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

\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 0<x<l,\; -\infty <t<\infty
\end{equation*}
with the Dirichlet condition in $x$ was studied. The Fourier method is
common used method to get existence of unbounded solutions and non existence
of solution dependent on coefficients $a$ and $b$.

Papers \cite{i4,i5}, the mixed problem for a first-order partial
differential equation with the involution was investigated. The Fourier
method was used to find a classical solution of the mixed problem for a
first-order differential equation with involution. Application of the
Fourier method was substantiated using refined asymptotic formulas obtained
for eigenvalues and eigenfunctions of the corresponding spectral problem.
The Fourier series representing the formal solution was transformed using
certain techniques, and the possibility of its term-by-term differentiation
was proved.

 The paper \cite{i6} was devoted to the study of first order linear
problems with involution and periodic boundary value conditions. First, it
was proved a correspondence between a large set of such problems with
different involutions to later focus attention to the case of the
reflection. Then, different cases for which a Green's function can be
obtained explicitly, it was derived several results in order to obtain
information about its sign. More general existence and uniqueness of
solution results were established.

 In  \cite{i7,i8}, the basic properties of systems of
eigenfunctions and associated functions for one kind of generalized spectral
problems for a second-order and a first-order ordinary differential
operators.

In  \cite{i9}, the notion of regularity of boundary conditions for
the simplest second-order differential equation with a deviating argument
was introduced. The Riesz basis property for a system of root vectors of the
corresponding generalized spectral problem with regular boundary conditions
(in the sense of the introduced definition) was established. Examples of
irregular boundary conditions to which the theory of Il'in basis property
can be applied were given.

 In  \cite{i10}, a nonclassical operator $L$ in $
L_2(-1,1) $, generated by the differential expression with
shifted argument
\begin{equation}
Lu:=-u''(-x),\quad -1<x<1  \label{1}
\end{equation}
and the boundary conditions
\begin{equation}
\alpha _{j}u'(-1)+\beta _{j}u'(1)+\alpha _{j1}u(-1)+\beta
_{j1}u(1)=0,\quad j=1,2  \label{2}
\end{equation}
was considered. For the spectral problem corresponding to \eqref{1}
and \eqref{2}, the author introduces a concept of regular boundary conditions
\eqref{2}. In some sense, the definition is similar to that of strong (Birkhoff)
regular boundary conditions \eqref{2} for second-order ordinary differential
equations. The main result of the paper states that a system of
eigenfunctions and associated functions of the operator $L$ forms an
unconditional basis of the space $L_2(-1,1) $.

In the paper \cite{i11}, spectral problem for a model second-order
differential operator with an involution was considered. An operator was
given by the differential expression $Lu=-u''(-x)$
 and boundary conditions of general form. A criterion for the basis
property of the systems of eigenfunctions of this operator in terms of the
coefficients in the boundary conditions was obtained.

In this article, we study the mixed problem for an elliptic equation
with the involution
\begin{equation}
\begin{gathered}
-\frac{\partial ^{2}u(t,x) }{\partial t^{2}}
=( a(x)u_x(t,x) ) _x+\beta (a(-x)u_x(
t,-x) ) _x-\sigma u(t,x) +f(t,x) ,\\
-l<x<l, \; 0<t<T, \\
u(t,-l) =u(t,l) ,\quad
u_x(t,-l) =u_x(t,l) ,\quad 0\leq t\leq T, \\
u(0,x) =\varphi (x) ,\quad
u(T,x) =\psi (x) ,\quad -l\leq x\leq l, \\
\varphi (-l) =\varphi (l) ,\quad
\psi (-l) =\psi (l) ,\quad \varphi '(-l) =\varphi '(l) ,\quad
\psi '(-l) =\psi '(l) ,
\end{gathered}  \label{2.1}
\end{equation}
where $u(t,x) $ is unknown function, $\varphi (x),\psi (x) ,a(x) $,
and $f(t,x) $ are sufficiently smooth functions,
$a\geq a(x) =a(-x) \geq \delta >0$ 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 t<t+\tau \leq T}
\frac{\| \varphi (t+\tau )-\varphi (t)\| _{E}}{\tau ^{\alpha }},
\\
\| \varphi \| _{C_{0T}^{\alpha }([0,T]
,E) }=\| \varphi \| _{C([0,T]
,E) }+\sup_{0\leq t<t+\tau \leq T}\frac{(t+\tau ) ^{\alpha
}(T-t) ^{\alpha }\| \varphi (t+\tau )-\varphi
(t)\| _{E}}{\tau ^{\alpha }},
\end{gather*}
respectively. Here, $C([0,T] ,E) $ stands for the
Banach space of all abstract continuous functions $\varphi (t)$ defined on
$[0,T] $ with values in $E$ equipped with the norm
\begin{equation*}
\| \varphi \| _{C([0,T] ,E)}=\max_{0\leq t\leq T}\| \varphi (t)\| _{E}.
\end{equation*}

\begin{definition} \label{def2.1}\rm
An operator $A$ densely defined in a Banach space $E $ with domain $D(A)$
is called positive in $E$, if its spectrum $\sigma_{A}$ lies in the interior
 of the sector of angle $\varphi $, $0<\varphi<\pi $, symmetric with respect
to the real axis, and moreover on the edges of this sector
$S_{1}(\varphi ) =\{\rho e^{i\varphi }:0\leq \rho
\leq \infty \}$ and
$S_2(\varphi ) =\{\rho e^{-i\varphi}:0\leq \rho \leq \infty \}$,
and outside of the sector the resolvent
$(\lambda -A) ^{-1}$ is subject to the bound (see \cite{i13})
\begin{equation*}
\| (A-\lambda )^{-1}\| _{E\rightarrow E}\leq \frac{M}{1+| \lambda | }.
\end{equation*}
The infimum of all such angles $\varphi $ is called the spectral angle of
the positive operator $A$ and is denoted by $\varphi (A)=\varphi (A,E)$.
The operator $A$ is said to be strongly positive in a Banach space $E$ if
$\varphi (A,E)<\frac{\pi }{2}$.
\end{definition}

Throughout this article, we will indicate with $M$ positive constants
which can be different from time to time and we are not interested in
precise. We will write $M(\alpha ,\beta ,\cdots )$ to stress the fact that
the constant depends only on $\alpha ,\beta ,\cdots $.

With the help of the positive operator $A$, we introduce the fractional
spaces $E_{\alpha }=E_{\alpha }(E,A) ,0<\alpha <1$, consisting
of all $v\in E$ for which the following norm is finite \cite{i13}:
\begin{equation}
\| v\| _{E_{\alpha }}
=\| v\| _{E}+\sup_{\lambda >0}\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 0<t<T,\; u(0)=\varphi , \; u(T)=\psi
\label{sarsenbi}
\end{equation}
in a Hilbert space $H=L_2[-l,l]$ with the unbounded operator $A=A^{x}$
defined by formula $\eqref{aaaa} $. Here, $f(t)=f(t,x) $ and
$u(t) =u(t,x) $ are known
and unknown abstract functions respectively and they are defined on
$(0,T)$ with values in $H=L_2[-l,l]$, $\varphi =\varphi (x) $,
$\psi =\psi (x) $, and $a=a(x) $ are given smooth
elements of $H=L_2[-l,l]$.

The main result of present paper is the following theorem on stability
estimates of \eqref{2.1} in spaces $C([0,T)],L_2[-l,l]) $
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{2.1}.

\begin{theorem} \label{thm2.1}
Assume that $\delta -a| \beta |\geq 0,\varphi (x) ,\varphi _{xx}(x) ,\psi (
x) ,\psi _{xx}(x) \in L_2[-l,l]$ and
$f(t,x) \in C_{0T}^{\alpha }([0,T],L_2[-l,l])$. Then the solution
of  \eqref{2.1} satisfies stability estimates
\begin{equation*}
\| u\| _{C([0,T],L_2[-l,l]) }
\leq M(\delta ,\sigma ,\beta ,l) [\| \varphi \|_{L_2[-l,l]}
+\| \psi \| _{L_2[-l,l]}+\|f\| _{C([0,T],L_2[-l,l]) }],
\end{equation*}
and the coercive stability estimates
\begin{align*}
&\| u_{tt}\| _{C_{0T}^{\alpha }([0,T],L_2[-l,l])}
 +\| u_{xx}\| _{C_{0T}^{\alpha }([0,T],L_2[-l,l])} \\
&\leq M(\delta ,\sigma ,\alpha ,\beta ,l)
\big[\|\varphi _{xx}\| _{L_2[-l,l]}+\| \psi _{xx}\|
_{L_2[-l,l]}+\| f\| _{C_{0T}^{\alpha }([0,T],L_2[-l,l])}\big]\,.
\end{align*}
\end{theorem}

\begin{theorem} \label{thm2.2}
Assume  $\delta -a| \beta |\geq 0$,
$\varphi (x) ,\varphi _{xx}(x) ,\psi (x) ,\psi _{xx}(x) \in L_2[-l,l]$ and
\begin{gather*}
(a(x)\psi_x(x) ) _x+\alpha (a(-x)\psi _x(-x) ) _x-\sigma \psi (x) +f(T,x)
=0,\\
(a(x)\varphi _x(x) ) _x+\alpha (a(-x)\varphi _x(-x) ) _x-\sigma \varphi (
x) +f(0,x) =0
\end{gather*}
 and
 $f(t,x) \in C^{\alpha }([0,T],L_2[-l,l])$. Then the solution of 
\eqref{2.1} satisfies coercive stability estimate
\begin{equation*}
\| u_{tt}\| _{C^{\alpha }([0,T],L_2[-l,l])}+\|
u_{xx}\| _{C^{\alpha }([0,T],L_2[-l,l])}\leq M(\delta
,\sigma ,\alpha ,\beta ,l) \| f\| _{C^{\alpha
}([0,T],L_2[-l,l])}\,.
\end{equation*}
\end{theorem}

The proofs of Theorem \ref{thm2.1} and \ref{thm2.2} are based on the following
abstract Theorem on stability of problem \eqref{sarsenbi} in
$C([0,T ] ,E)$ space and coercive stability in $C^{\alpha }([0,T],E)$ and
$C_{0T}^{\alpha }([0,T],E)$ spaces and on self-adjointness and positive
definite\ of the unbounded operator $A=A^{x}$ defined by formula $
\eqref{aaaa} $ in $L_2[-l,l]$ space.

\begin{theorem}[\cite{8bbb}] \label{thm2.3}
Let $A$ be positive operator in a Banach space $E$ and
$f\in C_{0T}^{\alpha }([0,T] ,E)$ $(0<\alpha<1) $. Then, for the
solution of the boundary value problem \eqref{sarsenbi},
stability and coercive stability inequalities
\begin{gather*}
\| u\| _{C([0,T] ,E)}\leq M[\| \varphi \|
_{E}+\| \psi \| _{E}+\| f\| _{C([0,T] ,E)}] ,
\\
\begin{aligned}
&\| u''\| _{C_{0T}^{\alpha }([0,T],E)}+\| Au\| _{C_{0T}^{\alpha }([0,T] ,E)}\\
&\leq M\big[\| A\varphi \| _{E}+\| A\psi \| _{E}+\frac{1}{
\alpha (1-\alpha ) }\| f\| _{C_{0T}^{\alpha }([0,T
] ,E)}\big]
\end{aligned}
\end{gather*}
hold. Moreover, assume that $A\varphi -f(0)=0$, $A\psi -f(T)=0$ and $f\in
C^{\alpha }([0,T] ,E)$ $(0<\alpha <1) $. Then, for
the solution of the boundary value problem \eqref{sarsenbi}, the coercive
stability inequality
\begin{equation*}
\| u''\| _{C^{\alpha }([0,T] ,E)}+\|Au\| _{C^{\alpha }([0,T] ,E)}
\leq \frac{M}{\alpha (1-\alpha ) }\| f\| _{C^{\alpha }([0,T] ,E)}
\end{equation*}
holds.
\end{theorem}

 In the next Section, the self-adjointness and positive definiteness
 of the operator $A=A^{x}$ defined by formula \eqref{aaaa}
 in $L_2[-l,l]$ space will be studied.

\section{Self-adjointness and positive definiteness}

 \begin{theorem} \label{thm3.1}
Assume that $\delta -a| \beta
| \geq 0$. Then, the operator $A=A^{x}$ defined by formula 
\eqref{aaaa} is a self-adjoint and positive definite operator in 
$L_2[-l,l]$ space with the spectral angle $\varphi (A,H)=0$.
\end{theorem}

\begin{proof} 
We will prove the following identity
\begin{equation}
\langle A^{x}u,v\rangle =\langle u,A^{x}v\rangle ,u,v\in D(A^{x}),  \label{i1}
\end{equation}
and estimate
\begin{equation}
\langle A^{x}u,u\rangle \geq \sigma \langle u,\quad 
u\rangle ,u\in D(A^{x}).  \label{i2}
\end{equation}
Applying definition of the inner product and integrating by part, we obtain
\begin{align*}
\langle A^{x}u,v\rangle 
&=-\int_{-l}^{l}(a(x)u_x(x)) _xv(x)dx
-\beta \int_{-l}^{l}(a(-x)u_x(-x) ) _xv(x)dx\\
&\quad +\sigma \int_{-l}^{l}u(x) v(x)dx \\
&=-a(l)u_x(l)v(l)+a(-l)u_x(-l)v(-l)
 +\int _{-l}^{l}a(x)u_x(x)v_x(x)dx \\
&\quad +\beta [-a(-l)u_x(-l)v(-l)+a(l)u_x(l)v(l)] \\
&\quad +\beta\int_{-l}^{l}a(-x)u_x(-x) v_x(x)dx+\sigma
\int_{-l}^{l}u(x) v(x)dx.
\end{align*}
From conditions $a(x) =a(-x) $ and $u,v\in D(A^{x})$, it follows that
\begin{equation*}
-a(l)u_x(l)v(l)+a(-l)u_x(-l)v(-l)=0.
\end{equation*}
Then,
\begin{equation}
\begin{aligned}
\langle A^{x}u,v\rangle
&=\int_{-l}^{l}a(x)u_x(x)v_x(x)dx+\beta
\int_{-l}^{l}a(x)u_x(x) v_x(-x)dx\\
&\quad +\sigma \int_{-l}^{l}u(x) v(x)dx. 
\end{aligned} \label{i3}
\end{equation}
In a similar manner, one establishes formula
\begin{equation*}
\langle u,A^{x}v\rangle
=\int_{-l}^{l}a(x)u_x(x)v_x(x)dx+\beta
\int_{-l}^{l}u_x(x) a(-x)v_x(-x)dx
+\sigma \int_{-l}^{l}u(x) v(x)dx.
\end{equation*}
Therefore, from these formulas and condition 
$a(x) =a( -x) $ it follows identity \eqref{i1}. Now, we will prove the estimate 
\eqref{i2}. Applying the identity \eqref{i3}, we obtain
\begin{align*}
&\langle A^{x}u,u\rangle\\
&=\int_{-l}^{l}a(x)u_x(x)u_x(x)dx+\beta
\int_{-l}^{l}u_x(x) a(-x)u_x(-x)dx+\sigma
\int_{-l}^{l}u(x) u(x)dx \\
&\geq \sigma \langle u,u\rangle +\delta
\int_{-l}^{l}u_x(x)u_x(x)dx+\beta \delta
\int_{-l}^{l}a(-x)u_x(x) u_x(-x)dx.
\end{align*}
Using the Cauchy inequality, we obtain
\begin{align*}
\int_{-l}^{l}a(-x)u_x(x) u_x(-x)dx
&\leq a\Big(\int_{-l}^{l}| u_x(x) |^{2}dx\Big) ^{1/2}
\Big(\int_{-l}^{l}| u_x(-x) | ^{2}dx\Big) ^{1/2}\\
&=a\langle u_x,u_x\rangle .
\end{align*}
Since $\beta \geq -| \beta | $, we have 
\begin{equation*}
\beta \int_{-l}^{l}a(-x)u_x(x) u_x(-x)dx\geq
-| \beta | a\langle u_x,u_x\rangle .
\end{equation*}
Then
\begin{equation*}
\langle A^{x}u,u\rangle \geq \sigma \langle u,u\rangle
+(\delta -| \beta | a) \langle
u_x,u_x\rangle \geq \sigma \langle u,u\rangle .
\end{equation*}
The proof is complete.
\end{proof}

\section{Conclusion}\label{sec:3}

In the present study, mixed problem \eqref{2.1} for an elliptic equation
with the involution is investigated. 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 this problem are established.

Moreover, applying results of paper \cite{g15} and the present paper, the
nonlocal problem for an elliptic equation with the involution
\begin{equation}
\begin{gathered}
-\frac{\partial ^{2}u(t,x) }{\partial t^{2}}
=(a(x)u_x(t,x) ) _x+\beta (a(-x)u_x(t,-x) ) _x-\sigma u(t,x)
+f(t,x) ,\\
-l<x<l,\; 0<t<T, \\
u(t,-l) =u(t,l) ,\quad u_x(t,-l) =u_x(t,l) , \quad 0\leq t\leq T, \\
u(0,x)=u(T,x)+\varphi (x), \quad
u_{t}(0,x)=u_{t}(T,x)+\psi (x),\quad -l\leq x\leq l, \\
\varphi (-l) =\varphi (l) ,\quad 
\psi (-l) =\psi (l) ,\quad \varphi '(-l) =\varphi '(l) ,\psi '(-l) =\psi '(l)
\end{gathered}  \label{2.1as}
\end{equation}
can be studied. Here, $u(t,x) $ is an unknown function, 
$\varphi(x) ,\psi (x) ,a(x) $, and $f(t,x) $ are sufficiently smooth functions, 
$a\geq a(x) =a(-x) \geq \delta >0$ 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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