This work is devoted to the study of a family of singularly perturbed $q-$difference-differential equations in the complex domain of the form

Here, the action of a small complex perturbation parameter $ϵ\in D(0,{ϵ}_0)$, for some ${ϵ}_0>0$, is studied when considering the unknown function $u=u(t,z,ϵ)$. The element Q stands for a polynomial with complex coefficients.

The operator ${\sigma }_{q;t}$ stands for the dilation operator acting on t variable, i.e. ${\sigma }_{q;t}h\left(t\right)=h\left(qt\right)$. Here, $q>1$ is a fixed positive number. The previous Definitionition naturally extends to ${\sigma }_{q;t}^{r}h\left(t\right)=h\left(q^{r}t\right)$ for $r\in {\mathbb{Q}}_{+}$.

The map $P(t,z,ϵ,{\omega }_1,{\omega }_2)$ depends holomorphically on the perturbation parameter $ϵ$, it is a polynomial in t variable and holomorphic on a horizontal strip in the complex plane with respect to z. This function is polynomial with respect to ${\omega }_1^{1/k}$ and ${\omega }_2$ variables. Indeed, k will be crucial in the determination of the $q-$Gevrey order relating the formal and the analytic solutions of (1). The forcing term $f(t,z,ϵ)$ is of logarithmic type in the sense that for some polynomials with respect to t with coefficients being holomorphic functions on some horizontal strip times $D(0,{ϵ}_0)$. The concrete form of the forcing term is determined in Assumption (B1) in Section 3. Finally, $H(v_0,z,ϵ,v_1,v_2)$ is a linear map with respect to $v_1$, $v_2$, polynomial of degree at most two in $v_0$ and holomorphic on a horizontal strip with respect to z variable and on $D(0,{ϵ}_0)$ regarding the perturbation parameter $ϵ$.

The concrete form of (1) is precised in (9), together with the assumptions needed for the main results of the work.

The appearance of the operator ${\gamma }_{ϵ}^{*}$ is crucial in the asymptotic behavior of the analytic solutions with respect to $ϵ$ at the origin. Indeed, the action of the so-called monodromy operator around the origin with respect to $ϵ$, ${\gamma }_{ϵ}^{*}$, described in Section 2.1, has been proved to modify the form of the analytic solutions of functional equations in the literature, such as [15,23]. In addition to this, the operator ${\gamma }_{ϵ}^{*}$ is used to construct the fundamental solutions to linear systems of differential equations showing an irregular singularity at a point (see Levelt-Turrittin Theoremrem in [18], and the reference [3]). The appearance of a monodromy matrix in the previous situation extends the one considered in the regular singular framework which can be obtained by analytically continuing a holomorphic solution Definitioned on a disc out of the origin and turn it counterclockwise around the origin (see [7]). In a more abstract setting, the monodromy operator in the study of Picard-Vessiot rings has been proved to be important from an algebraic point of view [18].

There exists another notion of monodromy used in the study of the meromorphic solutions of $q-$difference equations, introduced in [19], which is not considered in the present study.

As a matter of fact, the appearance of the $q-$difference operator ${\sigma }_{q;t}$ and the monodromy operator ${\gamma }_{ϵ}^{*}$ can be reinterpreted as difference-$q-$difference actions on t and $ϵ$ by the identification of ${\gamma }_{ϵ}^{*}$ with the shift mapping of angles $\theta \mapsto \theta +2\pi$ in polar coordinates in the writing $u(t,z,ϵ)=v(t,z,r,\theta )$, with $ϵ=r{e}^{i\theta }$ by

In this sense, we refer to [21] and the references therein. In that work, the authors study systems of linear differential and difference equations $Y^{\prime }\left(x\right)=A\left(x\right)Y\left(x\right)$, together with ${\sigma }_{q;x}Y\left(x\right)=B\left(x\right)Y\left(x\right)$ or $Y(x+a)=B\left(x\right)Y\left(x\right)$ for some shift a. See also [20,22] among other researches in this direction. The study of delay operators and $q-$difference operators is also a point of main interest for applied researchers nowadays. See, for Example, the works [1,2,8,24] for the first operators, and [16,17] for the second type of operators.

The present work can be seen as a linearized $q-$analog of the problem recently studied by the second author in [15], where he provides the analytic and formal solutions to a problem of the form

(where ${\left\{P_j\right\}}_{j\in J_1}$, ${\left\{Q_j\right\}}_{j\in J_2}$ are finite sets of complex polynomials and H is some nonlinear map in all its arguments) in which the Fuchsian operator $t{\partial }_t$ is substituted by the $q-$difference operator. The procedure followed in [15] rests on the substitution of the main equation by a decoupled system of equations in the Borel plane, which can be solved recursively as it is a triangularized system of equations. On the contrary, the $q-$analog treated in the present work is more involved, since the coupling of equations in the present study does not allow to come up to a triangular problem. Indeed, the system is no longer reducible in general. A further advance with respect to the preceding work, is that the general irreducible case is solved, whereas a product Banach space of functions is needed in order to solve the problem. This is the first appearance of such function space in the list of our joint works. We have also included a simplified situation of (1) in which a decoupled system arises, and the solution can be obtained by backward substitution of the partial solutions of a triangular system. The last section (Section 9) is motivated by the more relaxed assumptions needed on the $q-$Gevrey order, being able to solve the main equation under less restrictive geometric conditions.

We have decided not to state our principal result in the most generality which would have introduced high technical difficulties hiding the main Propositionerties of the analytic and formal solutions and their asymptotic relations. In this respect, we have considered functions which remain constant with respect to time variable (see $b_{jk}$ and $c_{\ell }$ in (9)), while considering a forcing term of the form (2) instead of the more general

The monodromy terms included in the main equation turn out to have an impact on the q-Gevrey order of the asymptotic solution in the general case. In the more restrictive situation in which the associated system is reducible, i.e. triangular, this influence is not so strong, being able to consider less restrictive assumptions. Indeed, the general framework deals with an equation (9) where the q-difference operator ${\sigma }_{q;t}$ appears with rational powers strictly less than 1 in contrast to the usual settings found in the literature where only integer powers arise. In our restrictive reducible case, it is worth noticing that such integer powers are granted.

The procedure followed in the present study is to search for solutions of (1) in the form as a natural consequence not only of the form of the forcing term (2) (see Assumption (B1)), but also on the appearance of the monodromy operator within the structure of the equation. This previous motivation yields the pair $\{u_0,u_1\}$ to be a solution of an auxiliary system of equations. At this point, one can search for $u_j$ for $j\in \{0,1\}$ in the form of a so-called q-Laplace and inverse Fourier transform, which transforms the previous system into an auxiliary system of equations in two auxiliary unknown functions $\{{\omega }_0,{\omega }_1\}$, say

(see (17), (18)).

More precisely, the solution is written in the form for $L_d=[0,\infty )e^{id}$ for some $d\in \mathbb{R}$, and where $\Theta (·)$ stands for Jacobi Theta function (see Section 2.2).

The existence of the analytic solution of the auxiliary system (3) needs an accurate description of the geometry and further assumptions on it (see Assumption (D)). Indeed, strict minorations for the auxiliary polynomial $P_m\left(\tau \right)$ (see (19)) need to be provided (see Section 6). Such tight estimates are needed to compensate the additional terms that come from the presence of logarithmic terms in the problem. Notice that the lower bounds for such a polynomial reached in the works [5,14] are not sufficient in our new setting.

The solution of the system (3) is attained via a fixed point argument of certain operator acting on a Banach space product. As mentioned above, although the problem enbraces a wide family of functional equations, the appearance of logarithmic terms makes it necessary to adopt several restrictive assumptions on the elements involved in the problem, having a control on its geometry. For that reason, we have decided to include an illustrative concrete Example at the end of Section 6. The first main result of the work (Theorem 2) describes the form of the analytic solutions to (1). The asymptotic behavior of such solutions is also analyzed with respect to the perturbation parameter $ϵ$ at the origin, through the application of a $q-$analog of the well-known Ramis-Sibuya Theorem (see Theorem ($q-$RS)). Indeed, the $q-$exponential decrease at the intersection of two consecutive sectors in $ϵ$ uniformly on the rest of the variables (Theorem 3) is the key point to prove the existence of a formal solution to (1) in the form

In addition to this, for $j\in \{0,1\}$, we prove that $u_j$ admits ${\widehat{u}}_j$ as its $q-$Gevrey asymptotic expansion of some $q-$Gevrey order (see the main result of the present work, Theorem 4).

We have included a final section, Section 9, in which the form of the main problem is slightly simplified. Indeed, only one of the coefficients of the monodromy terms in the equation does not appear. This slight simplification leads us to essential simplifications in the geometric assumptions of the problem. More precisely, under these settings, the geometric Assumption (D) is no longer needed. Moreover, the system (3) turns out to be triangular, and therefore much easier to solve. Also, a product Banach space is no longer needed in order to solve the system, but two partial fixed point steps.

The work is structured as follows: In Section 2, we recall the main facts about the formal monodromy operator, review some integral transforms and their related Propositionerties which are involved in the transformation of the main problem into the auxiliary problem. We finally recall the notions of $q-$Gevrey asymptotic expansions and a $q-$Gevrey analog of Ramis-Sibuya Theorem. In Section 3 we state the main problem under study (9) and describe the concrete assumptions and constructions related to it. After this, we show the strategy we follow to solve the main problem in Section 4 and describe the auxiliary Banach spaces of functions involved in the construction of the solution of an auxiliary system in the next section. The analytic solution to the main problem is constructed in Section 7 from the solution of the auxiliary system in Section 6, where a geometric assumption linked to the solution is described. The formal solution and the asymptotic study relating analytic and formal solution (Theorem 4) is studied in detail in Section 8. The work concludes with Section 9, where the mentioned slight simplification of the main problem is considered.

Given $r>0$, $D(0,r)$ denotes the open disc centered at the origin and radius r, and $\overline{D}(0,r)$ stands for its closure. Given an open sector S with vertex at the origin, we say that $S_1$ is a subsector of S, and denote it by $S_1\prec S$, if $S_1$ is a bounded sector with vertex at the origin and $\overline{S_1}\setminus \left\{0\right\}\subseteq S$.

Given a complex Banach space $\mathbb{E}$, we write $\mathbb{E}\left[\right[z\left]\right]$ for the vector space of formal power series in the variable z, with coefficients in $\mathbb{E}$. For every open set $U\subseteq \mathbb{C}$, we write $\mathcal{O}(U,\mathbb{E})$ for the set of holomorphic functions in U with values in $\mathbb{E}$. We adopt the simplified notation $\mathcal{O}\left(U\right)$ whenever $\mathbb{E}=\mathbb{C}$.

Let $q\in \mathbb{C}$. We denote ${\sigma }_{q;t}$ the dilation operator acting on t variable, i.e. ${\sigma }_{q;t}f\left(t\right)=f\left(qt\right)$. This Definitionition is naturally extended to ${\sigma }_{q;t}^{r}f\left(t\right)=f\left(q^{r}t\right)$ for any positive rational number r.

In this section, we recall some known results regarding the formal monodromy operator. We also recall the Definitionition and some of the main Propositionerties about $q-$Laplace and inverse Fourier transform. The section concludes with a brief review on $q-$Gevrey asymptotic expansions and related results.

Let $D\ge 2$, $k\ge 1$ be integer numbers and $q\in \mathbb{R}$ with $q>1$.

Let $d_{\ell },{\mathsf{\Delta }}_{\ell }$ for $1\le \ell \le D-1$ and $d_D$ be non negative integers and ${\delta }_{\ell }$ be a non negative rational number for $1\le \ell \le D-1$. Let $\beta >0$ and fix $0<{\beta }^{\prime }<\beta$. We also choose $0<{ϵ}_0<1$, and $\mu >1$. These parameters are chosen to satisfy Assumption (A) below. We also assume that ${ϵ}_0>0$ is small enough in terms of the elements involved in the problem. The precise value will be precised in this section. We also fix polynomials $R_{\ell }$ for $1\le \ell \le D$ and Q satisfying the second part of Assumption (A) and Assumption (C).

The forcing term $f(t,z,ϵ)$ and the coefficients $c_{\ell }(z,ϵ)$ for $1\le \ell \le D-1$ and $b_{jk}(z,ϵ)$ for $j,k\in \{0,1\}$ are assumed to be holomorphic functions of certain nature and Definitioned on some domain to be precised (see Assumptions (B1) and (B2)).

The main problem under study in the present work is the equation

We make the following assumptions on the elements involved in the previous equation:

Assumption (A):

Assumption (B1): The forcing term is of the form where $f_0,f_1$ are polynomials on their first variable with coefficients being holomorphic functions on $H_{{\beta }^{\prime }}\times D(0,{ϵ}_0)$. For $h\in \{0,1\}$, we write where ${\Lambda }_h\subseteq {\mathbb{N}}_0$ is a finite set. In addition to this, $F_{h,m_h}(z,ϵ)$ is assumed to be of the form for some $\mathbb{R}\ni m\mapsto {\tilde{F}}_{h,m_h}(m,ϵ)$. The previous function is assumed to be holomorphic for $ϵ\in D(0,{ϵ}_0)$. We write

We assume moreover uniform bounds with respect to the perturbation parameter as follows for some $C_F>0$, valid for every $h\in \{0,1\}$ and $m_h\in {\Lambda }_h$ (in other words, $m\mapsto {\tilde{F}}_{h,m_h}\in E_{(\beta ,\mu )}$ with uniform holomorphic bounds with respect to the perturbation parameter, see Definition 4).

This entails that $F_{h,m_h}(z,ϵ)$ is holomorphic on $H_{{\beta }^{\prime }}\times D(0,{ϵ}_0)$, for every $0<{\beta }^{\prime }<\beta$. Observe from (8) and the previous construction that $F_h$ is built as the $q-$Laplace of order k and inverse Fourier transform of some function, along a direction $d\in \mathbb{R}$. Indeed, for $h\in \{0,1\}$. The expression of $F_h$ actually does not depend on the choice of the direction d since it is a polynomial in t.

Assumption (B2): The coefficients $c_{\ell }(z,ϵ)$ for $1\le \ell \le D-1$, and $b_{jk}(z,ϵ)$ for $j,k\in \{0,1\}$ are holomorphic functions Definitioned in $H_{{\beta }^{\prime }}\times D(0,{ϵ}_0)$, constructed as a inverse Fourier transform. More precisely, there exist $m\mapsto C_{\ell }(m,ϵ)$ and $m\mapsto {\tilde{b}}_{jk}(m,ϵ)$ holomorphic with respect to $ϵ\in D(0,{ϵ}_0)$ such that there exist ${\mathcal{C}}_B>0$ with for $j,k\in \{0,1\}$ and ${\mathcal{C}}_C>0$ such that for every $1\le \ell \le D-1$ (these functions belong to $E_{(\beta ,\mu )}$, see Definition 4 of Section 2.2, with uniform bounds with respect to the perturbation parameter). We Definitione

Regarding the polynomials $R_{\ell }$ and Q, we make the next

Assumption (C):

Moreover, there exist ${\mathfrak{D}}_1,{\mathfrak{D}}_2>0$ such that

There exist $\tilde{d}\in \mathbb{R}$ and some small enough $\varsigma >0$, to be determined by the geometric configuration of the problem, which do not depend on $m\in \mathbb{R}$ such that $arg(Q\left(im\right)/R_D\left(im\right))\in [\tilde{d}-\varsigma ,\tilde{d}+\varsigma ]$. Observe from the previous condition that $Q\left(im\right)\ne 0$ and $R_D\left(im\right)\ne 0$ for $m\in \mathbb{R}$. In addition to this, we assume that

In this section, we describe the steps followed to solve the main problem under study, stated in Section 3. The strategy is divided in different steps modifying the initial equation into an equivalent one, and searching for solutions of a special shape. All the elements involved in the construction of the main problem described in Section 3 are maintained in this section.

First, we search for a formal solution to (9) in the form motivated by the form of the forcing term in (9) and the action of the monodromy operator. Let us write

By plugging $u(t,z,ϵ)$ into (9), and taking into account Lemma 3, one can rewrite (9) in the form which is equivalent to