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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 221, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/221\hfil Second-order differential equations]
{Solution to second-order differential equations
with discontinuous right-hand side}

\author[A. M. Kamachkin, D. K. Potapov, V. V. Yevstafyeva \hfil EJDE-2014/221\hfilneg]
{Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva}  
% in alphabetical order

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

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

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

\thanks{Submitted January 16, 2014. Published October 21, 2014.}
\subjclass[2000]{34A34, 34A36, 34C11, 34C60}
\keywords{Discontinuous nonlinearity; phase trajectories; existence of solutions}

\begin{abstract}
 We consider an ordinary differential equation of second order with
 discontinuous nonlinearity relative to the phase variable.
 Phase trajectories are studied. We establish a theorem on the existence
 of a continuum set for nontrivial solutions and the theorem on the
 boundedness of solutions.
\end{abstract}

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


\section{Introduction and statement of problem}

Over a number of years differential equations with discontinuous
right sides have attracted researchers' attention. Equations with
discontinuous nonlinearities are of interest on both theoretical and
practical grounds. The problem on existence of solutions for the Sturm-Liouville
task with discontinuous nonlinearity is considered in \cite{carl8}--\cite{pot38}.
The applications of such problems are shown in \cite{pot18,pot40},
and other papers.
Periodic solutions of second-order differential equations with
discontinuous right sides are studied in \cite{jacquemard,nyzhnyk}.
This paper extends this research.

We study the existence of solutions to
the second-order ordinary differential equation with discontinuous nonlinearity
of the form
\begin{gather}
-u''=g(x,u(x)), \quad  x\in \mathbb{R}, \label{kp1} \\
g(x, u)=\begin{cases}
m_1 & \text{for }  u<f(x),\\
m_2 & \text{for }  u\geq f(x).
\end{cases}
\label{kp2}
\end{gather}
Here the function $g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$,
 $m_i$ ($i=1,2$) are constants in $\mathbb{R}$, the function
$f:\mathbb{R}\to\mathbb{R}$ is piecewise smooth and one-to-one.

Let us remark that systems of ordinary differential equations with
multiple-valued discontinuous nonlinearity of this type are investigated, for
instance, in \cite{kam}--\cite{vica}.

 From \eqref{kp1}, \eqref{kp2} we receive the equations
\begin{equation}
-u''=m_i \quad (i=1,2). \label{kp3}
\end{equation}

It follows from \eqref{kp3} that $u'=-m_ix+c_1$,
$u=-\frac{m_i}{2}x^2+c_1x+c_2$, where $c_1$, $c_2$ are real constants.
Then the phase curves on the plane
$(u,u')$ are defined by the equations $2m_i(c_2-u)=(u')^2-c_1^2$.

We assume that the function $u'=\psi(u)$, which graph is a curve without
contact for the phase trajectories of system \eqref{kp1}, \eqref{kp2},
is assigned to the function $u=f(x)$ on the phase $(uOu')$-plane.
In other words, the phase trajectories have the only
isolated points of tangency to this curve, i.e. the points of tangency do not
belong to segments of this curve.

Let, in particular, the discontinuity surface be the straight line $u=f(x)=kx+b$,
where coefficients $k, b \in \mathbb{R}$. Note that such switching surfaces
arise quite often in automatic control systems.
On the phase $(uOu')$-plane the straight line $u'=k$ is
a switching line. Its graph represents a curve without
contact for the phase trajectories of system \eqref{kp1}, \eqref{kp2}.

Let us consider all possible relations between parameters $m_1$ and $m_2$.

\section{Solution of the problem}


\emph{Case 1.} Suppose $m_i>0$ ($i=1,2$); then the phase trajectories consist of
parabola pieces. The branches
of the parabolas are directed aside opposite to the positive direction
of the  $Ou$-axis.
From any initial point on the half-plane $u'>k$ the representative
point reaches the switching line and then along the parabola on the half-plane
$u'<k$ it goes to infinity ($u\to-\infty$). If the initial point lies
on the half-plane $u'<k$, then the
representative point also tends to infinity ($u\to-\infty$). The phase
trajectories are called  respectively trajectories of ``parabola -- parabola''
type or ``parabola'' one.

\emph{Case 2.}
The similar situation happens when $m_i<0$ ($i=1,2$). In this case
the parabola branches lying on both half-planes are
directed towards the positive direction of the $Ou$-axis. From any initial point on
the half-plane $u'>k$ the
representative point moves along the parabolic trajectory to infinity
($u\to +\infty$).
If the initial point is on the half-plane $u'<k$, then
the representative point comes to the switching line along the parabolic
trajectory and after that it goes into infinity ($u\to +\infty$).
The types of the phase trajectories are the same as in the previous case.

Both cases considered above correspond to the condition $m_1m_2>0$.
Suppose $m_1$ and $m_2$ have opposite signs.

\emph{Case 3.} To be precise, let $m_1<0$, $m_2>0$.
Then from any initial point that belongs to the phase
plane and does not to the line $u'=k$ the representative
point comes to the straight line $u'=k$ along the parabolic trajectory.
The phase trajectories are of ``parabola'' type.

\emph{Case 4.}
Now we assume that $m_1>0$, $m_2<0$.
From any initial point on $u'>k$ the representative point
extends to infinity ($u\to +\infty$) along the parabola, if on $u'<k$,
then it also tends to infinity ($u\to -\infty$) along the parabola.
We have the phase trajectories of ``parabola'' type.

The two latter cases correspond to the condition $m_1m_2<0$.

\emph{Case 5.}
Let $m_1<0$, $m_2=0$. If the initial point belongs to the half-plane
$u'\geq k$ ($k>0$), then the representative point
moves to infinity ($u\to +\infty$) along the straight line parallel to
the $Ou$-axis.
However, if the initial point is on the half-plane $u'<k$, then the representative
point comes to the straight line $u'=k$ along the parabolic trajectory
and also goes to infinity ($u\to+\infty$) along the straight line $u'=k$.
Note that the straight line $u'=0$ is a set of equilibrium points.
Let $k<0$. If the initial point is on the half-plane $u'<k$, then
the trajectory has ``parabola -- straight line'' type and the representative
point tends to infinity ($u\to -\infty$) along $u'=k$. At the same time,
if the initial point is on the half-plane $u'>0$, then the representative
point goes to infinity ($u\to +\infty$) along the straight lines parallel to
the $Ou$-axis. Here the $Ou$-axis is a set of equilibrium points.
If the initial point is from the set $k<u'<0$, then the
representative point leaves for infinity along the lines
parallel to the $Ou$-axis, so that $u\to-\infty$.
We receive two types of the trajectories, namely,
``straight line'' or ``parabola -- straight line''.

\emph{Case 6.}
Further, let $m_1=0$ and $m_2>0$.
Obviously, the qualitative picture of splitting the phase plane into
trajectories is not changed
in comparison with the previous case, but pieces of the parabolic trajectories
lie on the upper half-plane. The types of the phase
trajectories are the same as above.
Really, if the initial point belongs to the half-plane $u'>k$ ($k>0$),
then the representative point comes to the line $u'=k$
along the parabolic trajectories and goes to infinity ($u\to +\infty$)
along this straight line.
On the other hand, from any initial point on the set $0<u'<k$
the representative point
tends to infinity ($u\to +\infty$) along the straight line parallel to
the $Ou$-axis.
If $u'<0$, then the representative point leaves for infinity
($u\to -\infty$) along the straight line parallel the $Ou$-axis.
Let $k<0$. If the initial point is on the half-plane $u'>k$,
then along the parabolic trajectory the representative point comes
to the line $u'=k$ and along this line it goes
to infinity ($u\to -\infty$).
But if the initial point is on the half-plane $u'<k$, then the
representative point moves to infinity ($u\to -\infty$) along the line parallel
to the $Ou$-axis.
The straight line $u'=k$ is a set of equilibrium positions when $k=0$.

\emph{Case 7.}
Let $m_1=0$, $m_2<0$.
This case differs from Case~6 in motion directions along the parabolic pieces
of trajectories. Indeed, let $k>0$. If the initial point belongs to the plane
$u'\geq k$, then the representative point
goes to infinity ($u\to +\infty$) along the parabolic trajectory.
The representative point moves to infinity ($u\to +\infty$) along the straight
lines parallel to the $Ou$-axis when $0<u'<k$.
Note that the straight line $u'=0$ is a set of
equilibrium points. The representative point goes to
infinity ($u\to -\infty$) along the straight lines parallel to the $Ou$-axis
when $u'<0$.
Let $k<0$. If $u'\geq k$, then the representative point goes to infinity
($u\to +\infty$) along the  parabolic trajectory. If $u'<k$, then the
representative point goes to infinity ($u\to -\infty$) along the straight line
parallel to the $Ou$-axis.
The types of the trajectories are ``parabola'' or ``straight line''.

\emph{Case 8.}
Let $m_1>0$, $m_2=0$. This case differs from Case~5 in motion directions
along the parabolic trajectories. For example, let $k>0$. If the initial
point belongs to the half-plane $u'\geq k$, then the representative point
moves to infinity ($u\to +\infty$) along the straight lines parallel to
the $Ou$-axis.
If $u'<k$, then the representative point goes to infinity
($u\to -\infty$) along the parabolic trajectories.
Let $k<0$. If $u'>0$, then the representative point goes to infinity
($u\to +\infty$) along
the straight lines parallel to the $Ou$-axis.
Here the straight line $u'=0$ is a set of equilibrium points.
If $k\leq u'<0$, then the representative point goes to infinity
($u\to -\infty$) along
the straight lines parallel to the $Ou$-axis.
If the initial point is taken from the set $u'<k$, then the
representative point goes to infinity ($u\to -\infty$) along the parabolic
trajectories.
The types of the trajectories are ``parabola'' or ``straight line''.

\emph{Case 9.}
Let $m_1=m_2=0$. Let $k>0$. If the initial point belongs to either
$u'\geq k$ or $0<u'<k$, then the representative point moves to
infinity ($u\to +\infty$) along the straight line parallel to the $Ou$-axis.
The line $u'=0$ is a set of equilibrium points.
If the initial point is taken from $u'<0$, then the representative
point moves to infinity ($u\to -\infty$) along the straight lines parallel to
the $Ou$-axis. Now let $k<0$. If the initial point belongs to the half-plane
$u'>0$, then the representative point moves to infinity
($u\to +\infty$) along the straight lines parallel to the $Ou$-axis.
The straight line $u'=0$ is a set of equilibrium points.
If the initial point is taken from $k<u'<0$ or $u'\leq k$, then
the representative point moves to infinity ($u\to -\infty$).
The trajectories are of ``straight line'' type.

So, we have considered all cases of relations between the parameters $m_1$ and
$m_2$. All studied types of the trajectories take place under the conditions on
$f(x)$ imposed above. The function $f$ is not only linear. Thus the following
theorem on existence of solutions for problem \eqref{kp1}, \eqref{kp2} is fair.

\begin{theorem} \label{thm1} 
 Let $g:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$, and the function
$f:\mathbb{R}\to\mathbb{R}$ be piecewise smooth and
one-to-one. The graph of $f$ on the phase $(uOu')$-plane is
a curve without contact for the phase trajectories of system
\eqref{kp1}, \eqref{kp2}. Then there is a continuum set of nontrivial solutions
for problem \eqref{kp1}, \eqref{kp2} such that the
phase trajectories are the piecewise smooth curves consisting of the
pieces of parabolas and straight lines.
\end{theorem}

Notice that Theorem \ref{thm1} agrees with the results received in \cite{pot38} for
one-dimensional analog of the Gol'dshtik model for separated flows of
incompressible fluid.

The following corollary follows from  Theorem \ref{thm1}.


\begin{corollary} \label{coro1}
 Let the conditions of Theorem \ref{thm1} hold and in addition
$m_1m_2>0$, $f(x)=kx+b$, $k\neq 0$. Then for each point of the switching line
there exists a neighborhood such that switching of the phase trajectory pieces
in it does not lead to qualitative change of the phase trajectories in the
whole.
\end{corollary}

As established above, nontrivial solutions of problem \eqref{kp1},
\eqref{kp2} belong to the class of piecewise smooth functions. The phase
trajectories are ``sewed'' on continuity on the curve $u'=\psi(u)$ to
which the set $\{x\in \mathbb{R}: u(x)=f(x)\}$ is assigned.

\section{Boundedness}

Further, let the function $g:\Omega\times\mathbb{R}\to\mathbb{R}$,
where $\Omega\subset \mathbb{R}$ is a bounded connected set. Then the set of
points $\{(\psi(u),u): x\in\Omega, \; u(x)=f(x)\}$ has zero measure and is
closed with respect to the closed set $\Omega$. In particular, this is fair
for the set $\Omega=[x_1,x_2]$ ($x_1, x_2\in \mathbb{R}$, $x_1<x_2$).
We note that the similar result on properties of the ``separating'' set is
received in \cite{pot13, pot17} for equations of elliptic type with
discontinuous nonlinearities.

We have
$$
|g(x,u)|\leq\max \{|m_1|,|m_2|\}=m
$$
for any $x\in\Omega$ and $u\in \mathbb{R}$.
It follows from the inequality above and equation \eqref{kp1} that
$$
0\leq |-u''(x)|\leq m,
$$
where $m$ is a real non-negative number defined above.
So, the estimation for the differential operator of problem
\eqref{kp1}, \eqref{kp2} is received.

For $\Omega=[x_1,x_2]$, we get
$$
|u'(x_2)-u'(x_1)|=
\big|\int_{x_1}^{x_2}u''(x)dx\big|
\leq \int_{x_1}^{x_2}|u''(x)|dx\leq m(x_2-x_1)
$$
and
$$
|u'(x)|\leq m|x_2|+|c_1|=C_1.
$$
Note that such kind of estimations are also fair for any bounded closed
set $\Omega$.

From the form of solutions $u(x)$ on the set $\Omega=[x_1,x_2]$ it follows
that
$$
|u(x)|\leq\frac{m}{2}x_2^2+|c_1x_2|+|c_2|=C_2,
$$
which means boundedness of the solutions $u(x)$.

Thus the theorem on boundedness of solutions and their derivatives is received
for problem \eqref{kp1}, \eqref{kp2}.

\begin{theorem} \label{thm2} 
 Let $g:\Omega\times\mathbb{R}\to\mathbb{R}$,
where $\Omega$ is the bounded closed set in $\mathbb{R}$. Then the solutions
$u(x)$ of problem \eqref{kp1}, \eqref{kp2} are bounded on $\Omega$.
Also $u'(x)$ and $u''(x)$ are bounded
on the corresponding subsets of their existence of $\Omega$.
\end{theorem}

\noindent
{\bf Remark.}
 Notice  that solutions $u(x)$ of problem \eqref{kp1}, \eqref{kp2} are
bounded with respect to the norm in the corresponding functional spaces.

As an example, let us consider the norm in the Sobolev space
$H_{\circ}^1([x_1,x_2])$:
$$
\|u\|=\Big(\int_{x_1}^{x_2} |u'(x)|^2dx\Big)^{1/2}.
$$
We obtain
\begin{align*}
\|u(x)\|&=\Big(\int_{x_1}^{x_2}(c_1-m_ix)^2dx\Big)^{1/2}\\
&= \sqrt{c_1^2(x_2-x_1)-c_1m_i(x_2^2-x_1^2)+\frac{m_i^2}{3}(x_2^3-x_1^3)}\leq C_3.
\end{align*}
Since the space $H_{\circ}^1([x_1,x_2])$ is compactly embedded in
$C([x_1,x_2])$, we obtain (see, for example, \cite{bonanno8}):
$$
\|u\|_\infty\leq\frac{1}{\sqrt{2
\frac{2\operatorname{ess\,inf}_{[x_1,x_2]}1}{x_2-x_1}}} \|u\|.
$$
Thus,
$$
\|u(x)\|_\infty\leq\frac{\sqrt{x_2-x_1}}{2} \|u(x)\|\leq
\frac{\sqrt{x_2-x_1}}{2} C_3=C_4.
$$

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\end{document}