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functional-differential equations, quasilinear elliptic equations,
two points boundary value problems
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\begin{document}
\title[Functional-differential equation with dilation and compression]
{Functional-differential equation with dilation and compression of argument}
\author[Oleg Zubelevich]{Oleg Zubelevich\\ \\\tt
Department (\#803) of Differential Equations\\
Moscow State Aviation Institute\\
Volokolamskoe Shosse 4, 125993, Moscow, Russia\\
E-mail: ozubel@yandex.ru}
\address{Department (\# 803) of Differential Equations
Moscow State Aviation Institute
Volokolamskoe Shosse 4, 125993, Moscow, Russia}
\email{ozubel@yandex.ru}
\curraddr{2-nd Krestovskii Pereulok 12-179, 129110, Moscow, Russia}
%\date{}
%\thanks{Partially supported by grants RFBR 02-01-00400.}
\subjclass[2000]{34K10}
\keywords{functional-differential equations, quasilinear elliptic equations,
two points boundary value problems.}
\begin{abstract}
We consider a quasilinear 1-D equation of elliptic
type with compression/dilation of the argument and prove the existence
theorem.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}{Definition}[section]
\section{Introduction}
Various applications lead us to study boundary value problems for elliptic
functional-differential equations with shifts in the space variables.
These include problems from the mechanics of deformation of rigid body
\cite{1} and modern nonlinear optics \cite{2}.
Boundary value problems for elliptic functional-differential equations
have been studied in the articles \cite{6}, \cite{7} and others.
A theory of boundary value problems for elliptic differential-difference
equation in a bounded domain was constructed in \cite{8}.
Linear elliptic equations with dilation/compression of the argument
are studied in \cite{ross1}, \cite{ross2}.
Functional-differential equations with compression
of the argument in the one dimensional case have been studied by many authors \cite{9},
\cite{10}, \cite{11}. Most of these works have been concerned with problems
on the representation of solutions to the initial value problem, their
asymptotic behavior and stability.
In the present paper we consider an existence problem for a
quasilinear 1-D functional-differential
elliptic equation. The right side of this equation depends on two
arguments. The first argument contains a finite liner combination of
the compression/dilation operators and the second one contains
an infinite series of the dilation operators. The right side can be of
a high rate of growth in the second argument.
\section{Main theorem}
Let $M=(-1,1)$ be an interval in the set of real numbers
$\mathbb{R}=\{x\}$ and let
$N=\mathbb{R}\backslash M$ be its supplement, $\partial M=\{\pm 1\}$.
Define an operator $R_\lambda:L^r(M)\to L^r(M),\quad r\ge 2$ as follows.
First we assume a function $w\in L^r(M)$ to be extended by zero in $N$ and
then we put $$R_\lambda w=w(\lambda x),\quad \lambda=\mathrm{const}\in \mathbb{R}.$$
In the sequel it is convenient to extend all the functions of $x$ to $\mathbb{R}$ by zero.
Let $a=\sum_{i=1}^{\infty}|a_i|>0$ be a sum of convergent series with real
numbers $a_i$. Another operator to construct is
$$A=\sum_{i=1}^{\infty}a_iR_{\gamma_i},\quad |\gamma_i|\ge 1.$$
Let $\tilde C^1(\overline M)\subset C^1(\overline M)$ be a space of such
functions that vanish at $\partial M$.
Assume a function $f:\tilde C^1(\overline M)\times \mathbb{R}\to L^r(M)$ to be
continuous.
The main object of our study is the following functional-differential
problem in $M$
\begin{equation}
\label{main_problem}
-u''=f(u,Au),\quad
u\mid_{\partial M}=0,
\end{equation}
We denote a derivative by the stroke:
$$u'=\frac{du}{dx}.$$
Introduce a 1-degree of freedom Hamiltonian system with Hamiltonian
\begin{equation}\label{Hamiltonian}
H(p,q)=\frac{1}{2}p^2+a^{-1}V(aq),\quad V\in C^\infty(\mathbb{R}).
\end{equation}
Throughout of this paper we identify Hamiltonian (\ref{Hamiltonian})
and the corresponding Hamiltonian system.
Assume that Hamiltonian system (\ref{Hamiltonian}) has
a solution $(p,q)(x)$ such that $q(x)\ge 0,\quad x\in M$ and $q(\pm 1)=0$.
Next assume that for any $y\in \tilde C^1(\overline M)$ and $|z|\le q$ one has
\begin{equation}
\label{majorant_property}
|f(y,z)|\le V'(q)
\end{equation}
almost everywhere (a.e.) in $M$.
Let $H^{1,r}_0(M)$ be the Sobolev space of such functions that belong
to $L^r(M)$ with their
first derivatives
and that
have zero trace on $\partial M$.
\begin{theo}
\label{main_theo}
Problem (\ref{main_problem})
has a solution $$u\in \tilde H^{2,r}(M)=H^{2,r}(M)\bigcap
H^{1,r}_0(M),\quad r\ge 2.$$
\end{theo}
\begin{rem}Actually the conditions of Theorem \ref{main_theo} enumerated above
are
necessary. Indeed, if system (\ref{Hamiltonian})
does not have the solution $q(x)$ then
it provides by itself an example of nonexistence in a problem of type
(\ref{main_problem}).
\end{rem}
Consider some examples. Let for any $y\in \tilde C^1(\overline M)$
the function $f$ is estimated as follows:
$$|f(y,z)|\le c(1+2|z|e^{z^2})\quad \mathrm{a.e.,}$$
a positive constant $c$ does not depend on $y$ and $z$.
Then putting $$V(q)=c(q+e^{q^2})$$
we see that inequality (\ref{majorant_property}) holds true.
Though system (\ref{Hamiltonian}) is integrable, for a quantitative estimating
of corresponding integrals a numerical simulation is needed.
The simulation with given function
$V$ shows that for existence of the required solution $q(x)$
it is necessary and sufficient to have $$\sqrt{2ac}\le 1,02371\ldots$$
In the conclusion of this section note
that the function $f$ can contain not only dilation operators.
Consider a bounded operator
$$B=\sum_{|j|\le l}b_jR_{\beta_j}:\tilde C^1(\overline M)\to H^{1,\infty}(M)
,\quad b_j,\beta_j\in \mathbb{R},\quad l\in \mathbb{N}.$$
Taking $g(y,z)\in C(\mathbb{R}_y\times\mathbb{R}_z)$ we
construct the mapping $f$ as follows:
$$f(v,z)=g((Bv)',z).$$ This function maps the space $\tilde C^1(\overline M)\times
\mathbb{R}$ to the space $L^\infty(M)$ continuously.
Thus under the corresponding conditions on its growth it can be
substituted to the right side of (\ref{main_problem}) and Theorem
\ref{main_theo} will be fulfilled.
\section{Proof of Theorem \protect\ref{main_theo}}
It is convenient to consider our problems as 1-dimensional equations of
elliptic type. To stress this we rewrite system (\ref{Hamiltonian})
as follows:
\begin{equation}
\label{Hamiltonian:Laplace}
-\Delta q=V'(aq),\quad \Delta=\frac{d^2}{dx^2}.
\end{equation}
Since the solution $q(x)\ge 0,\quad x\in M$ by formula (\ref{majorant_property})
the function $V'(aq(x))$ is nonnegative and due to equation (\ref{Hamiltonian:Laplace}) we
see that $q(x)$ is concave. Furthermore, equation (\ref{Hamiltonian:Laplace})
is invariant under the change $x\mapsto-x$ thus the solution $q(x)$ is even:
$q(x)=q(-x)$. Particularly these arguments involve that
\begin{equation}
\label{max_q}
\max_{\overline M}q(x)=q(0).\end{equation}
\begin{lem}
\label{operators}
The operator
$$A:\tilde C^1(\overline M)\to C(\overline M)$$
is bounded.
\end{lem}
\proof
Indeed, it follows from the evident estimate:
$$\|Aw\|_{C(\overline M)}\le a\|w\|_{C(\overline M)}.$$
\endproof
\begin{lem}
\label{majorant:R}
Let $v\in L^r(M)$ and $|v(x)|\le q(x)$ almost everywhere (a.e.) in $M$. Then
$|R_\lambda v(x)|\le q(x),\quad |\lambda|\ge 1$ a.e. in $M$.
\end{lem}
\proof
Consider the case $\lambda\ge 1$, the case $\lambda\le -1$ follows from
the same arguments by replacing of the function $v(x)$ with the function
$v(-x)$.
From the assumption we have: $|v(\lambda x)|\le q(\lambda x)$. Recall that
all the functions are extended by zero outside $M$.
Thus it remains to
check that $q(\lambda x)\le q(x)$. For $\lambda x\notin M$ this
inequality is trivial. Assume that $\lambda x\in M$. By concavity of $q$
and inequality (\ref{max_q})
one has:
$$q(x)\ge tq(x/t)+(1-t)q(0)\ge tq(x/t)+(1-t)q(x),\quad 01,\quad 0<\delta<1,\quad r\ge 2.\end{equation}
The set $W\subset \tilde C^1(\overline M)$ is closed and convex. One the other hand,
by
embeddings (\ref{embedding}) and Lemma \ref{map:FW} the set $F(W)$
is precompact in $\tilde C^1(\overline
M)$. By the Schauder fixed point theorem, there exists such a
function $u\in W$ that $F(u)=u$.
To conclude the Proof it remains to notice that
$u\in \mathrm{Rank}\, F\subset \tilde H^{2,r}(M)$.
The Theorem is proved.
{\bf Acknowledgments.} The author wishes to thank Prof. A. L. Skubachevski{\u\i} for
his initial suggestion to consider functional-differential equations with
compression/dilation of the arguments.
Also the author thanks Dr. L. E. Rossovski{\u\i} for useful discussions.
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