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\title[Existence criterion for periodic solutions]
{Existence criterion for periodic solutions \\ to one class of systems of second order nonlinear ordinary differential equations}


\author{E. Mukhamadiev, A.N. Naimov}

\address{Ergashboy  Mukhamadiev,
\newline\hphantom{iii} Vologda State University,
\newline\hphantom{iii} Lenin str., 15,
\newline\hphantom{iii} 160000, Vologda, Russia
}
\email{emuhamadiev@rambler.ru}

\address{Alizhon Nabidjanovich Naimov,
\newline\hphantom{iii} Vologda State University,
\newline\hphantom{iii} Lenin str., 15,
\newline\hphantom{iii} 160000, Vologda, Russia
}
\email{naimovan@vogu35.ru}


\thanks{\sc E. Mukhamadiev, A.N. Naimov, Existence criterion for periodic solutions to one class of systems of second order nonlinear ordinary differential equations.}
\thanks{\copyright \ Mukhamadiev E., Naimov A.N. \ 2026}
\thanks{\rm The research is supported by Russian Science Foundation, project no.
 23-21-00032
 \\
 \indent\rm https://rscf.ru/project/23-21-00032/}
\thanks{\it Submitted January 12, 2025.}

\maketitle {\small
\begin{quote}
\noindent{\bf Abstract.} In this paper we study
  the existence of periodic solutions for a class of systems of second order ordinary differential equations with a separated  main nonlinear part. Taking into consideration the structure of  zero set for the main nonlinear part, we find new conditions ensuring an apriori estimate for periodic solutions.
Under the  apriori estimate, we formulate and prove a criterion for the existence of periodic solutions under any perturbation from a given class. The proof is made by using methods for calculating mapping degree of the vector field and employing the invariance of the existence of periodic solutions under a continuous varying of the main nonlinear part.
\medskip

\noindent{\bf Keywords:}  main nonlinear part, perturbation,
periodic solution, apriori estimate, guiding function,
mapping degree of the vector field.

\medskip
\noindent{\bf Mathematics Subject Classification:} {34B15, 45G15}

\end{quote}
}

\vskip20pt

\section{Introduction}

In this paper we study the existence of  $\om$-periodic solutions for
systems of nonlinear ordinary differential equations
\begin{equation}\label{f1}
x''=\prod_{j=1}^q\vp_j(x'-B_j(x))Q(x'- B(x)) + f(t,x,x'),
\quad t\in\mathds{R}, \quad x\in \mathds{R}^n.
\end{equation}
Here  $n$, $q$ are natural numbers   $n\geq 2$,
$\mathds{R}=(-\infty , +\infty)$, the mappings
$\vp_j: \mathds{R}^{n}\mapsto\mathds{R}$,
$B_j, Q, B: \mathds{R}^{n}\mapsto\mathds{R}^{n}$,
$f: \mathds{R}^{1+2n}\mapsto\mathds{R}^n$ are continuous and for some positive
   $\om$, $\a$, $\a_j$,
 $j=\overline{1, q}$,  where
 $\a_1+\ldots+\a_{q}+\a=m>1$,  they obey the following conditions

 \vskip5pt

 \begin{enumerate}
\item[1)] $\vp_j(\l y)\equiv\l^{\a_j}\vp_j(y)$,
$B_j(\l y)\equiv\l B_j(y)$, $j=\overline{1, q}$,
$B(\l y)\equiv\l B(y)$,
$Q(\l y)\equiv\l^{\a}Q(y)$ for all $\l>0$ (positive homogeneity condition);

 \vskip5pt


\item[2)] $\vp_j(y)>0$, $j=\overline{1, q}$ and   $Q(y)\neq 0$ for $y\neq 0$;

 \vskip5pt

 \item[3)] $f(t+\omega,y_1,y_2)\equiv f(t,y_1,y_2)$;

 \vskip5pt

 \item[4)] the growth order of $|f(t,y_1,y_2)|$ for large
$|y_1|+|y_2|$ is bounded by the limit
$$
\lim_{|y_1|+|y_2|\rightarrow\infty}\left(|y_1|+|y_2| \right)^{-m}
\max_{t\in [0, \omega]}\left|f(t,y_1,y_2) \right| = 0.
$$
 \end{enumerate}

The term
$$
\prod_{j=1}^q\vp_j(x'-B_j(x))Q(x'-B(x))
$$
in the right hand side of system \eqref{f1}, which is positive homogeneous in   $x$ and $x'$, is called the main nonlinear part, while  $f(t,x,x')$ is the perturbation. A solution $x\in C^2(\mathds{R};\mathds{R}^n)$ of system of equations  \eqref{f1} is called $\omega$-periodic if  $x(t+\omega)\equiv x(t)$.

In   \cite{1MN24}, \cite{2MN24}, \cite{3MN24} the existence of
$\om$-periodic solutions for systems of equations of the form
\begin{equation}\label{f2}
x''=P(t,x,x')+f(t,x,x'),
\quad t\in\mathds{R}, \quad x\in\mathds{R}^n,
\end{equation}
was investigated when the homogeneous nonlinear part $P$ and the perturbation $f$ satisfy Conditions~3,~4. The study was made in two steps. At the first step, there were found conditions, which  ensured an apriori estimate of $\om$-periodic solutions
$x$, i.e., the existence of a number $M_1>0$ independent of $x$  such that
\begin{equation}\label{f3}
\|x\|+\|x'\| < M_1,
\end{equation}
where
$\|x\|=\max\{|x(t)|: t\in [0, \omega] \}$.

At the second step, under conditions ensuring the apriori estimate, by
the methods for calculating the mapping degree of the vector field 
\cite{KZ75}, the existence of
$\om$-periodic solutions was established.


 Conditions ensuring   apriori estimate of \eqref{f3} were studied subject to the structure of  zero set of  main nonlinear
part  $P(t,y_1,y_2)$. There were considered cases, where the zero set consists of

\vskip5pt

\begin{enumerate}
	\item[a)] a single surface $y_2=B(t,y_1)$, and the system of equations
$x'=B(t,x)$ has no non--zero $\om$-periodic solutions;


\vskip5pt

\item[b)] a finitely many surfaces $y_2=B_j(t,y_1)$,
$j=\overline{1, q}$, between which there are no so--called switchings, which means that the autonomous system
$y' = P(t_0, x_0, y)$, $y\in\mathds{R}^n$,
for each fixed $t_0$ and $x_0$, has no non--stationary bounded
solution;

\vskip5pt

\item[c)] finitely many surfaces $y_2=B_j(t,y_1)$,
$j=\overline{1, q}$, between which the aforementioned switchings are possible.
\end{enumerate}

Cases a) and b) were studied in \cite{1MN24}, \cite{2MN24},
and Case c) for $n=2$ was considered in one example in  \cite{3MN24}.

System \eqref{f1} belongs to Case c).
By developing the approach from \cite{3MN24},  we find conditions   ensuring estimate   \eqref{f3} for any
perturbation of $f$. To derive the apriori estimate, we apply
the guiding function method and take into consideration the explicit structure of the set of zeros of   main nonlinear part. Under the   apriori estimate, we formulate and prove a criterion for the existence of $\om$-periodic solutions for any
perturbation of $f$. The proof applies and develops the methods of
\cite{2MN24}, \cite{NB24} and employs the invariance property of the existence of
$\om$-periodic solutions
under   continuous  varying of the main nonlinear part.


 In  \cite{KP18}, the existence of a periodic solution to nonlinear ordinary
differential equations of higher orders was investigated by means of the guiding function method.

\vskip20pt

\section{Main results}

Together with Conditions~1--4 we consider the following conditions:

\vskip5pt

\begin{enumerate}
	\item[5)] the system of equations $y'=Q(y)$, $y\in\mathds{R}^n$ has no
bounded non-zero solutions;

\vskip5pt


\item[6)] $B\in C^1(\mathds{R}^n\setminus\{0\};\mathds{R}^n)$ and
there exists a function $W\in C^1(\mathds{R}^n\setminus\{0\};\mathds{R})$
such that for any $y\in\mathds{R}^n\setminus\{0\}$ the   inequalities
\begin{equation*}
\langle B(y) , W'(y) \rangle > 0,\qquad \langle B_j(y) , W'(y) \rangle > 0,\qquad j=\overline{1, q},
\end{equation*}
hold, where $W'(y)$ is the derivative (gradient) of $W(y)$,
$\langle y , z \rangle = y_1z_1+\ldots+y_nz_n$ is the scalar product
in $\mathds{R}^n$;


\vskip5pt


\item[7)] $B_j(y)=A_jy$, $j=\overline{1, q}$ and $B(y)=Ay$,  where
$A$, $A_j$, $j=\overline{1, q}$ are mutually commuting matrices
and $\det\left(\exp(\om D)-I\right) \neq 0$ for  each convex combination
$D=\mu_0A+\mu_1A_1+\ldots+\mu_qA_q$.
\end{enumerate}

\vskip5pt

We introduce the family of mappings
$$
P_{\l}(y_1,y_2)=\prod_{j=1}^q\vp_j(y_2-B_{j,\l}(y_1))Q(y_2-B(y_1)),
\quad y_1,y_2\in\mathds{R}^n, \quad \l\in [0, 1],
$$
where $B_{j,\l}(y)=(1-\l)B_{j}(y)+\l B(y)$. Apriori estimate \eqref{f3} for $\om$-periodic solutions to system \eqref{f1} is implied by the  following theorem.

\begin{theorem}\label{th1}
Let Conditions 1, 2, 5 and one of Conditions 6 or 7 be satisfied.
Then there exist positive numbers $\si$ and $M$ such that
for each $\om$-periodic vector--function
$x\in C^2(\mathds{R};\mathds{R}^n)$ satisfying the inequality
$\|x\|+\|x'\|>M$, the    estimate
\begin{equation}\label{f4}
\|x''-P_{\l}(x,x')\| \geq \si \left( \|x\|+\|x'\| \right)^m
\end{equation}
holds  for each $\l\in [0, 1]$.
\end{theorem}

We  denote by $\g(Q)$, $\g(B)$ the mappings degrees of the vector fields $Q, B:\  \mathds{R}^{n}\mapsto\mathds{R}^{n}$ on the unit sphere
$|y|=1$ of the space $\mathds{R}^{n}$ \cite{KZ75}.



\begin{theorem}\label{th2}
Under the assumptions of Theorem \ref{th1},  there exist
$\om$-periodic solutions to system \eqref{f1}
for any perturbation $f$ if and only if
$\g(Q)\g(B)\neq 0$.
\end{theorem}

In the proof of Theorem~\ref{th2}, we apply and develop the methods from \cite{2MN24}, \cite{NB24} and employ the invariance of the existence of $\om$-periodic solutions to the system of equations
$$
x'' = P_{\l}(x,x')  + f(t,x,x'),
\quad t\in\mathds{R}, \quad x\in \mathds{R}^n,
$$
under  continuous varying of  $\l\in [0, 1]$.

Examples of positive  homogeneous mappings $Q$ satisfying
Condition 5 were provided in \cite{MN22}. For these mappings, an efficient algorithm for computing $\g(Q)$ was proposed.


\vskip20pt

\section{Proof of Theorem~\ref{th1}}

Suppose that  estimate \eqref{f4} fails.
Then there exist sequences
$\l_k\in [0, 1]$, $x_k\in C^2(\mathds{R};\mathds{R}^n)$,
$k=1, 2, \ldots, $ such that
\begin{align*}
&x_k(t+\om)\equiv x_k(t), \qquad  r_k:=\|x_k\|+\|x_k'\| \rightarrow \infty, \qquad k \rightarrow \infty,\\
&\|x_k''-P_{\l_k}(x_k,x_k')\| < \frac{1}{k} \Big(\|x_k\|+\|x_k'\|\Big)^m.
\end{align*}
We  consider the vector functions $y_k(t)=r_k^{-1}x_k(t)$, $t\in\mathds{R}$, $k=1, 2, \ldots$. For these vector functions we have
\begin{align}
&y_k\in C^2(\mathds{R};\mathds{R}^n), \quad    y_k(t+\om)\equiv y_k(t),  \quad \|y_k\|+\|y_k'\|=1, \nonumber \\
\label{f5}
&r_k^{1-m}y_k''(t)=P_{\l_k}(y_k(t),y_k'(t)) + o(1), \quad t\in\mathds{R}.
\end{align}
Without loss of generality, we can assume that
$\l_k\rightarrow\l_0$ and $\|y_k-y_0\|\rightarrow 0$
as $k\rightarrow\infty$.

If $y_0(t)\equiv 0$, then for the sequence of vector functions
$z_k(t)=y_k'(t_k+tr_k^{1-m})$,  $k=1, 2, \ldots $,
where $|y_k'(t_k)|=\|y_k'\|\rightarrow 1$ as $k\rightarrow\infty$,
we have
\begin{align*}
&|z_k(t)| \leq |z_k(0)|, \qquad z_k'(t)=P_{\l_0}(0, z_k(t)) + o(1), \quad t\in\mathds{R},\\
&|z_k(0)|\rightarrow 1, \qquad  k\rightarrow\infty.
\end{align*}
Passing to the limit, we obtain a non--zero bounded solution to the system of equations
\begin{equation}\label{f6}
z' = P_{\l_0}(0, z(t)), \quad  z\in\mathds{R}^n.
\end{equation}
On the other hand, one can directly verify that, due to Conditions~2 and~5, system \eqref{f6}
cannot have nonzero bounded solutions.
Therefore, $y_0(t)\not\equiv 0$. Proceeding then as in the proof of Theorem~1 in  \cite{1MN24}, we get
\begin{equation}\label{f7}
y_0(t) \neq 0, \quad t\in \mathds{R}.
\end{equation}
In what follows we can suppose that  $|y_k(t)|>\frac{1}{2}|y_0(t)|$ for all
$t$ and $k$.

Now we employ the fact that, due to Condition 5 and
  \cite[Thms. 2.2, 2.3]{M99}, for $Q$
there exists a so--called guiding function
$V\in C^1(\mathds{R}^n;\mathds{R})$ with the  properties
$$
V(\l y)\equiv\l^2V(y), \qquad
\langle  Q(y) , V'(y)  \rangle  > c_1 |y|^{\a+1}  > 0,
\qquad y\in\mathds{R}^n\setminus\{0\},
$$
where  $V'(y)$ is the derivative (gradient) of $V(y)$.
We calculate the scalar product of   both sides of \eqref{f5} with
$V'(y_k'(t)-B(y_k(t)))$
and integrate from $0$ to $\om$
\begin{align*}
&r_k^{1-m}\int\limits_0^{\om}\langle y_k''(t) ,
V'(y_k'(t)-B(y_k(t))) \rangle dt\\
& \qquad =\int\limits_0^{\om}\langle P_{\l_0}(y_k(t),y_k'(t)) ,
V'(y_k'(t)-B(y_k(t))) \rangle dt + o(1).
\end{align*}
We estimate the right hand side from below
\begin{align*}
&\int\limits_0^{\om}\langle P_{\l_0}(y_k(t),y_k'(t)) ,
V'(y_k'(t)-B(y_k(t))) \rangle dt \\
&\qquad \geq
c_1 \int\limits_0^{\om}\prod_{j=1}^q\vp_j(y_k'(t)-B_{j,\l_0}(y_k(t)))
|y_k'(t)-B(y_k(t))|^{\a+1}dt,
\end{align*}
and represent the integrand in the left hand side as
$$
\langle y_k'' , V'(y_k'-B(y_k)) \rangle
=\left( V(y_k'-B(y_k)) \right)' +
\langle B'(y_k)y_k' , V'(y_k'- B(y_k)) \rangle.
$$
As a result, taking into consideration the boundedness of $y_k$, $y_k'$  and passing to the limit, we obtain
$$
\int\limits_0^{\om}\prod_{j=1}^q\vp_j(y_k'(t)-B_{j,\l_0}(y_k(t)))
|y_k'(t)-B(y_k(t))|^{\a+1}dt \rightarrow 0, \quad  k\rightarrow\infty.
$$
Without loss of generality, we can suppose that for all  $t\in [0, \om]$
$$
\prod_{j=1}^q\vp_j(y_k'(t)-B_{j,\l_0}(y_k(t)))
|y_k'(t)-B(y_k(t))|^{\a+1} \rightarrow 0, \quad  k\rightarrow\infty.
$$
For each $\e>0$ for the measure of set
$$
E_{k,\e}=\{ t\in [0, \omega] : |y_k'(t)-B(y_k(t))|\geq\e,
|y_k'(t)-B_{j,\l_0}(y_k(t))|\geq\e, j=\overline{1, q}
\}
$$
we have
$\mes\left(E_{k,\e}\right)\rightarrow 0$, $k\rightarrow \infty$.

We fix $\e>0$ and introduce the sets
\begin{align*}
&E_{k,\e}^0=\{ t\in [0, \om] :
|y_k'(t)-B_{0,\l_0}(y_k(t))|<\e \},\qquad   B_{0,\l_0}=B,\\
&E_{k,\varepsilon}^l=\{ t\in [0, \om] :
 |y_k'(t)-B_{l,\l_0}(y_k(t))|<\e,
|y_k'(t)-B_{j,\l_0}(y_k(t))|\geq\e, j=\overline{0, l-1}
\}, \quad l=\overline{1, q}.
\end{align*}
The sets $E_{k,\e}$, $E_{k,\e}^0$, $\ldots$, $E_{k,\e}^q$ are mutually disjoint and their union coincides with  $[0, \om ]$.


If Condition 6 holds,   we choose a positive $c_2$ satisfying the inequalities
  $$\langle B(y), W'(y)  \rangle \geq c_2, \quad \langle B_j(y) , W'(y)  \rangle \geq c_2, \quad j=\overline{1, q}$$ for each
$y\in\mathds{R}^n$, $a_0\leq |y|\leq 1$,
where $a_0=0,5\min\{|y_0(t)|: t\in [0, \om]\}$. We then have
\begin{align*}
0=&W(y_k(\om)) - W(y_k(0))=\int\limits_0^{\om}\Big( W(y_k(t)) \Big)'dt =
\int\limits_0^{\om}\langle  y_k'(t) , W'(y_k(t)) \rangle dt \\
=&\sum_{j=0}^q\int\limits_{E_{k,\e}^j}\langle  B_{j,\l_0}(y_k(t)) ,  W'(y_k(t) \rangle dt
+ \sum_{j=0}^q\int\limits_{E_{k,\e}^j}\langle y_k'(t) - B_{j,\l_0}(y_k(t)) ,
 W'(y_k(t) \rangle dt \\
 &+  \int\limits_{E_{k,\e}} \langle  y_k'(t) , W'(y_k(t) \rangle dt
>  c_2\left(\om  - \mes\left(E_{k,\e}\right) \right)
 -\Big(\e\om + \mes\left(E_{k,\e}\right)\Big)
 \max_{a_0\leq |y|\leq 1}|W'(y)|.
\end{align*}
Taking into consideration the limit
$\mes\left(E_{k,\e}\right)\rightarrow 0$ as  $k\rightarrow \infty$,
we can choose
the number $\e>0$ and the index $k$  so that
$$
c_2\left(\om  - \mes\left(E_{k,\e}\right) \right)
 -\Big(\e\om + \mes\left(E_{k,\e}\right)\Big)
 \max_{a_0\leq |y|\leq 1}|W'(y)| > 0,
$$
which is the contradiction.

If Condition 7 holds,  we define
\begin{align*}
&D_{k,\e}(t)=\left\{
\begin{aligned}
&A, && t\in E_{k,\e}\cup E_{k,\e}^0,
\\
&A_l, && t\in E_{k,\e}^l, l\geq 1,
\end{aligned}
\right.\\
&h_{k,\e}(t)=y_k'(t)-D_{k,\e}(t)y_k(t).
\end{align*}
It is easy to verify that
\begin{align*}
&|h_{k,\e}(t)| < \e, \qquad
t\in E_{k,\e}^l, \qquad l=\overline{0, q},\\
&D_{k,\e}(t) \int\limits_0^s D_{k,\e}(\tau)d\tau =
\int\limits_0^s D_{k,\e}(\tau)d\tau D_{k,\e}(t),
\qquad t,s\in [0, \om ].
\end{align*}
In view of these properties for $y_k$ we have
\begin{align*}
&y_k(\om)=e^{\int\limits_0^{\om}D_{k,\e}(\tau)d\tau}y_k(0)+ \int\limits_0^{\om}e^{\int\limits_t^{\om}D_{k,\e}(\tau)d\tau}h_{k,\e}(t)dt,\\
&\left|\left(e^{\int\limits_0^{\om}D_{k,\e}(\tau)d\tau} - I\right) y_k(0)\right| < N\e \quad \text{for} \quad  k>k_0(\e),
\end{align*}
where $N$ is independent of $k$  and $\e$,
$$
\left|\left(
e^{\mu_{k,\e}A+\mu_{k,\e}^{(0)}A + \mu_{k,\e}^{(1)}A_1 + \ldots +
\mu_{k,\e}^{(q)}A_q} - I
\right) y_k(0)
\right| < N\e, \quad k>k_0(\e).
$$
Here
\begin{equation*}
 \mu_{k,\e}=\mes\left(E_{k,\e}\right),\qquad \mu_{k,\e}^{(l)}=\mes\left(E_{k,\e}^l\right),\qquad
\mu_{k,\e}+\mu_{k,\e}^{(0)} + \ldots + \mu_{k,\e}^{(q)} = \omega,\qquad \mu_{k,\e}\rightarrow 0,\quad k\rightarrow \infty.
\end{equation*}
Passing to the limit as  $k\rightarrow \infty$, we obtain
$$
\left|\left(
e^{\mu_{\e}^{(0)}A +\mu_{\e}^{(1)}A_1 + \ldots + \mu_{\e}^{(q)}A_q} - I
\right) y_0(0)
\right| \leq  N\e,
$$
where $\mu_{\e}^{(0)}+ \ldots + \mu_{\e}^{(q)} = \om$. Tending   $\e$ to zero, we arrive at the identities
$$
\left(
e^{\mu_{0}^{(0)}A + \mu_{0}^{(1)}A_1 +\ldots + \mu_{0}^{(q)}A_q} - I
\right) y_0(0) = 0, \qquad \mu_{0}^{(0)} + \ldots + \mu_{0}^{(q)} = \om.
$$
Since $y_0(0)\neq 0$ (by \eqref{f7}), we find
$$
\det\left(
e^{\mu_{0}^{(0)}A + \mu_{0}^{(1)}A_1 +\ldots + \mu_{0}^{(q)}A_q} - I
\right) = 0,
$$
and this contradicts Condition~7. The proof of Theorem~\ref{th1} is complete.


\vskip5pt

Let us verify that under the assumptions of Theorem \ref{th1},
 estimate \eqref{f3} is implied by \eqref{f4}. Indeed, for each
$\om$-periodic solution of   system \eqref{f1}, either $\|x\|+\|x'\|\leq M$,
or $\|x\|+\|x'\|>M$, and it follows  from \eqref{f4}  that
$$
\si\left( \|x\|+\|x'\| \right)^m  \leq  \|x''-P_{0}(x,x')\| =
\|f(\cdot,x,x')\|.
$$
By Condition~4,
$$
\|f(\cdot,x,x')\| <
\frac{\si}{2}\left( \|x\|+\|x'\| \right)^m  +  M_{f,\si}.
$$
Therefore, in the case   $\|x\|+\|x'\|>M$ the estimate
$$
\left( \|x\|+\|x'\| \right)^m <  \frac{2}{\si}M_{f,\si}
$$
holds, while in the general case
$$
\left( \|x\|+\|x'\| \right)^m  <  M^m  + \frac{2}{\si}M_{f,\si}.
$$

\vskip20pt

\section{Proof of Theorem~\ref{th2}}

Following the lines of \cite[Thm. 3]{1MN24}, we first  prove the invariance of the existence of
$\om$-periodic solutions to the system of equations
\begin{equation}\label{f8}
x'' = P_{\l}(x,x')  + f(t,x,x'),
\quad t\in\mathds{R}, \quad x\in \mathds{R}^n,
\end{equation}
under a continuous varying  of $\l\in [0, 1]$.

\begin{lemma}\label{lm1}
The system \eqref{f8} has $\om$-periodic solutions
for $\l=0$ and any perturbation $f$ if and only if
has $\om$-periodic solutions for $\l=1$ and any perturbation $f$.
\end{lemma}

\begin{proof}
Let $\l_1, \l_2\in [0, 1]$ and each vector function
$y\in C^1([0, \om];\mathds{R}^n)$ satisfies the inequality
$$
\|P_{\l_{1}}(y,y') - P_{\l_{2}}(y,y')\|
\leq  \frac{1}{4}\si
\Big(\|y\| + \|y'\|\Big)^m.
$$
Using  estimate \eqref{f4}, we are going to show that
for   system \eqref{f8}
with $\l=\l_1$  replaced by $\l=\l_2$ the existence of  $\om$-periodic solution is preserved
for any perturbation $f$. This will prove the lemma.



We take an arbitrary perturbation $\tilde{f}$. Each vector  function
$y\in C^1([0, \om];\mathds{R}^n)$ satisfies the inequality
$$
\|\tilde{f}(\cdot,y,y')\| \leq \frac{1}{4} \si
\Big(\|y\| + \|y'\|\Big)^m + M_{\tilde{f}, \si},
$$
where $M_{\tilde{f}, \si}$ is a positive number depending only on $\tilde{f}$
and $\si$. We choose a number
$$
L > \max \left(
M , \left(
2\si^{-1}M_{\tilde{f}, \si}
\right)^{\frac{1}{m}}
\right)
$$
and define the perturbation
$$
g_L(t,y_1,y_2)=\tilde{f}(t,y_1,y_2)+\eta (|y_1|+|y_2|)
\left(
P_{\l_{2}}(y_1,y_2) - P_{\l_{1}}(y_1,y_2)
 \right),
$$
where $\eta(s)\in C(\mathds{R})$, $0\leq \eta(s)\leq 1$ for all
$s\in\mathds{R}$, $\eta(s)=1$ for $|s|\leq L$ and $\eta(s)=0$ for $|s|\geq L+1$.


Let $x$ be a $\om$-periodic solution to system \eqref{f8} for
$\l=\l_{1}$ and perturbation $f=g_L$. We are going to verify that
$\|x\|+\|x'\|\leq L$; then it implies that
$x$ is also a $\om$-periodic solution to system \eqref{f8} for $\l=\l_{2}$ and the perturbation $f=\tilde{f}$.
Indeed, if $\|x\|+\|x'\| > L$,
then, according to estimate \eqref{f4} and the choice of the number $L$, we have
\begin{align*}
 \si\Big(\|x\|+\|x'\|\Big)^m <&
 \|x''- P_{\l_{1}}(x,x')\| \leq
\|\tilde{f}(\cdot,x,x')\| \\
&+ \|P_{\l_{1}}(x,x') - P_{\l_{2}}(x,x')\|
 \leq  0,5 \si \Big(\|x\|+\|x'\|\Big)^m
  + M_{\tilde{f}, \si}.
\end{align*}
This leads us to the contradiction
$$
\|x\|+\|x'\| \leq  \max \left(
2\sigma^{-1}M_{\tilde{f}, \si}
\right)^{\frac{1}{m}} < L.
$$
The proof is complete.
\end{proof}

Thus,  system \eqref{f1} has $\om$-periodic
solutions for any perturbation $f$ if and only if
the system of equations
\begin{equation}\label{f9}
 x'' = Q_1(x'-B(x))  + f(t,x,x'),
\quad t\in\mathds{R}, \quad x\in \mathds{R}^n,
\end{equation}
with $Q_1(y)=\vp_1(y)\cdot\ldots\cdot\vp_q(y) Q(y)$, has $\om$-periodic solutions for any perturbation $f$.
At the same time,  the condition holds

\vskip5pt
\begin{enumerate}
	\item[8)] the system of equations $y'=Q_1(y)$, $y\in\mathds{R}^n$ has no bounded non--zero solutions.
\end{enumerate}
\vskip5pt

Under Condition~6 the estimate
$\langle B(y) , W'(y) \rangle > 0$,  $y\in\mathds{R}^n\setminus\{0\},$
implies

\vskip5pt

\begin{enumerate}
	\item[9)] the system of equations $y'=B(y)$, $y\in\mathds{R}^n$  has no bounded non--zero solutions.
\end{enumerate}

If Condition 7 is satisfied, then $B(y)=Ay$. In this case, the mapping $B(y)$
by the formula $Ay+\l\e y$, $\l\in[0, 1]$ is reduced to the mapping
$Ay+\e y$ with matrix $A+\e I$, which has no purely imaginary eigenvalues.
Under this change, according to   \cite[Thm. 3]{1MN24},
the existence of $\om$-periodic solutions to system \eqref{f9}
under any perturbation $f$ is preserved. This is why, in what follows, without loss of generality, we can suppose that Condition 9 is  satisfied under Condition 7.
Now Theorem~\ref{th2} is implied by the next lemma.

\begin{lemma}\label{lm2}
System \eqref{f9} possesses $\om$-periodic solutions
for any perturbation $f$
if and only if
$\g(Q)\g(B)\neq 0$.
\end{lemma}

\begin{proof} \textit{Necessity.} Let $\g(Q)\g(B) = 0$. We are going to prove that for some
perturbation $f$,  system \eqref{f9} has no $\om$-periodic
solutions. In order to do this, we follow the lines of \cite[Thm. 2, Necessity]{NB24}.


By the known properties of mapping degree of vector fields  we have
$$
\g(Q_1(-B))=\g(Q)\g(-B)=(-1)^n\g(Q)\g(B)=0.
$$
Then by   \cite[Thm. 5.2]{KZ75} the mapping
   $Q_1(-B)$ can be continuously  continued inside the ball $|y|<1$
without zeros
\begin{align*}
F(y)=Q_1(-B(y)) \quad \text{for} \quad |y|=1 \quad \text{and} \quad F(y)\neq 0 \quad \text{for} \quad |y|<1.
\end{align*}
We let $g(y)=F(y)-Q_1(-B(y))$ for $|y|\leq 1$ and $g(y)= 0$
for $|y|>1$. It is easy to verify that
\begin{equation}\label{f10}
 |y_2| +  |Q_1(y_2-B(y_1))+g(y_1)|  > 0 \quad
 \text{for all} \quad y_1, y_2\in\mathds{R}^n.
\end{equation}

Let us show that for some $\omega_0>0$ the system of equations
\begin{equation}\label{f11}
x''=Q_1(x'-B(x))+g(x),
\quad x\in\mathds{R}^n,
\end{equation}
has no $\omega_0$-periodic solutions.
Indeed, otherwise there exists a sequence of
$\displaystyle\Big(\frac{1}{k}\Big)$-periodic solutions $x_k$, $k=1, 2, \ldots$
to system \eqref{f11}. Following lines of the proof of Theorem~\ref{th1},
it can be shown that
$$
\sup_{k\geq 1}\sup_{t\in\mathds{R}}
\Big(|x_k(t)|+|x_k'(t)|\Big) < \infty.
$$
Taking into consideration \eqref{f11}, we can assume that
the sequences of vector functions $x_k$, $x_k'$ converge uniformly
on each interval $[a , b]$
$$
\sup_{a\leq t \leq b}
\Big(|x_k(t)-x_0(t)|+|x_k'(t)-x_0'(t)|\Big) \rightarrow 0 , \qquad
k \rightarrow\infty.
$$
Moreover, due to the periodicity of $x_k$, $\displaystyle\Big(\frac{1}{k}\Big)$
for all $z_1, z_2\in\mathds{R}^n$, $k=1, 2, \ldots$
we have
$$
\frac{1}{k}\int\limits_0^{\frac{1}{k}}\Big(
\langle x_k'(t) , z_1 \rangle +
\langle Q(x_k(t),x_k'(t))+g(x_k(t)) , z_2 \rangle
\Big) dt = 0.
$$
Passing to the limit, we get
$$
\langle x_0'(0) , z_1 \rangle +
\langle Q(x_0(0),x_0'(0))+g(x_0(0)) , z_2 \rangle
  = 0
$$
for all  $z_1, z_2\in\mathds{R}^n$. This contradicts \eqref{f10}.

In   system \eqref{f11} we make the change $x(t)=r^{\b}y(r^{\nu}t)$,
where $\displaystyle r=\Big(\frac{\om}{\om_0}\Big)^{\frac{1}{\nu}}$,
$\b=0$, $\nu=1$ for $m=2$ and $\b=1$, $\displaystyle\nu=\frac{1-m}{m-2}$ for $m\neq 2$. This gives the system of equations
\begin{equation}\label{f12}
y''=Q_1(y'- r^{-\nu}B(y))+r^{-(\b+\nu)m}g(r^{\b}y),
\quad y\in\mathds{R}^n.
\end{equation}
If a solution $y(t)$ of  system \eqref{f12} is $\om$-periodic, then
the solution $x(t)=r^{\b}y(r^{\nu}t)$ of   system \eqref{f11}
is $\om_0$-periodic. By the above proven facts, this implies that
  system \eqref{f12} has no $\om$-periodic solutions.


By Conditions 8 and 9, the family of mappings $Q_1(y_2-((1-\l)r^{-\nu}+\l)B(y_1))$, $\l\in[0, 1]$ satisfies the assumptions of Theorem~3 in \cite{1MN24}. According to this theorem, since
  system \eqref{f12} has no $\omega$-periodic solutions,
then for some perturbation $f$   system \eqref{f9} also
has no $\om$-periodic solutions.


\textit{Sufficiency.}  Let $\g(Q)\g(B)\neq 0$. Then
$\g(Q_1(-B))\neq 0$ and all assumptions of Theorem~2 in \cite{2MN24} are satisfied by system \eqref{f9}. According to this theorem,  system \eqref{f9} has $\om$-periodic solutions for any perturbation $f$.  The proof is complete.
\end{proof}


\vskip20pt

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\noindent\hrr{https://doi.org/10.3103/S1066369X24700683}


\bibitem{2MN24}    E. Mukhamadiev, A.N. Naimov. \textit{On the solvability of a periodic problem for a system of second--order nonlinear ordinary differential equations} //
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\end{thebibliography}

\end{document}









