 and where d p ∈ R, for all 0 ≤ p ≤ ς − 1, are directions which satisfy the following constraints: Let q l (m) be the roots of the polynomials (136) defined by (137) and S dp , 0 ≤ p ≤ ς − 1 be unbounded sectors centered at 0 with directions d p and with small aperture. Let ρ > 0 be a positive real number. We assume that 1) There exists a constant M 1 > 0 such that 2) There exists a constant M 2 > 0 such that 3) There exist a family of unbounded sectors U dp with bisecting direction d p and bounded sectors S b dp with bisecting direction d p , with radius less than ρ, with aperture π κ + δ p , with 0 < δ p < ap(U dp ), for all 0 ≤ p ≤ ς − 1, with the property that S b dp ∩ S b d p+1 = ∅ for all 0 ≤ p ≤ ς − 1 (with the convention that d ς = d 0 ). 4) For all 0 ≤ p ≤ ς − 1, for all t ∈ T , all ǫ ∈ E p , we have that ǫt ∈ S dp,θ,ǫ 0 r T .
We define the forcing term f dp (t, z, ǫ) as (180) f dp (t, z, ǫ) := F −1 (m → F dp (ǫt, m, ǫ))(z) By construction, f dp (t, z, ǫ) represents a bounded holomorphic function on T ×H β ′ ×E p (provided that the radius r T of T satisfies the inequality ǫ 0 r T ≤ h ′ which will be assumed in the sequel).
In the next first main result, we construct a family of actual holomorphic solutions to the equation (175) for given initial data at t = 0 being identically equal to zero, defined on the sectors E p with respect to the complex parameter ǫ. We can also control the difference between any two neighboring solutions on the intersection of sectors E p ∩ E p+1 .
Using the estimates (187), we get that the function (T, z) → U dp (T, z, ǫ) = F −1 (m → U dp (T, m, ǫ))(z) defines a bounded holomorphic function on S dp,θ,h ′ ×H β ′ , for all ǫ ∈ D(0, ǫ 0 ) and any 0 < β ′ < β. For all 0 ≤ p ≤ ς − 1, we define u dp (t, z, ǫ) = U dp (ǫt, z, ǫ) = Taking into account the construction provided in 4) from Definition 8, the function u dp (t, z, ǫ) defines a bounded holomorphic function on the domain T × H β ′ × E p . Moreover, we have u dp (0, z, ǫ) ≡ 0 and using the properties of the Fourier inverse transform from Proposition 9, we deduce that u dp (t, z, ǫ) solves the main equation (175) on T × H β ′ × E p . Now, we proceed to the proof of the estimates (183). We detail only the arguments for the functions u dp since the estimates for the forcing terms f dp follow the same line of discourse as below with the help of the estimates (178) instead of (187).
In the last part of the proof, we show the estimates (184). Again, we only describe the arguments for the functions u dp since exactly the same analysis can be made for the forcing term f dp using the estimates (177) and (178) instead of (187) and (189).

There exist
We make the change of variable r k 2 = s in the integral (204) and we get We put ψ A,p (s) = exp(− ] as asymptotic expansion of Gevrey order k 2 κ + 1 = k 2 k 1 on some segment [0, δ ′ ] with 0 < δ ′ < (ρ ν,κ /2) k 2 . Hence, using again Lemma 9, we get two constants C ′ , M ′ > 0 with We deduce the existence of two constants for all ǫ ∈ E p ∩ E p+1 , all t ∈ T ∩ D(0, h A,p ), for some h A,p > 0. Gathering the last inequality (205) and (203) yields In conclusion, taking into account the above inequalities (202) and (206), we deduce from the decomposition (201) that for all t ∈ T with |t| < ( δ 1 δ 2 +ν ′ ǫ k 2 0 ) 1/k 2 and |t| ≤ h A,p for some constants δ 1 , δ 2 , h A,p > 0, |Im(z)| ≤ β ′ , for all ǫ ∈ E p ∩ E p+1 . Therefore the inequality (184) holds. ✷ 6 Existence of formal series solutions in the complex parameter and asymptotic expansion in two levels 6.1 Summable and multisummable formal series and a Ramis-Sibuya theorem with two levels In the next definitions we recall the meaning of Gevrey asymptotic expansions for holomorphic functions and k−summability. We also give the signification of (k 2 , k 1 )−summability for power series in a Banach space, as described in [1].

Definition 9
Let (E, ||.|| E ) be a complex Banach space and let E be a bounded open sector centered at 0. Let k > 0 be a positive real number. We say that a holomorphic function f : E → E admits a formal power seriesf (ǫ) = n≥0 a n ǫ n ∈ E[[ǫ]] as its asymptotic expansion of Gevrey order 1/k if, for any closed proper subsector W ⊂ E centered at 0, there exist C, M > 0 with If moreover the aperture of E is larger than π k + δ for some δ > 0, then the function f is the unique holomorphic function on E satisfying (207). In that case, we say thatf is k−summable on E and that f defines its k−sum on E. In addition, the function f can be reconstructed from the analytic continuation of the k 1 −Borel transform on an unbounded sector and by applying a k 1 −Laplace transform to it, see Section 3.2 from [1].
Definition 10 Let (E, ||.|| E ) be a complex Banach space and let 0 < k 1 < k 2 be two positive real numbers. Let E be a bounded open sector centered at 0 with aperture π k 2 + δ 2 for some δ 2 > 0 and let F be a bounded open sector centered at 0 with aperture π k 1 + δ 1 for some δ 1 > 0 such that the inclusion E ⊂ F holds.
A formal power seriesf (ǫ) = n≥0 a n ǫ n ∈ E[[ǫ]] is said to be ( In that case, the function f (ǫ) can be reconstructed from the analytic continuation of the k 1 −Borel transform off by applying successively some acceleration operator and Laplace transform of order k 2 , see Section 6.1 from [1].
In this section, we state a version of the classical Ramis-Sibuya theorem (see [13], Theorem XI-2-3) with two different Gevrey levels which describes also the case when multisummability holds on some sector. We mention that a similar multi-level version of the Ramis-Sibuya theorem has already been stated in the manuscript [32] and also in a former work of the authors, see [15].
Theorem (RS) Let 0 < k 1 < k 2 be positive real numbers. Let (E, ||.|| E ) be a Banach space over C and {E i } 0≤i≤ν−1 be a good covering in C * , see Definition 7. For all 0 ≤ i ≤ ν − 1, let G i be a holomorphic function from E i into the Banach space (E, ||.|| E ) and let the cocycle ∆ i (ǫ) = G i+1 (ǫ) − G i (ǫ) be a holomorphic function from the sector Z i = E i+1 ∩ E i into E (with the convention that E ν = E 0 and G ν = G 0 ). We make the following assumptions.
1) The functions G i (ǫ) are bounded as ǫ ∈ E i tends to the origin in C, for all 0 ≤ i ≤ ν − 1.
2) For some finite subset I 1 ⊂ {0, . . . ν−1} and for all i ∈ I 1 , the functions ∆ i (ǫ) are exponentially flat on Z i of order k 1 , for all 0 ≤ i ≤ ν − 1. This means that there exist constants K i , M i > 0 such that 3) For all i ∈ I 2 = {0, . . . , ν − 1} \ I 1 , the functions ∆ i (ǫ) are exponentially flat of order k 2 on Z i , for all 0 ≤ i ≤ ν − 1. This means that there exist constants K i , M i > 0 such that Then, there exist a convergent power series a(ǫ) ∈ E{ǫ} near ǫ = 0 and two formal serieŝ where G 1 i (ǫ) is holomorphic on E i and hasĜ 1 (ǫ) as asymptotic expansion of Gevrey order 1/k 1 on E i , G 2 i (ǫ) is holomorphic on E i and carriesĜ 2 (ǫ) as asymptotic expansion of Gevrey order 1/k 2 on E i , for all 0 ≤ i ≤ ν − 1.
Proof We consider two holomorphic cocycles ∆ 1 i (ǫ) and ∆ 2 i (ǫ) defined on the sectors Z i in the following way: We need the following lemma.
Proof We consider the family of functions u dp (t, z, ǫ), 0 ≤ p ≤ ς − 1 constructed in Theorem 1. For all 0 ≤ p ≤ ς − 1, we define G p (ǫ) := (t, z) → u dp (t, z, ǫ), which is by construction a holomorphic and bounded function from E p into the Banach space F of bounded holomorphic functions on (T ∩ D(0, h ′′ )) × H β ′ equipped with the supremum norm, where T is introduced in Definition 8, h ′′ > 0 is set in Theorem 1 and β ′ > 0 is the width of the strip H β ′ on which the coefficient c 0 (t, z, ǫ) and the forcing term f dp (t, z, ǫ) are defined with respect to z, see (176) and (180).
From the Theorem (RS) stated above in Section 6.1, we deduce the existence of a convergent power series a(ǫ) ∈ F{ǫ} and two formal seriesĜ 1 (ǫ),Ĝ 2 (ǫ) ∈ F[[ǫ]] such that G p (ǫ) owns the following decomposition where G 1 p (ǫ) is holomorphic on E p and hasĜ 1 (ǫ) as its asymptotic expansion of Gevrey order 1/k 1 on E p , G 2 p (ǫ) is holomorphic on E p and carriesĜ 2 (ǫ) as its asymptotic expansion of Gevrey order 1/k 2 on E p , for all 0 ≤ p ≤ ν − 1. We set u(t, z, ǫ) = m≥0 h m (t, z)ǫ m /m! := a(ǫ) +Ĝ 1 (ǫ) +Ĝ 2 (ǫ). the functions c 1,2 (ǫ), c 0 (t, z, ǫ) and c F (ǫ) are analytic w.r.t ǫ at 0, we know that On other hand, one can check by direct inspection from the recursion (219) and the expansions (220) that the seriesû(t, z, ǫ) = m≥0 h m (t, z)ǫ m /m! formally solves the equation (216). ✷ 7 Application. Construction of analytic and formal solutions in a complex parameter of a nonlinear initial value Cauchy problem with analytic coefficients and forcing term near the origin in C 3 In this section, we give sufficient conditions on the forcing term F (T, m, ǫ) for the functions u dp (t, z, ǫ) and its corresponding formal power series expansionû(t, z, ǫ) w.r.t ǫ constructed in Theorem 1 and Theorem 2 to solve a nonlinear problem with holomorphic coefficients and forcing term near the origin given by (264).

A linear convolution initial value problem satisfied by the formal forcing term F (T, m, ǫ)
Let k 1 ≥ 1 be the integer defined above in Section 5 and let D ≥ 2 be an integer. For 1 ≤ l ≤ D, let d l ,δ l ,∆ l ≥ 0 be nonnegative integers. We assume that for all 1 ≤ l ≤ D − 1. We make also the assumption that for all m ∈ R, all 0 ≤ l ≤ D − 1. Let β, µ > 0 be the integers defined above in Section 5. We consider sequences of functions m → C 0,n (m, ǫ), for all n ≥ 0 and m → F n (m, ǫ), for all n ≥ 1, that belong to the Banach space E (β,µ) and which depend holomorphically on ǫ ∈ D(0, ǫ 0 ). We assume that there exist constants K 0 , T 0 > 0 such that for all n ≥ 1, for all ǫ ∈ D(0, ǫ 0 ). We define C 0 (T, m, ǫ) = n≥1 C 0,n (m, ǫ)T n , F(T, m, ǫ) = n≥1 F n (m, ǫ)T n which are convergent series on D(0, T 0 /2) with values in E (β,µ) . Let c 0 (ǫ), c 0,0 (ǫ) and c F (ǫ) be bounded holomorphic functions on D(0, ǫ 0 ) which vanish at the origin ǫ = 0. We make the assumption that the formal series F (T, m, ǫ) = n≥1 F n (m, ǫ)T n , where the coefficients F n (m, ǫ) are defined after the problem (175) in Section 5 satisfies the next linear initial value problem for given initial data F (0, m, ǫ) = 0.
The existence and uniqueness of the formal power series solution of (225) is ensured by the following Proposition 17 There exists a unique formal series F (T, m, ǫ) = n≥1 F n (m, ǫ)T n solution of (225) with initial data F (0, m, ǫ) ≡ 0, where the coefficients m → F n (m, ǫ) belong to E (β,µ) for β, µ > 0 given above and depend holomorphically on ǫ in D(0, ǫ 0 ).
Proof From Proposition 4, we get that the coefficients F n (m, ǫ) of F (T, m, ǫ) are well defined, belong to E (β,µ) for all ǫ ∈ D(0, ǫ 0 ), all n ≥ 1 and satisfy the following recursion relation (226) (n + 1)F n+1 (m, ǫ) for all n ≥ max 1≤l≤D d l . ✷ 7.2 Analytic solutions for an auxiliary linear convolution problem resulting from a m k 1 −Borel transform applied to the linear initial value convolution problem Using the formula (8.7) from [31], p. 3630, we can expand the operators T δ l (k 1 +1) ∂ δ l T in the form where A δ l ,p , p = 1, . . . , δ l − 1 are real numbers, for all 1 ≤ l ≤ D. We define integers d l,k 1 ≥ 0 to satisfy Multiplying the equation (225) by T k 1 +1 and using (227), (228) we can rewrite the equation (225) in the form As above, we denote ψ k 1 (τ, m, ǫ) the formal m k 1 −Borel transform of F (T, m, ǫ) w.r.t T and ϕ k 1 (τ, m, ǫ) the formal m k 1 −Borel transform of C 0 (T, m, ǫ) with respect to T and ψ k 1 (τ, m, ǫ) the formal m k 1 −Borel transform of F(T, m, ǫ) w.r.t T , Using (224) we get that ϕ k 1 (τ, m, ǫ) ∈ F dp (ν,β,µ,k 1 ,k 1 ) and ψ k 1 (τ, m, ǫ) ∈ F dp (ν,β,µ,k 1 ,k 1 ) , for all ǫ ∈ D(0, ǫ 0 ), for all the unbounded sectors U dp centered at 0 and bisecting direction d p ∈ R introduced in Definition 8, for some ν > 0. Indeed, we have that ), ) By using the classical estimates (79) and the Stirling formula Γ(n/k 1 ) ∼ (2π) 1/2 (n/k 1 ) n k 1 − 1 2 e −n/k 1 as n tends to +∞, we get two constants A 1 , A 2 > 0 depending on ν, k 1 such that for all n ≥ 1, all ǫ ∈ D(0, ǫ 0 ). Therefore if A 2 < T 0 holds, we get the estimates Observe that d D,k 1 = 0. Using the computation rules for the formal m k 1 −Borel transform in Proposition 8, we deduce the following equation satisfied by ψ k 1 (τ, m, ǫ), We make the additional assumption that there exists an unbounded sector We choose the family of unbounded sectors U dp centered at 0, a small closed discD(0, ρ) (introduced in Definition 8) and we prescribe the sector S Q,R D in such a way that the following conditions hold.
2) There exists a constant M 2 > 0 such that By construction of the roots (236) in the factorization (235) and using the lower bound estimates (237), (238), we get a constant C P > 0 such that Proof The proof will follow the same lines of arguments as in Proposition 14. We give a thorough treatment of it only for the sake of completeness. We begin with a lemma which provides appropriate conditions in order to apply a fixed point theorem.