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     \hspace{80mm}\raisebox{1ex}{ISSN 1813-3304 }

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     {\huge СИБИРСКИЕ \ ЭЛЕКТРОННЫЕ} \\[2mm]
     {\huge МАТЕМАТИЧЕСКИЕ ИЗВЕСТИЯ} \\[2mm]
     {\large Siberian Electronic Mathematical Reports} \\[1mm]
     {\LARGE\tt{http://semr.math.nsc.ru}}\\[0.5mm]
%     {\small Институт математики им. С.Л.Соболева СО РАН}\\[-1mm]
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     \vspace{1mm}
     {\flushleft\it Том 16, стр. 144--144 (2019) \hspace{65mm}{\rm\small 517.926.4+517.982.4 \udcs}}
     \newline
     	{\rm\small DOI~10.33048/semi.2019.16.xxx}\hphantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa}{\rm\small 34C27, 34C46, 46F99\ \ \mscs }
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\title[Periodic Solutions of the Spatial Extension ]{Periodic Solutions of the Spatial Extension of a Conditionally Periodic System}
\author{{Yu.D. Kozlov}}%
\address{Yury Dm. Kozlov  % Контактные данные всех авторов и место работы указываются только на английском языке
\newline\hphantom{iii} Ural Federal University,
\newline\hphantom{iii} 19 Mira Str.,
\newline\hphantom{iii} 620002, Ekaterinburg, Russia}%
\email{jury.kozlov@urfu.ru}%

\thanks{\sc Kozlov Yu.D,
Periodic Solutions of the Spatial Extension of a Conditionally Periodic System]{Periodic Solutions of the Spatial Extension of a Conditionally Periodic System}}
\thanks{\copyright \ 2023 Kozlov Yu.D}
\thanks{\rm This work was supported by the Russian Academic Excellence Project
(agreement no. 02.A0 3.21.0006 of August 27, 2013, between the Ministry of Education and
Science of the Russian Federation and Ural Federal University).}
\thanks{\it Поступила 25 мая 2023 г., опубликована 31 декабря 2015 г.}%


\semrtop \vspace{1cm}
\maketitle {\small
\begin{quote}
\noindent{\sc Abstract. } We consider a linear  system of differential equations $x'=a(t)x-\mu x$  with a conditionally periodic matrix $a$ and a parameter
$\mu\in\mathbb C$. We prove that there exists  a nonempty set $M\subset\mathbb C$ such that for each $\mu\in M $ the spatial periodic extension of this system, which is a system of first order partial differential equations, has a generalized (in the framework  of Schwartz's theory of   distributions) periodic solution.\medskip

\noindent{\bf Keywords:} conditionally periodic system, quasi-periodic solution, periodic Schwartz's distribution, linear homogenous system.
 \end{quote}
}


\section*{\large 0. Introduction}

\subsection*{0.1 }

Consider the system of differential equations
$$
x'=a(t)x-\mu x\eqno (0.1_\mu)
$$
with a real conditionally periodic matrix $a(t)=A(et)$, where $A=A(\varphi)$ is a continuous $\omega_j$-periodic in $\varphi_j$ (j=1,\dots,m)  $n\times n$-matrix function,
$\varphi=(\varphi_1,\ldots,\varphi_m)\in \mathbb{R}^m$,  $\omega=(\omega_1,\ldots,\omega_m)$, $e=(1,\ldots,1)\in\mathbb R^m$  (not to be confused with the Euler number e),  $t\in \mathbb{R}$, $\mu\in \mathbb C$, and the frequencies $\beta_i=2\pi/\omega_i$ are rationally incommensurable.

Consider also the spatial periodic extension of this system% $(0.1_\mu)$
$$
\sum\limits_{j=1}^{m}\psi'_j=A(\varphi)\psi-\mu\psi,\eqno(0.2_\mu)
$$
where $\psi:\mathbb R^m\rightarrow\mathbb C^n$ is an unknown vector function. Remark that if $\psi$ is continuous and $\omega_j$-periodic in $\varphi_j$ $(j=1,\dots,m)$, then the conditionally periodic vector $x(t)=\psi(et)$ solves  system $(0.1_\mu)$.

The idea of the periodic extension of a  conditionally periodic function goes back to P. Bohl \cite{15},  who was the founder of the theory of conditionally periodic functions and who proved that each continuous conditionally periodic function has a unique periodic extension. 
%The first author who used  system $(0.2_0)$ to investigate the properties of system $(0.1_0)$, as far as we know, was

If  $m=1$, then there exists a non-empty set $M\subset\mathbb C$ such that for each $\mu\in M$ system $(0.1_\mu)$ has a periodic solution.

We assume that $m>1$. The problem is whether system $(0.1_\mu)$ has a  conditionally periodic solution for some $\mu\in \mathbb C$ (or system $(0.2_\mu)$ has a periodic one). This is well-known to be falls in general, even for $n=1$.  In this case,  $x(t)=c\exp{\int\limits_0^t(a(t)-\mu)dt}$ is a solution of system $(0.1_\mu)$ with $x(0)=c$, and $\int\limits_0^t(a(t)-\mu)dt$ may not be conditionally periodic even  though $\mu$ is the mean value of $a$.

The aforementioned problem is related  to that of the existence of  almost periodic solutions of a homogeneous system \cite{9, 10, 11, 12}.
In these papers, some conditions have been given which ensure that a bounded solution of the homogeneous almost periodic system itself is Bohr almost periodic  \cite{9, 10, 11} or  Besicovitch almost periodic \cite{12}.

Our approach  is completely different.
  
Firstly, we assume nothing, but that the matrix $A$ is sufficiently smooth. We prove that there exists a non-empty set $M\subset\mathbb C$ such that, for any $\mu\in M$,
there exists a periodic distribution $\psi$ satisfying  $(0.2_\mu)$. In a sequel to the present paper, we will apply this fact to prove that, for any $\mu\in M$, there exists a conditionally periodic distribution satisfying $(0.1_\mu)$.

Secondly, we consider a conditionally periodic system. This allows us to reduce the problem to the study of  difference equation $(0.6)$.
This method is unlike any
attempts to solve this problem that we are aware of.

Note that we used an inhomogeneous equation associated with $(0.6)$   to prove an analogue of the Massera theorem in \cite{22}. Namely, it was proved that
if an inhomogeneous conditionally periodic system has a bounded solution, then almost every system in its H-class has a Besicovich conditionally periodic bounded solution. The proof was based on the Schauder-Tikhonov fixed point theorem. Unfortunately, we were unable to apply this theorem in the case of a homogeneous system.


 Thirdly, since in the general case  system $(0.2_\mu)$ does not have a continuous periodic solution, we are looking for a solution in the space of periodic distributions, where apparently no one has looked for it yet.

\subsection*{0.2 }

We  now present some results in \cite{3} (see also \cite{4}), which we are going to use. Since this paper is hardly available,  we prove these results (Lemma 0.1) and Lemma 0.2 at the end of the paper.

Consider the system of integral equations
$$
 x(\varphi)=\int\limits_{0}^{\varphi_1-\varphi_{10}}A(\varphi-e\xi)x(\varphi-e\xi)\,d\xi+x_0(\hat\varphi-\hat e(\varphi_1-\varphi_{10})),\eqno (0.3)
$$
where we use the "hat" operator to remove the first coordinate of the vector. Thereby, $\hat{\varphi}=(\varphi_{2},...,\varphi_{m})$ and $\hat{e}=(1,\dots,1)\in\mathbb R^{m-1}$. For each continuous function $x^0\colon \mathbb{R}^{m-1}\rightarrow  \mathbb{R}^n$, this system has a unique continuous solution $x=x(\varphi;\varphi_{10},x^0)$.
This solution is
  $\omega_{k}$
  -periodic in
  $\varphi_{k}$
   if so is $x^0$
  $( k=2,\dots,m)$.
The vector
$x=x(et;\varphi_{10},x^{0})$
 is a solution of system $(0.1_0)$ with
 $x(e\varphi_{10};\varphi_{10},x^0)=x^{0}(\hat{e}\varphi_{10}).$

  Consider the  matrix equation associated with (0.3)
$$
 X(\varphi)=\int\limits_{0}^{\varphi_1-\varphi_{10}}A(\varphi-e\xi)X(\varphi-e\xi)\,d\xi+E,\eqno (0.4)
$$
where $E$ is the identity matrix. By $X(\varphi;\varphi_{10})$ we denote the solution of  this equation and $X_0(\varphi)=X(\varphi;0)$.

 %The following assertion was proved in \cite{3}(see also \cite{4}).


{\bf Lemma 0.1 }.
 {\it
  Let $A$ be a continuous $\omega_j$-periodic in $\varphi_j$ (j=1,\dots,m) matrix function. Then matrix equation } $(0.4)$ {\it has a unique continuous solution $X$, and the following properties are true.

  (a)  The matrix $X(\varphi;\varphi_{10})$ is   non-singular for all  $\varphi\in \mathbb{R}^{m}$ and $\varphi_{10}\in  \mathbb{R}$, and it 
	is
	$\omega_j$-periodic in $\varphi_j$ (j=2,\dots,m).


  (b) The solution of system }$(0.3)$ {\it can be represented in the form

  $x(\varphi;\varphi_{10},x^{0})=
  X(\varphi;\varphi_{10})x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))$.

(c)
  $X(\varphi_{1}+\omega_{1},\hat{\varphi};\varphi_{10})=
  X(\varphi;\varphi_{10})X(\varphi_{10}+\omega_{1},\hat{\varphi}-
  \hat{e}(\varphi_{1}-\varphi_{10});\varphi_{10})$; in particular,
 $X_{0}(\varphi_{1}+\omega_{1},\hat{\varphi})=
 X_{0}(\varphi)X_{0}(\omega_{1},\hat{\varphi}-\hat{e}\varphi_{1})$ $[$here the vector $\varphi$ has the form $(\varphi_{1},\hat\varphi)$$]$.

 (d) The matrix   $X_{0}(et)$ is the normalized fundamental matrix of system
    $(0.1_0)$.}


   % Using (0.4),  the following lemma can be proved in a standard way, so the proof is omitted.

    {\bf Lemma 0.2}. {\it Let the matrix  $A$ be $r$-times continuously differentiable on $\mathbb{R}^m$; then so is the matrix
    $X_{0}$}.

The matrix $X_0$ is a solution of a differential equation, which we are going to get now. Put $\varphi_1-\xi=\zeta$  in (0.4). Then
$$
 X_0(\varphi)=\int\limits_{\varphi_{10}}^{\varphi_1}A(\zeta,\hat\varphi+\hat e(\zeta-\varphi_1))X_0(\zeta,\hat\varphi+\hat e(\zeta-\varphi_1))\,d\zeta+E.
$$


Let the matrix $A$  be continuously differentiable; then so is $X_0$  . Differentiating both sides of the above equation w.r.t.
 $\varphi_{j}$ and summing w.r.t. $j$ from $1$ to $m$, we obtain
$$
\sum\limits_{j=1}^{m}( X_0)'_j=A(\varphi)X_0.\eqno(0.5)
$$

%Existence  of a periodic solution of system $(0.2_\mu)$ (and hence existence  of a conditionally periodic solution of system $(0.1_\mu))$ is connected with that of the system
The key role in our research is played by the system
$$
 X_0(\omega_1,\hat\varphi)L(\hat\varphi-\hat
 e\omega_1)=\lambda L(\hat\varphi).\eqno(0.6)
 $$

Suppose that $L:\mathbb{R}^{m-1}\rightarrow \mathbb{C}^n$ is a continuous  $\omega_j$-periodic in $\varphi_j$ $(j=2,\dots,m)$ vector function which satisfies (0.6) for some $\lambda\in \mathbb{C}$, $\lambda\neq0$, and let $\mu=\frac{1}{\omega_1}\ln\lambda$.
 We claim that
$$
\psi=X_0(\varphi)L(\hat\varphi-\hat
 e\varphi_1)\exp{(-\mu\varphi_1)}
$$
  is periodic, $\psi(et)$ is the conditionally periodic solution of $(0.1_\mu)$, and if $L$ is differentiable, then $\psi$ satisfies $(0.2_\mu)$.
 Indeed, $\psi$ is $\omega_j$-periodic in $\varphi_j$ (j=2,\dots,m) since $X$ and $L$ are so. Let us prove that it is periodic in $\varphi_1$. By item (c) of Lemma 0.1,  we have
$$
\begin{array}{l}
\psi(\varphi_1+\omega_1,\hat\varphi)=X_0(\varphi_1+\omega_1,\hat\varphi)L(\hat\varphi-\hat e(\varphi_1+\omega_1))e^{-\mu\omega_1}e^{-\mu\varphi_1}\\=X_0(\varphi)X_0(\omega_1,\hat{\varphi}-\hat e\varphi_1)L(\hat\varphi-\hat e(\varphi_1+\omega_1))\lambda^{-1}e^{-\mu\varphi_1}\\ \stackrel{(0.6)}{=}X_0(\varphi)L(\hat{\varphi}-\hat{e}\varphi_{1})\lambda\lambda^{-1}e^{-\mu\varphi_1}=\psi(\varphi). 
\end{array} \eqno(0.7)
$$

In view of item (d) of Lemma 0.1,  it is clear that $\psi(et)=X_0(et)L(\hat{0})e^{-\mu t}$ is the solution of $(0.1_\mu)$. That $\psi$ solves $(0.2_\mu)$ one can check by direct calculation using (0.5) and the relation
 $\sum\limits_{j=1}^{m}(L(\hat\varphi-\hat e\varphi_1) )'_j=0$.
 %if $L$ is continuously differentiable, then,
 
In the first section of this paper, we present the basic facts about periodic distributions. After that we show that if there exists a periodic distribution $L$ that solves (0.6) for some $\lambda\neq 0$, then $\psi=X_0(\varphi)L(\hat\varphi-\hat e\varphi_1)\exp{(-\mu\varphi_1)}$ 
solves system $(0.2_\mu)$.

 In the second section, we prove that there exists a non-empty set $\Lambda\subset \mathbb{C}$ such that, for each $\lambda\in\Lambda$, the distribution $L$ exists. And in the third section we prove Lemmas 0.1, 0.2.

%In the present paper, we show that there exists a set $\Lambda\subset \mathbb{C}$ such that for each $\lambda\in\Lambda$ there exists a periodic distribution $L$ that solves (0.6) . Then we prove that $\psi=X_0(\varphi)L(\hat\varphi-\hat
 %e\varphi_1)\exp{(-\mu\varphi_1)}$, where $\mu=\frac{1}{\omega_1}\ln\lambda$, is a generalized  periodic solution of system $(0.2_\mu)$.

\section*{\large 1. Periodic  distributions}

 The definition and  properties of  periodic distributions, which we use, can be found in the monographs
   \cite{7,8,1}. We introduce the vector periodic distributions following \cite{1} (see also \cite[Ch.1]{19}).
	
	Let  $ P_{n}^{q}(\omega)$ be   the Banach space of $\omega_j$-periodic in $\varphi_j$ $(j=1,\dots,m)$ q-times continuously differentiable  functions $y\colon \mathbb{R}^{m}\rightarrow \mathbb{C}^{n}$ with the norm
  $\|y\|_{ P_{n}^q(\omega)}=\max\limits_{|r|\leq q}\max\limits_{\varphi\in \mathbb{R}^m}\|D^{r}y(\varphi)\|$, where $\|\cdot\|$ is a  norm on a finite-dimensional space,  $r=(r_{1},\dots,r_{m})$,  $r_j$ are non-negative integers,
    $D^{r}y(\varphi)=\frac{\partial^{|r|}y(\varphi)}{\partial \varphi_{1}^{r_{1}}\dots \partial
   \varphi_{m}^{r_{m}}}$, and $|r|=r_{1}+\dots+r_{m}$.

    Let  $ P^q_{n^{2}}(\omega)$ be  the Banach space of periodic q-~times continuously differ\-en\-tiable $n\times n$ matrices with the  norm
  $\|A\|_{ P_{n^2}^q(\omega)}=\max\limits_{|s|\leq n}\|a_s\|_{ P_{n}^r(\omega)}$, where $a_s\in  P_{n}^{q}(\omega) $ $(s=1,\dots,n)$ are columns of $A$.
	
	Denote by $ P'_{n}{}^{q}(\omega)$ the space of linear continuous maps
	$x\colon P_{n}^q(\omega)\rightarrow \mathbb{C}^{n}$, where $x=(x_{1},\dots,x_{n})^T$, $x^T$ is the transpose of $x$, and $x_k$ are linear continuous functionals on $P_{1}^q(\omega)$.
	
	Every $x\in P'_{n}{}^{q}(\omega)$ determines the  linear continuous functional on $P_{n}^q(\omega)$ by the rule
$\langle x,y\rangle_{\omega}=\sum\limits_{k=1}^{n}\langle x_{k},y_{k}\rangle_{\omega}$, where $y=(y_1,\dots,y_n)^T\in P_{n}^{q}(\omega)$ and
$\langle x_{k},y_{k}\rangle_{\omega}$
  is the value of  $x_k$ at the point $y_k\in P_{1}^{q}(\omega)$.

  The space $P'_{n}{}^{q}(\omega)$,  endowed with the  norm $\|x\|_{P'_{n}{}^{q}(\omega)}=\sup\limits_{\|y\|_{ P_{n}^q(\omega)}=1}|\langle x,y\rangle_{\omega}|$,
  is the Banach space, and it is isomorphic  to the dual space of $P_{n}^q(\omega)$ \cite[p.267]{1}.

   Consider the countably normed space $P_{n}(\omega)=\bigcap_{q=0}^{\infty}P_{n}^q(\omega)$ and its dual $P'_{n}(\omega)=\bigcup_{q=0}^{\infty}P'_{n}{}^{q}(\omega)$. By definition, a sequence  $\{y^{k}\}$ in $P_n(\omega)$ converges to $y\in P_n(\omega)$ if
   $\|y^k-y\|_{ P_{n}^q(\omega)}\rightarrow 0$ for every non-negative integer $q$, and $\{x^{k}\}$ in $ P'_n(\omega)$ converges weakly to $x\in P'_n(\omega)$ if $\langle x^k-x,y\rangle_{\omega}\rightarrow 0$ for every $y\in P_n(\omega)$ as $k\rightarrow\infty$. The latter is equivalent to the fact that
  $x^k\rightarrow x$ weakly in some $P'_{n}{}^{q}(\omega)$ \cite[Ch.1]{19}.







The space $ P'_{1}(\omega)$ is isomorphic  to the space of the Schwartz periodic distributions \cite{1,7}. That is why $x\in P'_{1}(\omega)$
is called an {\it $\omega$-periodic distribution}, and therefore  $x\in P'_{n}(\omega)$ we call
 an {\it $\omega$-periodic vector distribution}  \cite{8,1}. Note that the term $\omega$-periodic is used for brevity. In fact, $x\in P'_{n}(\omega)$ is $\omega_j$-periodic in $\varphi_j$ $(j=1,\dots,m)$, i.e.,
$$
x(\varphi_1,\dots,\varphi_{j-1}, \varphi_j+\omega_j,\varphi_{j+1},\dots,\varphi_m)=x(\varphi).
$$

Indeed, by the definitions of equality and translation (see below on this page), we have

$$
\begin{array}{l}
\langle x(\varphi_1,\dots,\varphi_{j-1}, \varphi_j+\omega_j,\varphi_{j+1},\dots,\varphi_m),y(\varphi)\rangle_{\omega}\\
=\langle x(\varphi),y(\varphi_1,\dots,\varphi_{j-1}, \varphi_j-\omega_j,\varphi_{j+1},\dots,\varphi_m)\rangle_{\omega}
=\langle x(\varphi),y(\varphi)\rangle_{\omega}
\end{array}
$$ 
for any $y \in P_{n}(\omega)$ .


We are going to mainly deal  with the spaces of vector distributions $P'_{n}{}^{q}(\omega)$ and spaces of test functions $P_{n}^{q}(\omega)$. Suppose
 $x,z \in P'_{n}{}^{q}(\omega)$. Let us remember the definitions of
	$$
	\begin{array}{l}
	\mbox{{\it equality}}:~x=z~\Leftrightarrow~\langle x,y\rangle_{\omega}=\langle z, y\rangle_{\omega},~~\forall y\in P_{n}^{q}(\omega);\\
	\mbox{{\it derivative}}:~\langle x'_j,y\rangle_{\omega}=-\langle x,y'_j\rangle_{\omega},~~\forall y\in P_{n}^{q+1}(\omega);\\
	\mbox{{\it multiplication by  matrix}}~ A\in P^q_{n^{2}}(\omega):~~\langle A x,y\rangle_{\omega}=\langle x,A^T y\rangle_{\omega},~~~\forall y\in P_{n}^{q}(\omega);\\
	\mbox{{\it translation by}}~\tau:~\langle x(\varphi-\tau),y(\varphi)\rangle_{\omega}=\langle x(\varphi),y(\varphi+\tau)\rangle_{\omega},~~\forall y\in P_{n}^{q}(\omega),\\ \tau\in\mathbb R^m.
	\end{array}
	$$
	
	It was proved in the monographs  \cite[p.225]{7},  \cite[p.130]{8}  that every $x\in P'_{n}(\omega)$ has a unique Fourier-series expansion
$$
       x(\varphi)=\sum\limits_{k\in \mathbb{Z}^{m}}c_{k}\exp(i(k\beta\varphi)), \eqno(1.1)
       $$
      where\\
			$c_{k}=(\langle x_{1}(\varphi),\exp(-i(k\beta\varphi))\rangle_{\omega},\dots,\langle
       x_{n}(\varphi),\exp(-i(k\beta\varphi))\rangle_{\omega})^T/(\omega_{1}\dots\omega_{m}),$\\
$k=(k_{1},\dots,k_{m})\in \mathbb{Z}^{m}$, $\beta=(\beta_{1},\dots,\beta_{m})$,
       $\beta_{r}=2\pi/\omega_{r}$, $(k\beta\varphi)=\sum\limits_{j=1}^{m}k_{j}\beta_{j}\varphi_{j}$.

       Besides, there exist $K>0$ and $p\in \mathbb{N}$ such that
       $$
       \|c_{k}\|\leq K\|k\|^{p},~~k\in \mathbb{Z}^m.\eqno (1.2)
       $$
        Inversely, if there exist  $K>0$ and $p\in \mathbb{N}$ such that, for any $k\in \mathbb{Z}^m$, inequality (1.2) holds, then
       series (1.1) converges weakly and $x\in P'_{n}(\omega)$  \cite[pp.265,266]{1}, \cite [p.225]{7}, \cite[p.130]{8}.


		
 It was shown in the previous section  that if (0.6) has a  continuous periodic solution $L$, then $\psi(\varphi)=X_0(\varphi)L(\hat{\varphi}-\hat{e}\varphi_1)\exp(-\mu\varphi_1)$ is continuous and periodic. Now suppose that $L\in P'{}^{r}_{n}(\hat\omega)$  and $X_0\in P^{r}_{n^{2}}(\omega)$  (for some non-negative integer $r$); then $\psi\in P'{}^{r}_{n}(\hat\omega)$ for any $\varphi_1\!\in\! \mathbb{R}$.
 Likewise (see (0.7)), it can be proved that $\psi$ is $\omega_1$-periodic in $\varphi_1$. 
%$$
%\begin{array}{c}
%\psi(\varphi_1+\omega_1,\hat{\varphi})=
 %X_0(\varphi_1+\omega_1,\hat{\varphi})L(\hat{\varphi}-\hat e(\varphi_1+\omega_1))e^{-\mu(\varphi_1+\omega_1)}\\=
 %X_0(\varphi)X_0(\omega_1,\hat{\varphi}-\hat{e}\varphi_{1})L(\hat{\varphi}-\hat e(\varphi_1+\omega_1))e^{-\mu\omega_1}e^{-\mu\varphi_1}\\=
%X_0(\varphi)L(\hat{\varphi}-\hat{e}\varphi_{1})\lambda\lambda^{-1}e^{-\mu\varphi_1}= \psi(\varphi).
%\end{array}
%$$
 That is why
we are going to regard $\psi$ as the distribution given by $\langle\psi,y\rangle_{\omega}=\int\limits_0^{\omega_1}\langle\psi,y\rangle_{\hat{\omega}}\,d\varphi_1$,
    $y\in P_n^r(\omega)$. Let us prove that  $\psi\in P'_n{}^{r}(\omega)$.
		
		In this proof we   use a standard fact in calculus : {\it Let $y:[\alpha,\beta]\times K\rightarrow \mathbb{C}^n$ be  uniformly continuous for given $\alpha,\beta\in \mathbb{R}$  and $K\subset \mathbb{R}^{m-1}$,  let $\{\varphi_{1k}\}\subset[\alpha,\beta]$;
    then $\varphi_{1k}\rightarrow\varphi_{10}$ implies $y(\varphi_{1k},\hat{\varphi})\rightarrow y(\varphi_{10},\hat{\varphi})$ uniformly for $\hat{\varphi}\in K$ as $k\rightarrow\infty$.}


     {\bf Lemma 1.1.} {\it Let $A\in P^{r}_{n^{2}}(\omega)$, $L\in P'{}^{r}_{n}(\hat\omega)$, and  $r\geq0$; then  $\psi(\varphi)=X_0(\varphi)L(\hat{\varphi}-\hat{e}\varphi_1)\exp(-\mu\varphi_1)\in P'{}^{r}_{n}(\hat\omega)$ for any $\varphi_1\in \mathbb{R}$
    and the function $g(\varphi_1)=\langle\psi, y\rangle_{\hat{\omega}}$ is continuous  w.r.t. $\varphi_1\in[0,\omega_1]$ for every $y\in  P_n^r(\hat{\omega})$.}


\begin{proof}
   By Lemma 0.2, it follows that $X_0\in P_{n^2}^r(\omega)$.
  Hence,  $\psi\in P'^{r}_{n}(\hat{\omega})$ for each $\varphi_1\in \mathbb{R}$.  Put $X_{0\mu}(\varphi)=X_0(\varphi)\exp(-\mu\varphi_1)$. Then $\psi(\varphi)=X_{0\mu}(\varphi)L(\hat{\varphi}-\hat{e}\varphi_1)$, and  $g$ can be written as  $g(\varphi_1)= \langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\hat{\varphi}+
  \hat{e}\varphi_1)\rangle_{\hat{\omega}}$.
   The vector $v(\varphi)=X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\hat{\varphi}+
   \hat{e}\varphi_1)$  and its partial derivatives of $|j|{th}$ order ($|j|=1,\dots,r)$ are uniformly continuous on $[0,\omega_1]\times  \mathbb{R}^{m-1}$. Therefore, if $[0,\omega_1]\ni\varphi_{1k}\rightarrow \varphi_{10}\in[0,\omega_1]$ as $k\rightarrow\infty$,  then $v(\varphi_{1k},\hat{\varphi})$ and $D^{j}v(\varphi_{1k},\hat{\varphi})$ tend  uniformly w.r.t. $\hat{\varphi}\in \mathbb{R}^{m-1}$ to $v(\varphi_{10},\hat{\varphi})$ and
    $D^{j}v(\varphi_{10},\hat{\varphi})$ respectively. From this, remembering that $L$ is a continuous mapping, we conclude  that $g(\varphi_{1k})\rightarrow g(\varphi_{10})$. Hence, $g$ is continuous. 
\end{proof}

    {\bf Lemma 1.2.}{\it Let $A\in P^{r}_{n^{2}}(\omega)$, $L\in P'{}^{r}_{n}(\hat\omega)$, and  $r\geq0$; then $\psi\in P'_n{}^{r}(\omega)$. }

     
		\begin{proof}
		First, by using the fact that the mapping  $\langle\cdot ,\cdot\rangle_{\hat{\omega}}\colon P'{}_n^{r}(\hat{\omega})\times P_n^{r}(\hat{\omega})\rightarrow \mathbb{C}$ is bilinear and  separately continuous, we prove that   $\langle\psi,y\rangle_{\hat{\omega}}$ is continuous in $\varphi_1\in[0,\omega_1]$ if  $y\in P_n^{r}(\omega)$.
		
		It follows from Lemma 1.1 that the function $\psi:[0,\omega_1]\rightarrow P'{}^{r}_{n}(\hat\omega)$ is continuous, i.e.,
		%	Let $\varphi_{10}\in[0,\omega_1]$
      $[0,\omega_1]\ni\varphi_{1k}\rightarrow\varphi_{10}$ implies
     $$
      \langle\psi(\varphi_{1k},\cdot),z\rangle_{\hat{\omega}}\rightarrow
      \langle\psi(\varphi_{10},\cdot),z\rangle_{\hat{\omega}} \eqno (1.3)
      $$
       for every $z\in  P_n^{r}(\hat{\omega})$.

       Besides,  if  $y\in P_n^{r}(\omega)$, then $\|y(\varphi_{1k},\hat{\varphi})-y(\varphi_{10},\hat{\varphi})\|_{ P_n^{r}(\hat{\omega})}\rightarrow0$  since the derivatives
       $D^j y(\varphi)$, $0\leq |j|\leq r$ are uniformly continuous on $ \mathbb{R}^m$, and hence
       $$
      \langle x,y(\varphi_{1k},\cdot)\rangle_{\hat{\omega}}\rightarrow  \langle x,y(\varphi_{10},\cdot)\rangle_{\hat{\omega}} \eqno (1.4)
     $$
     for each $x\in P'{}^r_n(\hat{\omega})$.

      Therefore,   by the bilinear mapping continuity theorem \cite[Th. 2.17]{5}, in view of relations (1.3) and (1.4), we get
     $$
		\langle\psi(\varphi_{1k},\cdot),y(\varphi_{1k},\cdot)\rangle_{\hat{\omega}}
     \rightarrow\langle\psi(\varphi_{10},\cdot),y(\varphi_{10},\cdot)
     \rangle_{\hat{\omega}},~~y\in P_n^{r}(\omega).
		$$
     This proves that $\langle\psi,y\rangle_{\hat{\omega}}$ is continuous and hence integrable w.r.t. $\varphi_1\in[0,\omega_1]$.

     To conclude the proof, we note that
		$$|\int\limits_0^{\omega_1}\langle\psi,y\rangle_{\hat{\omega}}\,d\varphi_1|
     \leq\omega_1\sup\limits_{\varphi_1\in [0,\omega_1]}|\langle\psi,y\rangle_{\hat{\omega}}|\leq\omega_1 K\sup\limits_{\varphi_1\in[0,\omega_1]}\|y\|_{ P_n^{r}(\hat{\omega})}\leq\omega_1 K\|y\|_{ P_n^{r}(\omega)}.
		$$
		The existence of
     $K=\sup\limits_{\varphi_1\in[0,\omega_1]}\|\psi\|_{ P_n'{}^{r}(\hat{\omega})}$ follows from the Principle of Uniform Boundedness. Indeed, due to Lemma 1.1, $\langle\psi,y\rangle_{\hat{\omega}}$ is bounded in $\varphi_1\in[0,\omega_1]$ at each point $y\in P_n^{r}(\hat{\omega})$. Hence, the collection $\{\langle\psi,\cdot\rangle_{\hat{\omega}}: \varphi_1\in[0,\omega_1] \}$ of the continuous linear mappings from $ P_n^{r}(\hat{\omega})$ into $ \mathbb{C}$ is bounded.
The inequality just obtained proofs the lemma.
\end{proof}

{\it We say that $\psi\in P'_n{}^{r}(\omega)$ is a solution of system $(0.2_\mu)$ if
$$
 \langle\sum\limits_{i=1}^m \psi'_i,y\rangle_{\omega}=\langle A\psi-\mu\psi,y\rangle_{\omega} \eqno(1.5)
$$
for any  $y\in  P^{r+1}_{n}(\omega) $}.

{\bf Lemma 1.3.}
  {\it Let  $A\in P^{r}_{n^{2}}(\omega)$,   $L\in P'{}^{r}_{n}(\hat{\omega})$ be a solution of $(0.6)$ for some $\lambda\neq0$, $r\geq1$, and $\mu=\omega^{-1}_{1}\ln \lambda$; then the distribution $\psi(\varphi)=X_{0\mu}(\varphi)L(\hat{\varphi}-\hat{e}\varphi_{1})\in P'{}^{r}_{n}(\omega)$  satisfies system $(0.2_\mu)$.}

\begin{proof}
  It follows from Lemma 1.2 that  $\psi\in P'_n{}^{r}(\omega)$, and hence $\psi_j'\in P'_n{}^{r+1}(\omega)$, $j=1,\dots,m$.


 Let $y\in  P^{r+1}_{n}(\omega) $. We are going to use the relations
 $$
 \begin{array}{l}
   \int\limits_0^{\omega_1}\langle L(\hat{\varphi}-\hat{e}\varphi_1),(X_{0\mu}^T(\varphi)A^T(\varphi)-\mu X_{0\mu}^T(\varphi))y(\varphi)\rangle_{\hat{\omega}}d\varphi_1\\ = \int\limits_0^{\omega_1}\langle( A(\varphi)X_{0\mu}(\varphi)-\mu X_{0\mu}(\varphi))L(\hat{\varphi}-\hat{e}\varphi_1),y(\varphi)\rangle_{\hat{\omega}}d\varphi_1\\= \langle A\psi-\mu\psi,y\rangle_{\omega}
   \end{array}\eqno(1.6)
   $$
   and
   $$
   \begin{array}{l}
   \partial X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)/\partial\varphi_{1}=
   \sum\limits_{i=1}^m( X_{0\mu}^T)'_i(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\\= X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)A^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)-\mu X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1).
   \end{array}\eqno(1.7)
   $$
   The latter follows from (0.5).
	
So, remembering that $y\in  P^{r+1}_{n}(\omega) $, we have
   $$
   \begin{array}{l}
   \langle\sum\limits_{i=1}^m \psi'_i,y\rangle_{\omega}=-\langle X_{0\mu}(\varphi) L(\hat{\varphi}-\hat{e}\varphi_1),\sum\limits_{i=1}^m y'_i(\varphi)\rangle_{\omega}\\=-\int_0^{\omega_1}\langle L(\hat{\varphi}-\hat{e}\varphi_1),X_{0\mu}^T(\varphi)\sum\limits_{i=1}^m y'_i(\varphi)\rangle_{\hat{\omega}}\,d\varphi_1
\\=-\int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1) y'_1(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1\\-
   \int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\sum\limits_{i=2}^m y'_i(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1.
   \end{array}
   $$

Denote the first summand at the right in  the above relation by $I_1$,  the second by $I_2$, and transform $I_1$.
$$
   \begin{array}{l}
I_1=
   -\int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\partial y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)/\partial\varphi_{1}\rangle_{\hat{\omega}}\,d\varphi_1\\+
   \int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\sum\limits_{i=2}^m y'_i(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1\\=
    \int_0^{\omega_1}\langle L(\hat{\varphi}),\partial X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)/\partial\varphi_{1}y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1\\-
     \int_0^{\omega_1}\langle L(\hat{\varphi}),\partial( X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1))/\partial\varphi_{1}\rangle_{\hat{\omega}}\,d\varphi_1\\
   %  \end{array}
 %  $$
 % $$
 %  \begin{array}{l}
     + \int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\sum\limits_{i=2}^m y'_i(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1,\\
      I_2=-\int_0^{\omega_1}\langle L(\hat{\varphi}),X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\sum\limits_{i=2}^m  y'_i(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}d\varphi_1.
 \end{array}
   $$
   Hence,
   $$
   \begin{array}{l}
I_1+I_2=
    \int_0^{\omega_1}\langle L(\hat{\varphi}),\partial X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)/\partial\varphi_{1}y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1\\-
     \int_0^{\omega_1}\langle L(\hat{\varphi}),\partial( X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\varphi_1,\hat{\varphi}+
     \hat{e}\varphi_1))/\partial\varphi_{1}\rangle_{\hat{\omega}}\,d\varphi_1.
 \end{array}
   $$
    We denote the first integral in the previous relation by $J_1$ [and the second by $J_2$] and transform it using (1.7):
     $$
   \begin{array}{l}
	J_1= \int_0^{\omega_1}\langle L(\hat{\varphi}),(X_{0\mu}^T(\varphi_1,\hat{\varphi}+
     \hat{e}\varphi_1)A^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\\
  -\mu X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1))y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}\,d\varphi_1=%\\
    \int_0^{\omega_1}\langle L(\hat{\varphi}-\hat{e}\varphi_1),(X_{0\mu}^T(\varphi)A^T(\varphi)\\-\mu X_{0\mu}^T(\varphi))y(\varphi)\rangle_{\hat{\omega}}\,d\varphi_1\stackrel{(1.6)}{=}%\\
    \langle A\psi-\mu\psi,y\rangle_{\omega}.
   \end{array}
   $$

To transform $J_2$, we use differentiation of $\langle\cdot,\cdot\rangle_{\hat{\omega}}$  w.r.t. parameter $\varphi_1$ \cite[p.105]{7}:\\
  $\langle L(\hat{\varphi}),\partial( X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1))/\partial\varphi_{1}\rangle_{\hat{\omega}}$\\= $\partial\langle L(\hat{\varphi}), X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1))\rangle_{\hat{\omega}}/\partial\varphi_1$.\\
	As a result, we get
$$
   \begin{array}{l}
   J_2=\int_0^{\omega_1}\partial \langle L(\hat{\varphi}), X_{0\mu}^T(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)y(\varphi_1,\hat{\varphi}+\hat{e}\varphi_1)\rangle_{\hat{\omega}}/\partial\varphi_{1}\,d\varphi_1\\=
   \langle L(\hat{\varphi}),(X_{0\mu}^T(\omega_1,\hat{\varphi}+\hat{e}\omega_1)y(\omega_1,\hat{\varphi}+\hat{e}\omega_1)-X_{0\mu}^T(0,\hat{\varphi})y(0,\hat{\varphi}))\rangle_{\hat{\omega}}\\=
   \langle X_{0\mu}(\omega_1,\hat{\varphi})L(\hat{\varphi}-\hat{e}\omega_1)-X_{0\mu}(0,\hat{\varphi})L(\hat{\varphi}), y(0,\hat{\varphi})\rangle_{\hat{\omega}}.
   \end{array}
   $$
   Remembering that $L$ satisfies (0.6),  $X_{0\mu}(\omega_1,\hat{\varphi})=X_{0}(\omega_1,\hat{\varphi})\lambda^{-1}$,\\ and  $X_{0\mu}(0,\hat{\varphi})=X_{0}(0,\hat{\varphi})=E$, we get $J_2=0$.

   Summarizing all the above, we get (1.5).  
	\end{proof}
	
         \section*{ \large 2. Generalized solutions of systems (0.6) and $\bf{(0.2}_\mu)$}

 Consider the system

   $$
   X_0(\omega_1,\hat \varphi)L(\hat \varphi-\alpha_s)=\lambda L(\hat \varphi), \eqno (2.1)
   $$
   where $\alpha_s=(p_{2s}\omega_2/q_{2s},\dots,p_{ms}\omega_m/q_{ms})$,
   $p_{js}\in \mathbb{Z}$,  $q_{js}\in \mathbb{N}$. Due to the incommensurability of the frequencies $\beta_i=2\pi/\omega_i$,   we can choose  $p_{js},q_{js}$ such that $\alpha_s\to\hat e\omega_1$ as $s\to\infty$.
	
	First, we want to prove that, for each $s\in\mathbb N$, there exists a distribution $L_s$ which solves this system for some $\lambda_s\in \mathbb C$.  Then we prove that the sequence $\{L_s\}$ has a week partial limit $L$ which satisfies (0.6) for some $\lambda\in\mathbb C$.

   {\bf Lemma 2.1}. {\it Let $A\in P^0_{n^2}{}(\omega)$; then, for any $s\in \mathbb{N}$,
   there exist a distribution $L_s\in P'_n{}^0(\hat{\omega})$
    and a complex number $\lambda_s\neq0$ which satisfy $(2.1)$}.
   \begin{proof}


    Let us endeavour to satisfy (2.1) by the distribution
   $$
   L_s(\hat\varphi)=\sum\limits_{j=0}^{\nu_s-1}f_{sj}\delta(\hat\varphi-j\alpha_s), \eqno (2.2)
   $$
    where $\nu_s$ is the least common multiple of $q_{js}$  $(j=2,\dots,m)$, $f_{sj}$ are constant column vectors, and
    $\delta\in P'_1{}^0(\hat{\omega})$ is the Dirac distribution: $\langle f_{sj}\delta(\hat\varphi-\hat{\varphi}^0),y\rangle_{\hat{\omega}}=f^T_{sj}y(\hat{\varphi}^0)$, $y\in P^0_n(\hat\omega)$. 
		
		Note that, in our case, $\delta$ is a periodic functional since it is defined on the space of periodic functions.
    
	{\it	Recall that  $L_s$ satisfies (2.1) if, for any $y\in P^0_n(\hat\omega)$, the following relation holds}
    $$
   \langle X_0(\omega_1,\hat \varphi)L_s(\hat \varphi-\alpha_s),y\rangle_{\hat{\omega}}=\langle\lambda_sL_s(\hat \varphi) ,y\rangle_{\hat{\omega}}.
    $$

   Upon substituting $L_s$ into the left and right sides of this equation, we find
	$$
   \begin{array}{l}
	\langle X_0(\omega_1,\hat \varphi)L_s(\hat \varphi-\alpha_s),y(\hat\varphi)\rangle_{\hat{\omega}}=
	\langle L_s(\hat \varphi-\alpha_s),(X_0(\omega_1,\hat \varphi))^Ty(\hat\varphi)\rangle_{\hat{\omega}}\\=
	\langle L_s(\hat \varphi),(X_0(\omega_1,\hat \varphi+\alpha_s))^T y(\hat \varphi+\alpha_s)\rangle_{\hat{\omega}}\\=
	\sum\limits_{j=0}^{\nu_s-1}\langle f_{sj}\delta(\hat\varphi-j\alpha_s),(X_0(\omega_1,\hat \varphi+\alpha_s))^T y(\hat \varphi+\alpha_s)\rangle_{\hat{\omega}}\\=
	\sum\limits_{j=0}^{\nu_s-1}f_{sj}^T(X_0(\omega_1,(j+1)\alpha_s))^T y((j+1)\alpha_s)
	=\sum\limits_{j=0}^{\nu_s-1}(X_0(\omega_1,(j+1)\alpha_s)f_{sj})^T y((j+1)\alpha_s)
  \end{array}  
		$$
		
		and 
		$$\langle\lambda_sL_s(\hat \varphi) ,y(\hat\varphi)\rangle_{\hat{\omega}}=
		\lambda_s\sum\limits_{j=0}^{\nu_s-1}\langle f_{sj}\delta(\hat\varphi-j\alpha_s), y(\hat \varphi)\rangle_{\hat{\omega}}=
		\lambda_s\sum\limits_{j=0}^{\nu_s-1}f_{sj}^Ty(j\alpha_s)$$.
		
		Thus,
		
		
	$$
	\sum\limits_{j=0}^{\nu_s-1}(X_0(\omega_1,(j+1)\alpha_s)f_{sj})^Ty((j+1)\alpha_s)=
   \lambda_s\sum\limits_{j=0}^{\nu_s-1}f^T_{sj}y(j\alpha_s).
	$$
%Observing that $y(\nu_s\alpha_s)=y(\hat{0})$, we come to conclusion that
 This relation holds  if the vectors $f_{sj}$ satisfy the system
  $$
    \left\{
    \begin{array}{lcl}
X_0(\omega_1,\alpha_s)f_{s0}&=&\lambda_sf_{s1}\\
X_0(\omega_1,2\alpha_s)f_{s1}&=&\lambda_sf_{s2}\\
\dots&\dots&\dots\\
X_0(\omega_1,(\nu_s-1)\alpha_s)f_{s\nu_s-2}&=&\lambda_sf_{s \nu_s-1}\\
X_0(\omega_1,\nu_s\alpha_s)f_{s\nu_s-1}&=&\lambda_sf_{s0}.\\
\end{array}
\right .\eqno (2.3)
$$

Indeed, let $y\in P^0_n(\hat\omega)$. Then it follows from (2.3) that

$$
    \left\{
    \begin{array}{lcl}
(X_0(\omega_1,\alpha_s)f_{s0})^T y(\alpha_s)&=&\lambda_s(f_{s1})^Ty(\alpha_s)\\
(X_0(\omega_1,2\alpha_s)f_{s1})^T y(2\alpha_s)&=&\lambda_s(f_{s2})^T y(2\alpha_s)\\
\dots&\dots&\dots\\
(X_0(\omega_1,(\nu_s-1)\alpha_s)f_{s\nu_s-2})^T y((\nu_s-1)\alpha_s)&=&\lambda_s(f_{s \nu_s-1})^T y((\nu_s-1)\alpha_s)\\
(X_0(\omega_1,\nu_s\alpha_s)f_{s\nu_s-1})^T y(\nu_s\alpha_s)&=&\lambda_s(f_{s0})^T y(\nu_s\alpha_s).\\
\end{array}
\right.
$$
 
Putting $y(\nu_s\alpha_s)=y(\hat{0})$ in the right-hand side of the last equation and summing these equations, we obtain the desired relation.

It follows from (2.3) that
$$
X_0(\omega_1,\nu_s\alpha_s)X_0(\omega_1,(\nu_s-1)\alpha_s)\dots
X_0(\omega_1,\alpha_s)f_{s0}=\lambda_s^{\nu_s}f_{s0}.\eqno (2.4)
$$
Consider some root $G_s=(X_0(\omega_1,\nu_s\alpha_s)X_0(\omega_1,(\nu_s-1)\alpha_s)\dots
X_0(\omega_1,\alpha_s))^{\frac{1}{\nu_s}}$, some eigenvalue
  $\lambda_s$  of $G_s$, and  the corresponding eigenvector $f_{s0}$; then
   $\lambda_s$ and $f_{s0}$
  satisfy (2.4).
   Substituting  $\lambda_s$ and $f_{s0}$ into (2.3),  we successively find the  vectors
  $f_{sk}, k=1,\dots,\nu_s-1$. Hence, we get the distribution $L_s$  which is a solution of (2.1). It can be seen from  (2.2) that $L_s\in P'_n{}^0(\hat{\omega})$ since $f_{sj}\delta(\hat\varphi-j\alpha_s)\in P'_n{}^0(\hat{\omega})$.
	\end{proof}

{\bf Lemma 2.2}. {\it Let $A\in P_{n^2}^{0}(\omega)$ and let $\lambda_s$ be an eigenvalue of $G_s$, $s\in\mathbb N$; then there exists a convergent  subsequence  $\{\lambda_{s_k}\}$ of the sequence $\{\lambda_{s}\}$ with nonzero limit.}

\begin{proof}
The matrix function $X_0(\omega_1,\cdot)$ is non-singular, continuous, and $\hat\omega$-periodic in $\hat\varphi\in\mathbb{ R}^{m-1}$.
 Hence, there exist $\alpha,\beta>0$ such that for any   $ x\in \mathbb{R}^n$ and $\hat\varphi\in \mathbb{R}^{m-1}$
  we have
  $\alpha\|x\|\leq\|X_0(\omega_1,\hat\varphi)x\|\leq\beta\|x\|$.
  From these inequalities we get
  $\|G_s^{\nu_s}\|\leq\beta^{\nu_s}$ and
   $\|G_s^{-\nu_s}\|\leq\alpha^{-\nu_s}$ ( the matrix norm induced by the  vector norm $\left\|\cdot\right\|$ is also denoted by $\left\|\cdot\right\|$).
   Since $\lambda_s^{\nu_s}$ and $\lambda_s^{-\nu_s}$ are eigenvalues of the matrices $G_s^{\nu_s}$ and $G_s^{-\nu_s}$ respectively, it follows that $|\lambda_s^{\nu_s}|\leq\beta^{\nu_s}$ and $|\lambda_s^{-\nu_s}|\leq\alpha^{-\nu_s}$, and hence
	$\alpha\leq|\lambda_{s}|\leq\beta$, $s\in\mathbb N$. On the strength of that there exists a partial limit $\lambda$ of $\{\lambda_s\}$ such that
	$\alpha\leq|\lambda|\leq\beta$. Hence, the desired  subsequence exists. 
	\end{proof}

Now we want to show that there exist sequences $\{\lambda_{s}\}$  and $\{L_{s}\}$ such that   $\{L_{s}\}$  has a weak  limit $L$ which is a solution of (0.6) with $\lambda=\lim \lambda_{s}$.

Consider the Furier-series of $L_{s}$:
 $$
L_s(\hat\varphi)=\sum\limits_{\hat k\in \mathbb{Z}^{m-1}}t_{s\hat k}\exp(i(\hat
k\hat\beta\hat\varphi)),
$$
where
 $t_{s\hat k}=
 (\omega_2\ldots\omega_m)^{-1}\sum\limits_{j=0}^{\nu_s-1}f_{sj}\exp(-ij(\hat
 k\hat\beta\alpha_s))$ and
$(\hat k\hat\beta\alpha_s)=\sum\limits_{r=2}^m k_r\beta_r p_{sr}\omega_r/q_{sr}$.

{\bf Lemma 2.3}. {\it For each $s\in \mathbb{N}$, the mapping
  $\hat{k}\rightarrow t_{s\hat k}$ is  $\nu_s$-periodic w.r.t. $ k_r$ $(r=2,\dots,m)$}.

\begin{proof}
Recall that $\hat k=(k_2,\dots,k_m)$, $\beta_r=2\pi/\omega_r$, $\nu_s$ is the least common multiple of $q_{rs}$,  and  $\alpha_s=(p_{2s}\omega_2/q_{2s},\dots,p_{ms}\omega_m/q_{ms})$.  Since $\nu_s/q_{rs}\in \mathbb{Z}$ $(r=2,\dots,m)$,  we have
 $$
\begin{array}{lcl}
t_{s\,(k_2\dots,k_r+\nu_s,\dots,k_m)}=
\sum\limits_{j=0}^{\nu_k-1}f_{sj}\exp(-ij(\hat k\hat\beta\alpha_s+\nu_s(2\pi/\omega_r)p_{rs}\omega_r/q_{rs}))\\=
\sum\limits_{j=0}^{\nu_k-1}f_{sj}\exp(-ij(\hat k\hat\beta\alpha_s))= t_{s\hat k}.
\end{array}
$$
\end{proof}

 {\bf Lemma 2.4}. {\it For any $\hat j\in \mathbb{Z}^{m-1}$, in addition to $L_s$ and $ \lambda_{s}$, the pair
  $L_s^{\hat j}(\hat\varphi)=
   \sum\limits_{\hat k\in \mathbb{Z}^{m-1}}t_{s\hat k-\hat j}\exp(i(\hat
k\hat\beta\hat\varphi))$ and
  $\lambda_s^{\hat j}= \lambda_s\exp(-i(\hat j\hat\beta \alpha_s))$ also satisfies the system $(2.1)$.}

\begin{proof}
 Making a change $\hat{k}-\hat{j}=\hat{r}$ in the Fourier-series of $L_s^{\hat j}$, we obtain
 $L_s^{\hat j}=L_s\exp{i(\hat \beta\hat j\hat\varphi)}$. Then putting $L_s=L_s^{\hat j}\exp(-i(\hat \beta\hat j\hat\varphi))$
into (2.1), we conclude that the lemma is true.  \end{proof}

 {\bf Lemma 2.5.} {\it Let, for any
 $\hat{k}\in \mathbb{Z}^{m-1}$,  there exists
     $t_{\hat{k}}=\lim\limits_{s\rightarrow\infty}t_{s\hat{k}}$, and let there exists $C>0$ such that
      for all $s\in \mathbb{N}$ and $\hat{k}\in \mathbb{Z}^{m-1} $ the inequality
     $\|t_{s\hat{k}}\|\leq C$ holds. Then $L_s\rightarrow L=\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1} }t_{\hat{k}}\exp(i(\hat{k}\hat{\beta}\hat{\varphi}))$ and $L_s(\hat\varphi-\alpha_s)\rightarrow L(\hat\varphi-e\omega_1)$ weakly
      in $ P'^r_{n}(\hat{\omega})$  for $r\geq[\frac{m-1}{2}]+1$ $([t]$ stands for the integral part of $t)$.}

\begin{proof}
 Let $y\in  P_{n}^{r}(\hat{\omega})$, $y_k$ be the Fourier constants of $y$, and $r\geq[\frac{m-1}{2}]+1 $;
   then $\sum\limits_{\hat{k}\in\bf Z^{m-1} }\|y_{\hat{k}}\|$ converges   \cite {21}.
  We say that $\langle L_s, y\rangle_{\hat{\omega}}=\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1}  }t^T_{s\hat{k}}y_{-\hat{k}}$ and $\langle L, y\rangle_{\hat{\omega}}=\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1}  }t^T_{\hat{k}}y_{-\hat{k}}$. The proofs of these relations are similar since the boundedness of $\{t_{s\hat{k}}\}$ implies $\|t_{\hat{k}}\|\leq C$ for $\hat{k}\in \mathbb{Z}^{m-1}$ , so we prove the first one. It holds if  $y=y^h=\sum\limits_{\|\hat{k}\|\leq h }y_{\hat{k}}\exp(i(\hat{k}\hat{\beta}\hat{\varphi}))$ is a trigonometrical polynomial \cite[p.132]{8}. Besides, the series $\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1} }t^T_{s\hat{k}}y_{-\hat{k}}$ converges
 because  $|t^T_{s\hat{k}}y_{-\hat{k}}|\leq C\|y_{-k}\|$. Therefore,  we can take the limit in $\langle L_s, y^h\rangle_{\hat{\omega}}=\sum\limits_{\|\hat{k}\|\leq h}t^T_{s\hat{k}}y_{-\hat{k}}$ as $h\rightarrow\infty$ and get the desired relation.



 To prove the weak convergence  of  $\{L_s\}$ to $L=\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1} }t_{\hat{k}}\exp(i(\hat{k}\hat{\beta}\hat{\varphi}))$, we consider $\langle L_s-L,y\rangle_{\hat{\omega}}=\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1}
  }(t_{s\hat{k}}-t_{\hat{k}})^Ty_{-\hat{k}}$. For given $\varepsilon>0$ we find $M_1$, $M_2$ such that $\sum\limits_{\|\hat{k}\|>M_1 }\|y_{\hat{k}}\|<\varepsilon/(4C)$ and
  $\sum\limits_{\|\hat{k}\|\leq M_1 }\|t_k-t_{ks}\|<\varepsilon/(2C_1)$ for $s>M_2$, where $C_1=\max\{\|y_{\hat{k}}\|:\|\hat{k}\|<M_1\}$. Then
	$$
	|\sum\limits_{\hat{k}\in \mathbb{Z}^{m-1}
  }(t_{s\hat{k}}-t_{\hat{k}})^Ty_{-\hat{k}}|<C_1\sum\limits_{\|\hat{k}\|\leq M_1 }\|t_k-t_{ks}\|+2C\sum\limits_{\|\hat{k}\|>M_1 }\|y_{\hat{k}}\|<\varepsilon
	$$
	for $s>M_2$. Hence, $L_s\rightarrow L$ weakly as $s\rightarrow\infty$. By the same argument, we have
	$$
	\begin {array}{l}
	\langle L_s(\hat\varphi-\alpha_s)-L(\hat\varphi-e\omega_1),y\rangle_{\hat{\omega}}=\\
  \sum\limits_{\hat{k}\in \mathbb{Z}^{m-1}
}(t_{s\hat{k}}\exp(-i(\hat{k}\hat{\beta}\alpha_s))-t_{\hat{k}}\exp(-i(\hat{k}\hat{\beta}\omega_1)))^Ty_{-\hat{k}}\rightarrow0
\end{array}
$$
as $s\rightarrow\infty$.

 Owing to  the weak* sequential  completeness of  $ P'^r_{n}(\hat{\omega})$, we obtain
  $L\in P'^r_{n}(\hat{\omega})$ since $L_s\in P'^0_{n}(\hat{\omega})\subset P'^r_{n}(\hat{\omega})$. 
\end{proof}

{\bf Theorem 2.1}. {\it Let $A\in P^{r_0}_{n^2}(\omega)$ and
  $ r_0=[\frac{m-1}{2}]+1$; then there exist a distribution
   $L\in P'^{r_0}_{n}(\hat\omega)$  and
     $\lambda\neq0$ satisfying $(0.6)$.}
    \begin{proof}
 There exists a sequence   $\{L_s\}$ such that  
 $$
 \|t_{s\hat{0}}\|=1\geq\|t_{s\hat{k}}\|  \eqno(2.5)
 $$
  for $\hat{k}\neq\hat{0}$ and $s\in \mathbb{N}$. Indeed,  due to Lemma 2.3, for each $s\in \mathbb{N}$, there exists  $t_{s\hat{k}_s}$
such that $\|t_{s\hat{k}_s}\|\geq\|t_{s\hat{k}}\|$, $\hat{k}_s\neq \hat{k}$. Owing to Lemma 2.4, the distribution $L_s^{-\hat{k}_s}/\|t_{s\hat{k}_s}\|=\sum\limits_{\hat k\in \mathbb{Z}^{m-1}}\tilde{t}_{s\hat k}\exp(i(\hat
k\hat\beta\hat\varphi))$ satisfies (2.1) for $\lambda=\lambda_s \exp(i(\hat k_s\hat\beta \alpha_s))$; then $\tilde{t}_{s\hat k}=t_{s\hat k+\hat k_s}/\|t_{s\hat{k}_s}\|$ satisfy (2.5).

Let the Fourier constants of $L_s$ satisfy (2.5). Taking a subsequence if ne\-ces\-sary,  we may assume, in view of Lemma 2.2, that
$\lambda_{s}\rightarrow\lambda\neq0$. It follows from (2.5) that for each $\hat{k}\in \mathbb{Z}^{m-1}$   the sequence $\{t_{s\hat k}\}$ is bounded. Hence, there exists a sequence $\{s_p\}$ of positive integers such that    $\lim\limits_{p\rightarrow\infty}t_{s_p\hat{k}}=t_{\hat{k}}$ exists and $\left\|t_{\hat{k}}\right\|\leq 1$   for each $\hat{k}\in \mathbb{Z}^{m-1}$. By Lemma 2.5, we get 
$L_{s_p}\rightarrow L$ weakly in $ P'^{r_0}_{n}(\hat\omega)$.
Besides,  by Lemma 0.2,
       $X_{0}(\omega_{1},\cdot)\in P_{n^{2}}^{r_0}(\hat{\omega})$. Therefore, $X_{0}(\omega_{1},\cdot)L_{s_p}\rightarrow X_{0}(\omega_{1},\cdot)L$ weakly in $ P'^{r_0}_{n}(\hat\omega)$.  Then taking limit in (2.1), we get    $X_0(\omega_1,\hat\varphi)L(\hat\varphi-\hat{e}\varphi_1)=
       \lambda L(\hat\varphi)$. Moreover, $L\neq0$ since  $t_{\hat0}\neq0$. 
			\end{proof}
			
			Consider the set of sequences $\{\lambda_s\}$, where $\lambda_s$ is an eigenvalue of $G_s$ and denote by $\Lambda$ the set of partial limits of these sequences.



{\bf Theorem 2.2.} {\it Let $\lambda\in\Lambda$, $\mu=\omega_{1}^{-1}\ln\lambda$, and $A\in P^{r_0}_{n^2}(\omega)$; then there exists a periodic distribution
$\psi\in P^{r_0}_{n}(\omega)$ that satisfies  $(0.2_\mu)$.}
\begin{proof}

 If $A\in P^{r_0}_{n^2}(\omega)$, then, by Theorem 2.1, there exists  $L\in P'^{r_0}_{n}(\hat\omega)$ satisfying (0.6). Therefore, 
 Theorem 2.2 follows from  Lemma 1.3. 
\end{proof}




\section*{\large 3. Proof of Lemmas 0.1 and 0.2}
{\bf Lemma 0.1}.

\begin{proof}

System (0.5) has the unique continuous solution
$$
 \begin{array}{l}
X(\varphi;\varphi_{10})=
E+\int\limits_{0}^{\varphi_{1}-\varphi_{10}}A(\varphi-e\xi)\,d\xi\\+
\sum\limits_{k=2}^{\infty}\int\limits_{0}^{\varphi_{1}-\varphi_{10}}\int\limits_{0}^{\varphi_{1}-\varphi_{10}-\xi_{k}}\ldots\int\limits_{0}^{\varphi_{1}-\varphi_{10}-\xi_{2}-\ldots-\xi_{k}}
A(\varphi-
e\xi_{k}) A(\varphi-e(\xi_{k}+\xi_{k-1}))\dots\\
A\left(\varphi-e\sum\limits_{i=1}^{k}\xi_{i}\right)\,d\xi_{1}\dots\,d\xi_{k},
\end{array}\eqno(3.1)
$$
which one can get by the method of successive approximations. Let $\alpha,\beta\in\mathbb{R}$ be arbitrary. Then this series converges uniformly w.r.t. $\varphi_1\in[\alpha,\beta],$ $\hat{\varphi}\in\mathbb{R}^{m-1}$ because it has the majorant
$$
\sum\limits_{s=0}^\infty\|A\|_{P_{n^2}^0(\omega)}^s|\varphi_1-\varphi_{10}|^s/s! .
$$
This solution is $\omega_j$-periodic in $\varphi_j$ $(j=2,\dots,m)$ since so is the right part of (3.1). Besides,\\
 $X(\varphi+et;\varphi_{10})=
 \int\limits_{0}^{\varphi_{1}+t-\varphi_{10}}A(\varphi+e(t-\xi))X(\varphi+e(t-\xi));\varphi_{10})\,d\xi+E\\=
 [t-\xi=\zeta]=\int\limits_{\varphi_{10}-\varphi_{1}}^{t}A(\varphi+e\zeta)X(\varphi+e\zeta;\varphi_{10})\,d\zeta+E$.\\
 Therefore,
$$
dX(\varphi+et;\varphi_{10})/dt=A(\varphi+et)X(\varphi+et;\varphi_{10})\eqno(3.2)
$$
 and   $X(\varphi_{10},\hat{\varphi};\varphi_{10})=E$. Consequently, the matrix $X(\varphi+et,\varphi_{10})$, being a fundamental matrix of the system
$$
x'=A(\varphi+et)x,
$$
is non-singular for  $\varphi\in\mathbb R^m$ and $\varphi_{10}\in\mathbb R$. In particular,  $X_0(et)$ is the normalized fundamental matrix of system $(0.1_0)$.

The fact that the function $x(\varphi;\varphi_{10},x^{0})=
  X(\varphi;\varphi_{10})x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))$ is a solution of system (0.3) we prove by substituting it into this system. Taking into account  that  $x(\varphi-e\xi;\varphi_{10},x^{0})=
  X(\varphi-e\xi;\varphi_{10})x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))$, we get
	$$
	\begin{array}{l}
 X(\varphi;\varphi_{10})x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))\\=\int\limits_{0}^{\varphi_1-\varphi_{10}}A(\varphi-e\xi)X(\varphi-e\xi;\varphi_{10})x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))\,d\xi+x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10}))\\=(\int\limits_{0}^{\varphi_1-\varphi_{10}}A(\varphi-e\xi)X(\varphi-e\xi;\varphi_{10})\,d\xi+E)x^{0}(\hat{\varphi}-\hat{e}(\varphi_{1}-\varphi_{10})).
\end{array}
$$
This  is true since $X(\varphi;\varphi_{10})$ satisfies (0.4).
  
	To prove (c), note that $X(t+\omega_1,\hat{\varphi}+\hat{e}t;\varphi_{10})$ solves matrix equation (3.2). Hence, 
	$$
	X(t+\omega_1,\hat{\varphi}+\hat{e}t;\varphi_{10})=X(t,\hat{\varphi}+\hat{e}t;\varphi_{10})Q. \eqno (3.3)
	$$
	Putting $t=\varphi_{10}$ in this relation, we get
  $Q=X(\varphi_{10}+\omega_1,\hat{\varphi}+\hat{e}\varphi_{10};\varphi_{10})$. Then substituting $\varphi_{1}$ for $t$ and $\hat{\psi}$ for $\hat{\varphi}+\hat{e}\varphi_{1}$ in (3.3),  we get
  $X(\varphi_{1}+\omega_1,\hat{\psi};\varphi_{10})=
  X(\varphi_{1},\hat{\psi};\varphi_{10})X(\varphi_{10}+\omega_1,\hat{\psi}+\hat{e}(\varphi_{10}-\varphi_1);\varphi_{10})$. 
	\end{proof}
	{\bf Lemma 0.2}.
	
	\begin{proof}
	
	Let the matrix  $A$ be $r$-times continuously differentiable on $\mathbb{R}^m$. Then the matrix X, being a solution of system (3.2), is $r$-times continuously differentiable by  the theorem on differentiability of the solution w.r.t. parameter  $\varphi$   \cite[p.126]{23}. 
	
	\end{proof}
  
 
	
	It is noteworthy that if $\lambda_0\in \Lambda$ and $\mu_0=\omega_{1}^{-1}\ln\lambda_0$, then some system in the hull of $(0.1_{\mu_0})$ has a bounded solution. It follows from this that $\mu_0$ belongs to the Sacker-Sell spectrum of this system  and $\lambda_0$ belongs to the spectrum of the monodromy operator of this system. Besides, system $(0.1_{\mu_0})$ has a generalized conditionally periodic solution.  But these will be the objects of another paper.

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В данной статье мы обсуждаем некоторые проблемы коммутативной
алгебры.

\begin{theorem}
Формулировка теоремы.
\end{theorem}

\begin{proof}
Текст доказательства.
\end{proof}

\bigskip
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