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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 284, 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 2015 Texas State University.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2015/284\hfil Well-posedness of an elliptic equation]
{Well-posedness of an elliptic equation with involution}
\author[A. Ashyralyev, A. M. Sarsenbi \hfil EJDE-2015/284\hfilneg]
{Allaberen Ashyralyev, Abdizhahan M. Sarsenbi}
\address{Allaberen Ashyralyev \newline
Department of Elementary Mathematics Education, Fatih University,
Istanbul, Turkey. \newline
Department of Applied Mathematics,
ITTU, Ashgabat, Turkmenistan}
\email{aashyr@fatih.edu.tr}
\address{Abdizhahan M. Sarsenbi \newline
Department of Mathematical Methods and Modeling, M. Auezov SKS
University, \newline
Shimkent, Kazakhstan.\newline
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan}
\email{abzhahan@gmail.com}
\thanks{Submitted July 25, 2015. Published November 11, 2015.}
\subjclass[2010]{35J15}
\keywords{Elliptic equation; Banach space; self-adjoint;
positive definite; \hfill\break\indent stability estimate; involution}
\begin{abstract}
In this article, we study a mixed problem for an elliptic equation
with involution. This problem is reduced to boundary
value problem for the abstract elliptic equation in a Hilbert space
with a self-adjoint positive definite operator. Operator tools
permits us to obtain stability and coercive stability estimates in
H\"older norms, in $t$, for the solution.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\section{Introduction}\label{sec:1}
Elliptic equations have important applications in a wide range
of applications such as physics, chemistry, biology and ecology and other
fields. In mathematical modeling, elliptic equations are used together with
boundary conditions specifying the solution on the boundary of the domain.
Dirichlet and Neumann conditions are examples of classical boundary
conditions. The role played by coercive inequalities (well-posedness) in the
study of local boundary-value problems for elliptic and parabolic
differential equations is well known (see, e.g., \cite{g1,g2} and
the references therein). Mathematical models of various physical, chemical,
biological or environmental processes often involve nonclassical conditions.
Such conditions are usually identified as nonclassical boundary conditions
and reflect situations when a data on the domain boundary can not be
measured directly, or when the data on the boundary depends on the data
inside the domain. Well-posedness of various classical and nonclassical
boundary value problems for partial differential and difference equations
has been studied extensively by many researchers with the operator method
tool (see \cite{i3,g4,g11,g10,g15,g12,aa12,g13,bb12,g7,g3,g5,g6,g8,g9}).
The theory of functional-differential equations with the involution has
received less attention than functional-differential equations.
Except for a few works \cite{i3,i2,i1}
parabolic differential and difference equations
with the involution are not studied enough in the literature.
For example, in \cite{i1}, the mixed problem for a parabolic
partial differential equation with the involution with respect to $t$
\begin{equation*}
u_{t}(t,x) =au_{xx}(t,x) +bu_{xx}(-t,x), \quad 00$ and $\sigma >0$ is a sufficiently large number.
Here, we study problem \eqref{2.1} for an
elliptic equation with the involution by using the operator tool
in monograph \cite{8bbb}. We establish stability estimates in the
$C([0,T],L_2[-l,l])$ norm, and coercive stability estimates in the
$C^{\alpha }([0,T],L_2[-l,l])$ and $C_{0T}^{\alpha }([0,T],L_2[-l,l])$
norms for the solution of this problem.
\section{Preliminaries and statement of main results}
\label{sec:2}
To formulate our results, we introduce the Hilbert $
L_2[-l,l] $ of all integrable functions\ $f$ defined on $[-l,l]$, equipped
with the norm
\begin{equation*}
\| f\| _{L_2[-l,l]}=\Big(\int_{-l}^{l}|f(x)|^{2}dx\Big) ^{1/2}.
\end{equation*}
We introduce the inner product in $L_2[-l,l]$ by
\begin{equation*}
\langle u,v\rangle =\int_{-l}^{l}u(x)v(x)dx.
\end{equation*}
In this article, $C^{\alpha }([0,T] ,E) $ and
$C_{0T}^{\alpha }([0,T] ,E)$ $(0<\alpha <1)$ stand for Banach spaces
of all abstract continuous functions $\varphi (t)$
defined on $[0,T] $ with values in $E$ satisfying a H\"older
condition for which the following norms are finite
\begin{gather*}
\| \varphi \| _{C^{\alpha }([0,T],E) }
=\| \varphi \| _{C([0,T] ,E) }+\sup_{0\leq t0}\lambda ^{1-\alpha }\|
A\exp \{ -\lambda A\} v\| _{E}. \label{b6}
\end{equation}
Finally, we introduce a differential operator $A^{x}$ defined by the formula
\begin{equation}
A^{x}v(x)=-(a(x)v_x(x) _x-\beta (a(-x)v_x(
-x) ) _x+\sigma v(x) \label{aaaa}
\end{equation}
with the domain $D(A^{x})=\{ u,u_{xx}\in L_2[-l,l]:u(-l)
=u(l) ,u'(-l) =u'(l)\} $.
We can rewrite problem \eqref{2.1} in the following
abstract form
\begin{equation}
-u_{tt}(t)+Au(t)=f(t) ,\quad 00$ and $\sigma >0$ is a sufficiently large
number. The stability estimates in $C([0,T],L_2[-l,l])$ norm and coercive
stability estimates in $C^{\alpha }([0,T],L_2[-l,l])$ and $C_{0T}^{\alpha
}([0,T],L_2[-l,l])$ norms for the solution of problem
\eqref{2.1as} can be established. Finally, applying the result of the monograph
\cite{8bbb}, the high order of accuracy two-step difference schemes for the
numerical solution of mixed problems \eqref{2.1} and \eqref{2.1as} can be
presented. Of course, the stability estimates for the solution of these
difference schemes have been established without any assumptions about the
grid steps.
\subsection*{Acknowledgements}
The authors are thankful to the anonymous reviewers for their valuable
suggestions and comments, which improved this article.
This work is supported by the Grant No. 5414/GF4 of the Committee of
Science of Ministry of Education and Science of the Republic of Kazakhstan.
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\end{document}