\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 04, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2016/04\hfil Non-existence of periodic solutions]
{Non-existence of periodic solutions
to non-autonomous second-order differential equation
with discontinuous nonlinearity}

\author[A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva
 \hfil EJDE-2016/04\hfilneg]
{Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva}


\address{Alexander M. Kamachkin \newline
Saint Petersburg State University,
7-9, University emb., 199034  St. Petersburg, Russia}
\email{a.kamachkin@spbu.ru}

\address{Dmitriy K. Potapov \newline
Saint Petersburg State University,
7-9, University emb., 199034  St. Petersburg, Russia}
\email{d.potapov@spbu.ru}

\address{Victoria V. Yevstafyeva \newline
Saint Petersburg State University,
7-9, University emb., 199034  St. Petersburg, Russia}
\email{v.evstafieva@spbu.ru}

\thanks{Submitted June 1, 2015. Published January 4, 2016.}
\subjclass[2010]{34A34, 34A36, 34H05, 93C73}
\keywords{Non-autonomous differential equation; discontinuous nonlinearity;
\hfill\break\indent periodic solution; non-existence of solutions}

\begin{abstract}
 We consider a second-order differential equation with discontinuous
 nonlinearity and sinusoidal external influence, and obtain conditions
 for the non-existence of periodic solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\allowdisplaybreaks


\section{Introduction}

Relay control systems have been studied for a long time
(see, e.g.~\cite{tsypkin}--\cite{kam}). Nevertheless
automatic control systems with relay nonlinearity and
external influence are of interest nowadays \cite{vica2015}, since there are
still open questions. In this article the automatic system is described by a
second-order differential equation
with time-independent coefficients. There are both a control function
and an external influence function in the right-hand side of the equation.
We consider the signum function as a relay control model.
Since the time of  Andronov \cite{andronov} this model have been used in
automatic control systems. The system may be free from
periodic modes in case if there is no external influence. Moreover, it is possible
that the system does not have periodic modes when relay
control is absent. However stabilization of the system may occur under
both external influence and control.
By stabilization we mean existence of stable oscillations of the
certain configuration and given period. If we can not affect on
the dynamics of the object and the parameters of the external influence, then
at switching the step height $c$ of function $u=\frac{c}{2}\operatorname{sgn} x$
influences significantly on the system dynamics.
Taking account the stabilization type above, we set the task for choosing
parameter $c$ that depends on the other parameters of the automatic control system.

In recent years the second-order differential equations with
discontinuous nonlinearities have been considered
in \cite{jacquemard}--\cite{pot43}.
Applied problems for such equations are discussed in \cite{pot18,pot40}.
Control problems for systems with distributed parameters and
discontinuous nonlinearity are studied in \cite{pot36}.
On signification of the research for non-existence of the solutions to the
problem with singular potential, see \cite{catrina}.
Classification of discontinuities for real-valued functions is well described
in \cite{radulescu} giving as an example for the function with discontinuity
of the first kind at zero the signum function to be defined as follows:
$$
\operatorname{sgn}x=\begin{cases}
-1 & \text{if }  x<0,\\
0  & \text{if }  x=0,\\
1  & \text{if }  x>0.
\end{cases}
$$

This article continues research stated above.
We study the problem when there are no
periodic solutions of the automatic control system with
a relay and a sine function as external influence.
The system dynamics is described by the equation
\begin{equation}
x''+a_1x'+a_0x=\frac{c}{2}\operatorname{sgn}x+\beta\sin(\gamma t),
\label{f1}
\end{equation}
where $a_1$, $a_0$, $\beta$ are real constants, parameter
$c>0$, the external influence frequency $\gamma>0$. Function $\operatorname{sgn}x$
describes the relay control.

We note that equation \eqref{f1} is investigated
when $a_1=a_0=0$ and $c<0$ in \cite{jacquemard}.
The autonomous equations of the form \eqref{f1} are studied
in \cite{nyzhnyk}--\cite{pot18}.
In \cite{pot40}, we consider nonperiodic external influence
for the one-dimensional Lavrent'ev model described by the equation
$-x''=\mu \operatorname{sgn} x$, where parameter $\mu>0$ means a vorticity.
Hence this paper develops \cite{jacquemard}--\cite{pot40}.

\section{Solution of the problem}

Let  $\lambda_1$, $\lambda_2$ be the real roots of the equation
$\lambda^2+a_1\lambda+a_0=0$ and $\lambda_1\neq 0$, $\lambda_2\neq 0$,
$\lambda_1\neq\lambda_2$, i.e. $a_0\neq 0$ and $a_1^2>4a_0$.
Also, let $a_1\neq 0$ and $\lambda_1<\lambda_2$.
Then the general solution of equation \eqref{f1} has the form
\begin{equation}
x(t)=C_1e^{\lambda_1t}+C_2e^{\lambda_2t}\pm C_0+Q_1\cos(\gamma t)+
Q_2\sin(\gamma t),
\label{f2}
\end{equation}
where $C_1$, $C_2$ are the constants we have to define,
$$
C_0=\frac{c}{2a_0},\quad
Q_1=-\frac{a_1\beta\gamma}{(\gamma^2-a_0)^2+a_1^2\gamma^2},\quad
Q_2=-\frac{\beta(\gamma^2-a_0)}{(\gamma^2-a_0)^2+a_1^2\gamma^2}.
$$
Since $a_1\neq 0$ and $\gamma>0$, then $(\gamma^2-a_0)^2+a_1^2\gamma^2\neq 0$
and so that solution \eqref{f2} exists.
Thus the following theorem holds.


\begin{theorem} \label{thm1} 
 Let $a_1\neq 0$, $a_0\neq 0$, $a_1^2>4a_0$, $c>0$,
$\beta\in\mathbb R$, and $\gamma>0$.
Then equation \eqref{f1} has a solution of the form \eqref{f2}
on half-planes $x>0$ and $x<0$ from phase plane $(xOx')$.
\end{theorem}

We assume that  \eqref{f1} has a periodic solution with period
$T$. If we are interested in the periodic solution with a desirable period,
we set period $T$.
The closed trajectory relating to a jump on axis
$Ox_2$ at switching of $u=\frac{c}{2}\operatorname{sgn} x$ corresponds to
the periodic solution on plane $(x_1Ox_2)$, where $x_1=x$, $x_2=x'$.
The closed trajectory consists of two phase trajectory pieces by virtue of the
different systems
\begin{equation}
\begin{aligned}
x_1'&=x_2,\\
x_2'&=-a_0x_1-a_1x_2\pm\frac{c}{2}+\beta\sin(\gamma t).
\end{aligned}\label{ff1}
\end{equation}
Let $T=t_1+t_2$, where $t_1$ corresponds with the trajectory part at
$x_1>0$, and $t_2$ corresponds with the trajectory part at $x_1<0$.
Axis $Ox_2$ is a straight line on which there is a gluing together of the closed
trajectory pieces and the other trajectories by virtue of systems \eqref{ff1}.
Therefore we suppose that the closed trajectory consists of two pieces that are
located on half-planes $x_1>0$ or $x_1<0$.

Let the point with coordinates
$x_1=0$, $x_2=x_0'$ belong to the closed
trajectory the image point passes for $T$.
We take this point as initial. Then, taking into account that
$$
x'(t)=\lambda_1C_1e^{\lambda_1t}+\lambda_2C_2e^{\lambda_2t}-
\gamma Q_1\sin(\gamma t)+\gamma Q_2\cos(\gamma t),
$$
we can find $C_1$ and $C_2$ corresponding to interval $[0,t_1]$.

On interval $[0,t_1]$, we denote $C_1=C_1^1$, $C_2=C_2^1$ and, on interval
$[0,t_2]$, we do $C_1=C_1^2$, $C_2=C_2^2$ respectively. We obtain
$$
C_1^1=C_2-C_0-Q_1,\;\;
C_2^1=\frac{x_0'+\lambda_1(C_0+Q_1)-\gamma Q_2}{\lambda_2-\lambda_1},
$$
where $x_0'$ is unknown. Then we calculate
\begin{equation}
\begin{gathered}
x(t_1)=C_1^1e^{\lambda_1t_1}+C_2^1e^{\lambda_2t_1}+C_0+Q_1\cos(\gamma t_1)+
Q_2\sin(\gamma t_1)=0, \\
x'(t_1)=\lambda_1C_1^1e^{\lambda_1t_1}+\lambda_2C_2^1e^{\lambda_2t_1}-
\gamma Q_1\sin(\gamma t_1)+\gamma Q_2\cos(\gamma t_1).
\end{gathered} \label{f6}
\end{equation}
These values are initial for the solution on interval $[0,t_2]$
\begin{gather*}
x(0)=0=C_1^2+C_2^2-C_0+Q_1, \\
x'(0)=x'(t_1)-c=\lambda_1C_1^2+\lambda_2C_2^2+\gamma Q_2,
\end{gather*}
from where it follows that
$$
C_1^2=C_0-C_2^2-Q_1, \quad
C_2^2=\frac{x'(t_1)-c-\lambda_1(C_0-Q_1)-\gamma Q_2}{\lambda_2-\lambda_1}.
$$
Completing the circle along the closed trajectory and remembering the gluing
together on axis $Ox_2$, we have
\begin{equation}
\begin{gathered}
x(t_2)=0=C_1^2e^{\lambda_1t_2}+C_2^2e^{\lambda_2t_2}-C_0+Q_1\cos(\gamma t_2)+
Q_2\sin(\gamma t_2), \\
x'(t_2)=\lambda_1C_1^2e^{\lambda_1t_2}+\lambda_2C_2^2e^{\lambda_2t_2}-
\gamma Q_1\sin(\gamma t_2)+\gamma Q_2\cos(\gamma t_2)=x_0'-c.
\end{gathered} \label{f9}
\end{equation}
Constants $C_1^1$, $C_2^1$ in \eqref{f6} contain $x_0'$ and constants
$C_1^2$, $C_2^2$ in \eqref{f9} contain $x'(t_1)$.
Further, we transform equalities \eqref{f6}, \eqref{f9} after
excluding $x_0'$ and $x'(t_1)$ from them. Thus
$Q_1\cos(\gamma t_1)+Q_2\sin(\gamma t_1)=\sin(\gamma t_1+\delta)$,
where $\delta=\arctan (Q_1/Q_2)$, $Q_2\neq 0$,
i.e. $\beta\neq 0$, $\gamma^2\neq a_0$.
After tedious transformations, we get
two transcendental equations with respect to $t_1$ and $t_2$,
\begin{equation}
\begin{aligned}
&\frac{1}{\lambda_2-\lambda_1}\Big(-\frac{\lambda_1}{\lambda_2-\lambda_1}
e^{\lambda_1t_1}+\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2t_1}\Big)
\Big[(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0)e^{\lambda_1t_1}\\
&- (\lambda_2-\lambda_1) (C_0+Q_1)e^{\lambda_1t_1}
-(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0) e^{\lambda_2t_1}
+(\lambda_2-\lambda_1)C_0\\
&+(\lambda_2-\lambda_1)\sin(\gamma t_1+\delta)\Big]
 +\frac{1}{\lambda_2-\lambda_1} (e^{\lambda_1t_2}-e^{\lambda_2t_2})\\
&\times \Big[\frac{\lambda_1}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0)e^{\lambda_1t_1} 
-\lambda_1(C_0+Q_1)e^{\lambda_1t_1}\\
&-\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0)e^{\lambda_2 t_1}+\gamma
\cos(\gamma t_1+\delta)\Big]\\
&=\frac{1}{\lambda_2-\lambda_1}(c+\lambda_1(C_0-Q_1)+\gamma Q_2)e^{\lambda_1t_2}
+(C_0-Q_1)e^{\lambda_1t_2}\\
&\quad - \frac{1}{\lambda_2-\lambda_1}
 (c+\lambda_1(C_0+Q_1)+\gamma Q_2)e^{\lambda_2t_2}
 - C_0+\sin(\gamma t_2+\delta),
\end{aligned}\label{f10}
\end{equation}
and
\begin{equation}
\begin{aligned}
&(e^{\lambda_1t_1}-e^{\lambda_2t_2})^{-1}[(\gamma Q_2-\lambda_2 Q_1-
\lambda_2 C_0)e^{\lambda_1t_1}-(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0)
e^{\lambda_2t_1}\\
&+ (\lambda_2-\lambda_1)C_0+(\lambda_2-\lambda_1)\sin(\gamma t_1+\delta)]-c\\
&=\Big(-\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1t_2}+
\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2t_2}\Big)
\Big\{\Big(-\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1t_1}+
\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2t_1}\Big)\\
&\quad \times (e^{\lambda_1t_1}-e^{\lambda_2t_1})^{-1}
 \Big[(\gamma Q_2-\lambda_2 Q_1-
\lambda_2 C_0)e^{\lambda_1t_1} \\
&\quad -(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0) e^{\lambda_2t_1} 
+ (\lambda_2-\lambda_1)C_0+(\lambda_2-\lambda_1)\sin(\gamma t_1
 +\delta)\Big] \\
&\quad +\frac{\lambda_1}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0) e^{\lambda_1t_1}
 - \lambda_1(C_0+Q_1)e^{\lambda_1t_1} \\
&\quad -\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1 Q_1-\lambda_1 C_0)e^{\lambda_2t_1}+
\gamma\cos(\gamma t_1+\delta)\Big\}\\
&\quad + \frac{\lambda_1}{\lambda_2-\lambda_1}(c+\lambda_1(C_0-Q_1)+\gamma Q_2)
e^{\lambda_1t_2}+\lambda_1(C_0-Q_1)e^{\lambda_1t_2}\\
&\quad -\frac{\lambda_2}{\lambda_2-\lambda_1}(c+\lambda_1(C_0-Q_1)+\gamma Q_2)
e^{\lambda_2t_2}+\gamma\cos(\gamma t_2+\delta).
\end{aligned} \label{f11}
\end{equation}
If the system \eqref{f10}, \eqref{f11}
is solvable for $t_1$, $t_2$ satisfying \eqref{f1},
then there is a periodic solution with period $T=t_1+t_2$.
However it is very difficult to solve system  \eqref{f10}, \eqref{f11}.
We suppose that the period of the appearing oscillations $T$ is given, i.e.
$t_2=T-t_1$. Next we write out equation \eqref{f10}
depending only on $t_1$ and group the expressions at identical multipliers
in it. Then we have
\begin{align}
&-\frac{\lambda_1}{\lambda_2-\lambda_1}(\gamma Q_2-\lambda_2(C_0+Q_1))
e^{2\lambda_1t_1}+\Big[\frac{\lambda_2}{\lambda_2-\lambda_1}(\gamma Q_2-
\lambda_2 (C_0+Q_1)) \nonumber \\
&+\frac{\lambda_1}{\lambda_2-\lambda_1}(\gamma Q_2-\lambda_1(C_0+Q_1))\Big]
e^{(\lambda_2+\lambda_1)t_1}-\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1(C_0+Q_1))e^{2\lambda_2t_1}  \nonumber \\
&- \lambda_1C_0e^{\lambda_1t_1}+\lambda_2C_0e^{\lambda_2t_1}
 -\lambda_1\cos\delta e^{\lambda_1t_1}\cos(\gamma t_1)
 -\lambda_1\sin\delta e^{\lambda_1t_1}\cos(\gamma t_1) \nonumber\\
&+\lambda_2\cos\delta
e^{\lambda_2t_1}\sin(\gamma t_1)+\lambda_2\sin\delta e^{\lambda_2t_1}
\cos(\gamma t_1) \nonumber \\
&-\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_2T}
(\gamma Q_2-\lambda_2(C_0+Q_1))e^{(\lambda_1-\lambda_2)t_1} \nonumber\\
&-\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_1T}
(\gamma Q_2-\lambda_1(C_0+Q_1))e^{(\lambda_2-\lambda_1)t_1} \nonumber \\
&+ e^{\lambda_1T}\gamma\cos\delta e^{-\lambda_1t_1}\cos(\gamma t_1)-
e^{\lambda_1T}\gamma\sin\delta e^{-\lambda_1t_1}\sin(\gamma t_1) 
\label{f12} \\
&-(c+\lambda_2(C_0-Q_1)+\gamma Q_2)e^{\lambda_1T}e^{-\lambda_1t_1}+
(c+\lambda_1(C_0-Q_1)+\gamma Q_2)e^{\lambda_2T}e^{-\lambda_2t_1}\nonumber\\
&-\frac{1}{\lambda_2-\lambda_1}(\sin(\gamma T)\cos\delta+
\cos(\gamma T)\sin\delta)\cos(\gamma t_1) \nonumber\\
&-\frac{1}{\lambda_2-\lambda_1}(\sin(\gamma T)\sin\delta
 -\cos(\gamma T)\cos\delta)\sin(\gamma t_1) \nonumber\\
& +\Big\{\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1T}
[\gamma Q_2-\lambda_2(C_0+Q_1)]+
\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2T}
[\gamma Q_2-\lambda_1(C_0+Q_1)] \nonumber\\
& +\frac{1}{\lambda_2-\lambda_1}C_0\Big\}=0. \nonumber
\end{align} 
Also, we write out equation \eqref{f11} for $t_1$ and group the terms as above.
We obtain
\begin{align}
&\Big\{-c+\gamma Q_2-\lambda_2(C_0+Q_1)-\theta_1(\gamma Q_2-\lambda_2(C_0+Q_1))
 \nonumber \\
&-\frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T}
 (\gamma Q_2-\lambda_1(C_0+Q_1))-
\theta_2\frac{\lambda_1}{\lambda_2-\lambda_1}(\gamma Q_2-\lambda_1(C_0+Q_1)) 
 \nonumber \\
&+\theta_2\lambda_1(C_0+Q_1)-\frac{\lambda_2^2}{(\lambda_2-\lambda_1)^2}
e^{\lambda_2T} (\gamma Q_2-\lambda_1(C_0+Q_1))\Big\}e^{\lambda_1t_1} \nonumber\\
&+\Big\{c-\gamma Q_2+\lambda_1(C_0+Q_1)+\theta_1(\gamma Q_2-\lambda_1(C_0+Q_1)) 
 \nonumber\\
&+\frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_1T}
(\gamma Q_2-\lambda_2(C_0+Q_1))+
\theta_2\frac{\lambda_2}{\lambda_2-\lambda_1}(\gamma Q_2-\lambda_1(C_0+Q_1))  
 \nonumber\\
&+\frac{\lambda_1^2}{\lambda_2-\lambda_1}e^{\lambda_1T}(C_0+Q_1)-
\frac{\lambda_1^2}{(\lambda_2-\lambda_1)^2}e^{\lambda_1T}
(\gamma Q_2-\lambda_1(C_0+Q_1))\Big\}e^{\lambda_2t_1} \nonumber\\
&+\big[(\lambda_2-\lambda_1)\cos\delta-\theta_1(\lambda_2-\lambda_1)\cos\delta+
\theta_2\gamma\sin\delta\big]\sin(\gamma t_1)\nonumber \\
&+[(\lambda_2-\lambda_1)\sin\delta-\theta_1(\lambda_2-\lambda_1)\sin\delta-
\theta_2\gamma\cos\delta]\cos(\gamma t_1)\nonumber \\
&+\Big[\frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T}
(\gamma Q_2-\lambda_2(C_0+Q_1))-
\frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T}
(\gamma Q_2-\lambda_1(C_0+Q_1))\nonumber \\
&+ \frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2T}(C_0+Q_1)\Big]
e^{(2\lambda_1-\lambda_2)t_1} \nonumber\\
&+\big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}C_0e^{\lambda_2T}+
\frac{\lambda_2}{\lambda_2-\lambda_1}(c+\lambda_1(C_0-Q_1)+\gamma Q_2)
e^{\lambda_2T}\big]e^{(\lambda_1-\lambda_2)t_1}
  \label{f13}  \\
&+ \Big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}C_0e^{\lambda_1T}+
\frac{\lambda_1}{\lambda_2-\lambda_1}(c+\lambda_1(C_0-Q_1)+\gamma Q_2)
e^{\lambda_1T} \nonumber\\
&+ \lambda_1(C_0-Q_1)e^{\lambda_1T}\Big]e^{(\lambda_2-\lambda_1)t_1} \nonumber\\
&+ \big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}\cos\delta e^{\lambda_1T}+
\gamma\sin\delta\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1T}\big]
e^{(\lambda_2-\lambda_1)t_1}\sin(\gamma t_1)\nonumber\\
&+\big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}\sin\delta e^{\lambda_1T}-
\gamma\cos\delta\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1T}\big]
e^{(\lambda_2-\lambda_1)t_1}\cos(\gamma t_1)\nonumber\\
&+\big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}\cos\delta e^{\lambda_2T}+
\gamma\sin\delta\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2T}\big]
e^{(\lambda_1-\lambda_2)t_1}\sin(\gamma t_1) \nonumber\\
&+\big[\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}\sin\delta e^{\lambda_2T}-
\gamma\cos\delta\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2T}\big]
e^{(\lambda_1-\lambda_2)t_1}\cos(\gamma t_1) \nonumber\\
&+\gamma(\cos(\gamma T)\cos\delta-\sin(\gamma T)\sin\delta)
e^{\lambda_1t_1}\cos(\gamma t_1) \nonumber\\
&-\gamma(\cos(\gamma T)\cos\delta-\sin(\gamma T)\sin\delta)e^{\lambda_2t_1}
\cos(\gamma t_1) \nonumber\\
&+\gamma(\cos(\gamma T)\sin\delta+\sin(\gamma T)\cos\delta)
 e^{\lambda_1t_1}\sin(\gamma t_1) \nonumber\\
&-\gamma(\cos(\gamma T)\sin\delta+\sin(\gamma T)\cos\delta)
 e^{\lambda_2t_1}\sin(\gamma t_1) \nonumber\\
&+ \Big\{(\lambda_2-\lambda_1)C_0-\theta_1(\lambda_2-\lambda_1)C_0-
\frac{\lambda_1}{\lambda_2-\lambda_1}[c+\lambda_1(C_0-Q_1)+\gamma Q_2]
e^{\lambda_1T} \nonumber\\
&-\frac{\lambda_2}{\lambda_2-\lambda_1}[c+\lambda_1(C_0-Q_1)+
\gamma Q_2]e^{\lambda_2T}-
\lambda_1(C_0-Q_1)e^{\lambda_1T}\Big\}=0, \nonumber
\end{align}
where
\[
\theta_1=\frac{\lambda_1^2}{(\lambda_2-\lambda_1)^2}e^{\lambda_1T}+
\frac{\lambda_2^2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T},\quad
\theta_2=-\frac{\lambda_1}{\lambda_2-\lambda_1}e^{\lambda_1T}-
\frac{\lambda_2}{\lambda_2-\lambda_1}e^{\lambda_2T}.
\]
The expressions in braces from \eqref{f12}, \eqref{f13},
which represent the last terms, are the constants independent of $t_1$.
Obviously, equations \eqref{f12}, \eqref{f13} have equal solutions with 
respect to $t_1$.
Next we find out the conditions under which it is possible. We use the approach
based on assumption that the coefficients of equations are equal to each other.
We add up the expressions at multipliers
$e^{(\lambda_2+\lambda_1)t_1}$,
$e^{\lambda_1t_1}$, $e^{\lambda_2t_1}$ from \eqref{f12} and set
equal to the corresponding multipliers $e^{\lambda_1t_1}$, $e^{\lambda_2t_1}$
from \eqref{f13}. Then we obtain
\begin{equation}
\begin{aligned}
&-\lambda_1C_0+\frac{\lambda_1}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1(C_0+Q_1))e^{\lambda_2t_1}\\
&= -c+\gamma Q_2-\lambda_2(C_0+Q_1)
  -\theta_1(\gamma Q_2-\lambda_2(C_0+Q_1))\\
&- \frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T}
(\gamma Q_2-\lambda_1(C_0+Q_1))\\
& -\theta_2\frac{\lambda_1}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1(C_0+Q_1))+\theta_2\lambda_1(C_0+Q_1)\\
&-\frac{\lambda_2^2}{(\lambda_2-\lambda_1)^2}e^{\lambda_2T}
(\gamma Q_2-\lambda_1(C_0+Q_1)),
\end{aligned}\label{f14}
\end{equation}
and
\begin{equation}
\begin{aligned}
&\lambda_2C_0+\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_2(C_0+Q_1))e^{\lambda_1t_1}\\
&= c-\gamma Q_2+\lambda_1(C_0+Q_1)+ \theta_1(\gamma Q_2-\lambda_1(C_0+Q_1)) \\
&\quad + \frac{\lambda_1\lambda_2}{(\lambda_2-\lambda_1)^2}e^{\lambda_1T}
(\gamma Q_2-\lambda_2(C_0+Q_1)) \\
&\quad + \theta_2\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1(C_0+Q_1))+\frac{\lambda_1^2}{\lambda_2-\lambda_1}
e^{\lambda_1T}(C_0+Q_1)\\
&\quad -\frac{\lambda_1^2}{(\lambda_2-\lambda_1)^2}e^{\lambda_1T}
(\gamma Q_2-\lambda_1(C_0+Q_1)),
\end{aligned}\label{f15}
\end{equation}
where $e^{\lambda_2t_1}$ is given by \eqref{f14}. From \eqref{f15}, we obtain
$e^{\lambda_1t_1}$. Then we equate constants in
\eqref{f12}, \eqref{f13}. We have the first condition on the parameters
of the system
\begin{equation}
\begin{aligned}
&(2\gamma Q_2+c)(\lambda_1e^{\lambda_1T}+\lambda_2e^{\lambda_2T})-
2\lambda_1\lambda_2 Q_1(e^{\lambda_1T}+e^{\lambda_2T})\\
&+ C_0(\lambda_1^2e^{\lambda_1T}+\lambda_2^2e^{\lambda_2T})-
C_0((\lambda_2-\lambda_1)^2-1)=0.
\end{aligned} \label{f16}
\end{equation}
Further, we consider the terms in \eqref{f12}, \eqref{f13}
without the sine and cosine functions.
We set equal the coefficients at multiplier $e^{(\lambda_1-\lambda_2)t_1}$.
Then
\begin{equation}
2\lambda_1\lambda_2Q_1-\gamma Q_2(\lambda_1+\lambda_2)=
\lambda_2(c+\lambda_1C_0).
\label{f17}
\end{equation}
Next we equate the coefficients at multiplier $e^{(\lambda_2-\lambda_1)t_1}$.
Then
\begin{equation}
\lambda_1\lambda_2Q_1-\gamma Q_2(\lambda_1+\lambda_2)=
\lambda_1c+\lambda_1(\lambda_1+1)(C_0-Q_1).
\label{f18}
\end{equation}
It should add the condition following from
equalities of the sum of the rest terms at sine and
cosine multipliers from \eqref{f12}, \eqref{f13}
to conditions \eqref{f16}--\eqref{f18}.
These conditions are redundant. First, we have proceeded from the
necessary conditions for the existence of
the periodic solution to the initial system with period
$T$ we do given. Secondly, parameters $a_1$, $a_0$ (i.e. $\lambda_1$,
$\lambda_2$), and $\beta$, $\gamma$ are
the parameters influencing on the choice of $c$.
All these parameters are in conditions \eqref{f17}, \eqref{f18}, since
$Q_1=Q_1(a_1,a_0,\beta,\gamma)$, $Q_2=Q_2(a_1,a_0,\beta,\gamma)$.
In particular, from \eqref{f17} and \eqref{f18}, we have
\begin{equation}
c=\gamma(\lambda_1+\lambda_2)Q_2
\Big(\frac{1}{2}+\frac{1+\lambda_2-\lambda_1}{1+\lambda_1-\lambda_2}\Big)
\label{f19}
\end{equation}
provided that $\lambda_2-\lambda_1\neq 1$ or $a_1^2-4a_0\neq 1$.
From \eqref{f16}, we obtain
\begin{equation}
\begin{aligned}
&2\gamma Q_2(\lambda_1e^{\lambda_1T}+\lambda_2e^{\lambda_2T})-
2\lambda_1\lambda_2 Q_1(e^{\lambda_1T}+e^{\lambda_2T})\\
&=c\Big\{\frac{1}{2\lambda_1\lambda_2}
[(\lambda_2-\lambda_1)^2-1]-
\frac{1}{2\lambda_1\lambda_2}
(\lambda_1^2e^{\lambda_1T}+\lambda_2^2e^{\lambda_2T})-
\lambda_1 e^{\lambda_1T}\\
&\quad -\lambda_2 e^{\lambda_2T} \Big\}.
\end{aligned}\label{f21}
\end{equation}
From \eqref{f21}, it is easy to express $c$ with respect to $T$.

Now we return to \eqref{f14}, \eqref{f15}. From \eqref{f14}, we get
\begin{equation}
\begin{aligned}
e^{\lambda_2t_1}
&=\Big[\frac{\lambda_1}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_1(C_0+Q_1))\Big]^{-1} \\
&\quad\times  \Big\{\lambda_1 C_0-c+\gamma Q_2-\lambda_2(C_0+Q_1)\\
&\quad -e^{\lambda_2T}\frac{\lambda_2}{(\lambda_2-\lambda_1)^2}
[2\lambda_2\gamma Q_2-(\lambda_2^2+\lambda_1^2)(C_0+Q_1)]\Big\}.
\end{aligned}\label{f22}
\end{equation}
From \eqref{f15}, we have
\begin{equation}
\begin{aligned}
e^{\lambda_1t_1}
&=\Big[\frac{\lambda_2}{\lambda_2-\lambda_1}
(\gamma Q_2-\lambda_2(C_0+Q_1))\Big]^{-1} \\
&\quad\times \Big\{-\lambda_2 C_0+c-\gamma Q_2+\lambda_1(C_0+Q_1)\\
&\quad + e^{\lambda_1T}\frac{\lambda_1^2}{\lambda_2-\lambda_1}(C_0+Q_1)\}.
\end{aligned} \label{f23}
\end{equation}
Thus the following two conditions must be satisfied
\begin{equation}
\gamma Q_2-\lambda_1(C_0+Q_1)\neq 0,\quad
\gamma Q_2-\lambda_2(C_0+Q_1)\neq 0.
\label{f24}
\end{equation}
Since $\lambda_1\neq\lambda_2$, conditions \eqref{f24} hold and
do not turn into equalities if
\begin{equation}
c\neq\frac{2\gamma\beta\lambda_1\lambda_2(2\lambda_2+\lambda_1)-
2\lambda_2\gamma^3\beta}{\gamma^4+\gamma^2(\lambda_1^2+\lambda_2^2)+
\lambda_1^2\lambda_2^2},\quad
c\neq\frac{2\gamma\beta\lambda_1\lambda_2(2\lambda_1+\lambda_2)-
2\lambda_1\gamma^3\beta}{\gamma^4+\gamma^2(\lambda_1^2+\lambda_2^2)+
\lambda_1^2\lambda_2^2}
\label{f25}
\end{equation}
for the first and second conditions of \eqref{f24} respectively.
In \eqref{f25}, the condition
$$
\gamma^4+\gamma^2(\lambda_1^2+\lambda_2^2)+\lambda_1^2\lambda_2^2\neq 0
$$
is carried out automatically owing to the conditions imposed on $\lambda_1$,
$\lambda_2$, i.e. for all $\gamma>0$, equality to zero does not take place.
Besides, the right-hand sides of \eqref{f22}, \eqref{f23} have to be more than 
unit if $\lambda_1$, $\lambda_2$ are positive, and to belong to interval
$(0,1)$ if $\lambda_1$, $\lambda_2$ are negative.

So, we formulate the conditions guaranteeing non-existence of the
periodic solutions to system \eqref{f1} with given period $T$.

\begin{theorem} \label{thm2} 
 Let $a_1\neq 0$, $a_0\neq 0$, $0<a_1^2-4a_0\neq 1$,
$c>0$, $\beta\in\ {\mathbb R}\backslash\{0\}$, $\gamma>0$, and $\gamma^2\neq a_0$.
Then equation \eqref{f1} has no periodic solutions with given period
$T$, where $T=t_1+t_2$, $t_1$ is defined by \eqref{f22}, \eqref{f23} under
\eqref{f25} if \eqref{f19}, \eqref{f21} are not fair.
\end{theorem}


If the other coefficients of \eqref{f12}, \eqref{f13}
are equated, we obtain the other equalities for
$t_1$, $t_2$ and therefore the other ratios linking
$c$ with parameters $a_1$, $a_0$, $\beta$, $\gamma$.
In the space of parameters $c$, $a_1$, $a_0$, $\beta$, $\gamma$
the set of such ratios allows us
to allocate domains for the possible existence of the periodic solutions with
given period $T$ and domains for which such solutions do not exist.

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\end{document}