Multisummability of formal solutions for a family of generalized singularly perturbed moment differential equations

The main aim of this work is to give a step forward in the theory of summability of formal solutions to functional equations in the complex domain. More precisely, we deal with the so-called moment (partial) differential equations, in which the operators known as moment derivatives act on the unknown function.

The main advances in the present study are twofold: on the one hand, we provide an integral representation of the moment derivatives of generalized sums of formal power series which can be extended to an infinite sector satisfying some generalized exponential growth. In addition to this, we describe the dependence of such moment derivatives with respect to three elements, namely the moment sequence, the sequence involved in the asymptotic representation of the sum, and also the generalized exponential growth at infinity (Theorem 2). Consequently, a novel definition of moment derivatives acting on generalized multisums of formal power series makes sense, and induces many possible applications in the theory, as the following one. On the other hand, we apply the previous result to achieve the main result of this research, namely the multisummability of the formal solutions to certain family of singularly perturbed moment differential equations (Theorem 3). More precisely, we prove that the formal solution to

is multisummable along a certain appropriate multidirection $(d_{1},d_{2})\in\mathbb{R}^{2}$ with respect to the perturbation parameter $\varepsilon$ if and only if the forcing term $\hat{f}(z,\varepsilon)\in\mathbb{C}[[z,\varepsilon]]$ and the initial conditions $\hat{\psi}_{j}(\varepsilon)\in\mathbb{C}[[\varepsilon]]$, for $0\leq j\leq p-1$ are multisummable along the same multidirection. Here, $\varepsilon$ stands for a small complex parameter, $a(z)$ is a holomorphic function near the origin, $m_{2}$ is a sequence of moments, and $k,p$ are positive integers with $k<p$. The precise description of the elements involved in the problem is given in Section 5.

The study of moment differential equations is motivated by the versatility of the moment derivative operator, whose definition was initially put forward by W. Balser and M. Yoshino in [2] for formal power series. Let $m=(m(p))_{p\geq 0}$ be a sequence of positive real numbers. Then, the moment derivative $\partial_{m,z}:\mathbb{C}[[z]]\to\mathbb{C}[[z]]$ is defined by $\partial_{m,z}(z^{p})=\frac{m(p)}{m(p-1)}z^{p-1}$ for every positive integer $p$ and $\partial_{m,z}(1)=0$, defining moment derivatives for elements in $\mathbb{C}[[z]]$ by linearity. It is natural to extend the previous definition to a holomorphic function on some disc at the origin by identifying the function with its Taylor series at $z=0$. In addition to this, the authors proved that one can extend the definition of moment differentiation to the generalized sums in a direction of a formal power series, as the sum of the moment derivative of that formal power series (see Corollary 1 and Definition 10, [13]). In this work, we further extend it to the generalized multisums of a formal power series (Corollary 1 and Definition 10).

The most suitable choice for $m$ is to be a moment sequence of certain Laplace-like operator. It is clear that the sequence $m_{1}=(p!)_{p\geq 0}$ gives rise to the usual derivative when considering the moment operator $\partial_{m_{1},z}$. Apart from that choice for $m$, many other derivations which appear in concrete applications can be represented as a moment derivative. For example, for every $k>0$, the sequence $m_{1/k}=(\Gamma(1+p/k))_{p\geq 0}$ is associated with the Caputo fractional derivative $\partial_{z}^{1/k}$ via the relation $(\partial_{m_{1/k},z}\hat{f})(z^{1/k})=\partial_{z}^{1/k}(\hat{f}(z^{1/k}))$, valid for every $\hat{f}(z)\in\mathbb{C}[[z]]$ (see [16], Remark 3, for further details). Recently, many applications of Caputo derivatives appear in the literature such as [3, 8], also in the study of the asymptotic periodic solutions of evolution equations [21], numerical studies, etc. Let $q\in(0,1)$. The sequence $m_{q}=([p]_{q}!)_{p\geq 0}$, with $[p]!_{q}=[1]_{q}[2]_{q}\cdots[p]_{q}$ is known as the sequence of $q-$factorials, where $[\ell]_{q}=\sum_{j=0}^{\ell-1}q^{j}$. It determines the moment differentiation which coincides with the $q-$derivative $D_{q,z}$ given by $D_{q,z}z^{p}=[p]_{q}z^{p-1}$ for every $p\in\mathbb{N}$. This moment differentiation is quite related to the dilation operator, appearing in the study of $q-$difference equations which is of great interest in the scientific community with interesting advances in the knowledge of the asymptotic behavior of the solutions of $q-$difference equations (see [9, 15] among others, and the references therein).

The interest of moment functional equations has increased in the last decade, and recent achievements have been reached in this concern. A first step was given in the seminal work [2], where the authors study the formal solutions and Gevrey estimates of their coefficients of linear moment partial differential equations with constant coefficients. The development of a more general theory through the construction of kernels for generalized summability by J. Sanz in [22] allows to enlarge the class of moment sequences considered, in the framework of strongly regular sequences. The sequence $m_{1/k}$ above belongs to such a family for every $k>0$, whereas $m_{q}$ does not. The so-called $1+$ level, appearing in the asymptotic study of difference equations is also related to a strongly regular sequence [4, 5].

After the seminal work [2], the second author gave answer to the problem of analyticity of such problems [16], via splitting of the characteristic equation with respect to one of its variables. We also refer to [17] for a further study in the homogeneous situation, while dropping the condition of the convergence of the initial data. Further knowledge on the solutions to moment partial differential equations with constant coefficients is given in [18] to study the growth properties and summability of the formal solutions. It is also worth mentioning the family of partial differential equations studied in [10], where the coefficients of the equation under study belong to certain functional spaces associated with functions whose derivatives are uniformly bounded in terms of some strongly regular sequence.

The last advances in the theory of the asymptotic behavior of solutions to moment functional equations have been obtained recently regarding the summability of certain families of moment integro-differential equations [12], and also Maillet-type theorems [11, 24]. We mention the recent works by P. Remy in the study of partial differential equations [19] and integro-differential equations [20] of a similar nature as those considered in these works.

Recent results on generalized multisummability of formal power series concerning different (comparable and nonequivalent) levels associated with ultraholomorphic classes achieved in [6] are applied in the present study to achieve asymptotic properties of the solutions to a singularly perturbed moment differential equation. That concept of multisummability as long as previous results achieved by the authors in [13] have been the key points used to describe generalized multisummability of the formal solution of the main equation (1). As mentioned above, novel integral representations and accurate estimates of the moment derivatives of generalized sums of formal power series are needed, arriving to the coherent definition of moment derivation of the generalized multisum of a formal power series. More precisely, the first main result of the present study, Theorem 2, resorts to an appropriate deformation path which is split in order to provide upper bounds for the moment derivative of a sectorial holomorphic function, quite related to the multisummability process. As a consequence, Corollary 1 and Definition 10 encompass the notion of moment derivation of the multisum of a formal power series along some multidirection. As an application of these results, Section 5 characterizes multisummability of the formal solution to (1) in terms of that of the initial data and the forcing term (Theorem 3). The proof of this last result is based on the properties of formal Borel transform which transform the problem into an auxiliary moment partial differential equation, which is easier to handle.

The paper is structured as follows. We fix notation in Section 2, followed by Section 3, where we recall the definition and main results on strongly regular sequences, generalized summability and multisummability of formal power series. In that section, some technical results needed in the sequel are also proved. The main purpose of Section 4 is to state Theorem 2, leading to a coherent definition of moment derivative of the generalized multisum of a formal power series. Section 5 describes an application (Theorem 3) of the previous results in the framework of singularly perturbed moment differential equations.

By $\mathbb{N}$ we shall denote the set of all positive integers, i.e., $\{1,2,\dots\}$ and $\mathbb{N}_{0}=\mathbb{N}\cup\{0\}$.

$\mathcal{R}$ stands for the Riemann surface of the logarithm.

For all $r>0$ and $z_{0}\in\mathbb{C}$, $D(z_{0},r)$ stands for the open disc in the complex plane centered at $z_{0}$ and with radius $r$. For any fixed $\theta>0$ and $d\in\mathbb{R}$ a subset of $\mathcal{R}$ defined as

is an open infinite sector with vertex at the origin, bisecting direction $d$ and opening $\theta$. In cases where the opening is not specified, we simply write $S_{d}$. For every $r>0$, we write $S_{d}(\theta;r):=S_{d}(\theta)\cap D(0,r)$. A sectorial region $G_{d}(\theta)$ is a subset of $\mathcal{R}$ such that there exists $r>0$ for which $G_{d}(\theta)\subset S_{d}(\theta;r)$ and for any $0<\theta^{\prime}<\theta$ there exists $0<r^{\prime}<r$ such that $S_{d}(\theta^{\prime};r^{\prime})\subset G_{d}(\theta)$.

We put $\hat{S}_{d}(\theta;r):=S_{d}(\theta)\cup D(0,r)$. Analogously, we write $\hat{S}_{d}(\theta)$ (resp. $\hat{S}_{d}$) whenever the radius $r>0$ (resp. the radius and the opening $r,\theta>0$) can be omitted. We write $S\prec S_{d}(\theta)$ if $S$ is an infinite sector with vertex at the origin such that $\bar{S}\subset S_{d}(\theta)$, where the closure is considered with respect to $\mathcal{R}$. Similarly, for two sectorial regions $G_{d}(\theta)$ and $G_{d^{\prime}}(\theta^{\prime})$ we write $G_{d}(\theta)\prec G_{d^{\prime}}(\theta^{\prime})$ whenever $G_{d}(\theta)\subset G_{d^{\prime}}(\theta^{\prime})$ and relation $\prec$ holds for the sectors appearing in the definitions of both sectorial regions.

If $(\mathbb{E},\|\cdot\|_{\mathbb{E}})$ is a complex Banach space, by $\mathcal{O}(U,\mathbb{E})$ we denote the set of all functions holomorphic on the open set $U\subset\mathbb{C}$ with values from $\mathbb{E}$. For $\mathbb{E}=\mathbb{C}$ we simply write $\mathcal{O}(U)$. The set of all formal power series in $t$ with coefficients in $\mathbb{E}$ is denoted by $\mathbb{E}[[z]]$.

The purpose of this section is to recall the main facts on the theory of generalized summability and also one of the equivalent notions of generalized multisummability of a formal power series along certain multidirection, developed in [6]. We first remind the main elements regarding the theory of strongly regular sequences and related properties together with the theory of generalized summability, for the sake of completeness. These definitions and the detailed constructions can be found in [23] and the references therein.

In this section, we focus our attention on the concept of a moment derivative and recall some properties associated with this notion. We also state some new results to be applied in the work.

The notion of a generalized derivative operator allows to consider functional problems under greater generality. More precisely, we deal with the following formal operator.

Usual derivatives are recovered when considering the moment sequence $(p!)_{p\geq 0}$, which is associated with the kernel function $e(z)=z\exp(-z)$. In addition to this, moment derivatives can be read in terms of Caputo $\alpha$-fractional derivatives $\partial_{z}^{\alpha}$ as follows. Let $\alpha$ be a positive rational number. The moment sequence $m_{\alpha}:=(\Gamma(1+\alpha p))_{p\geq 0}$ is associated with the kernel function $e(z)=\frac{1}{\alpha}z^{\frac{1}{\alpha}}\exp(-z^{\frac{1}{\alpha}})$ and the fractional derivative of order $\alpha$ is defined on formal power series in $z^{\alpha}$ by

Therefore, one has

for every $\hat{f}\in\mathbb{E}[[z]]$.

We may also observe that $q-$derivatives defined by

for some fixed $q\in(0,1)$ can be interpreted in terms of the moment derivatives associated with the moment sequence $([p]_{q}!)_{p\geq 0}$, with $[p]_{q}!=[1]_{q}[2]_{q}\cdots[p]_{q}$ and $[j]_{q}=\sum_{h=0}^{j-1}q^{h}$.

We also have the following result.

Moment differentiation can be naturally extended to holomorphic functions on some neighborhood of the origin by identifying the function with its Taylor series at the origin. However, this formal differentiation does not preserve convergence unless some regularity property is assumed for the sequence of moments. However, if one departs from a strongly regular sequence $\mathbb{M}$ which admits a nonzero proximate order, and considers a pair of kernel functions associated with it and then constructs the corresponding sequence of moments $m$, then it holds that $\mathbb{M}$ and $m$ generate the same ultraholomorphic space of functions (i.e., they are equivalent sequences) and $m$ is indeed a strongly regular sequence (see Remark 3.8, [10]). In [13], the definition of moment differentiation was also extended to the generalized sum along a direction of a formal power series as the generalized sum along that same direction of the formal moment derivative of the initial formal power series. For that purpose, the moment derivative was also provided for any function defined on some neighborhood of the origin with holomorphic extension to an infinite sector and with certain generalized exponential growth at infinity (as described in Definition 3). Indeed, the first part of Theorem 3 [13] reads as follows.

At this point, we give a step forward in order to define the moment derivatives on functions defined on some sectorial region, and which can be extended under certain generalized exponential growth to infinity.