\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 80, pp. 1--5.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/80\hfil Hyers-Ulam stability]
{Hyers-Ulam stability for second-order linear differential
 equations with boundary conditions}

\author[P. G\u{a}vru\c{t}\u{a}, S.-M. Jung, Y. Li
  \hfil EJDE-2011/80\hfilneg]
{Pasc G\u{a}vru\c{t}\u{a}, Soon-Mo Jung, Yongjin Li}  % in alphabetical order

\address{Pasc G\u{a}vru\c{t}\u{a} \newline
Department of Mathematics, University Politehnica of Timisoara,
Piata Victoriei, No. 2, 300006 Timisoara, Romania}
\email{pgavruta@yahoo.com}

\address{Soon-Mo Jung \newline
Mathematics Section, College of Science and Technology,
Hongik University, 339-701 Jochiwon, Korea}
\email{smjung@hongik.ac.kr}

\address{Yongjin Li \newline
Department of Mathematics, Sun Yat-Sen University,
 Guangzhou 510275,  China}
\email{stslyj@mail.sysu.edu.cn}

\thanks{Submitted April 26, 2011. Published June 20, 2011.}
\subjclass[2000]{34K20, 26D10}
\keywords{Hyers-Ulam stability, differential equation}
\thanks{Yongjin Li is the corresponding author}

\begin{abstract}
  We prove the Hyers-Ulam stability of linear differential
  equations of second-order with boundary conditions or
  with initial conditions.  That is, if $y$ is an approximate
  solution of the  differential equation $y''+ \beta (x) y = 0$
  with $y(a) = y(b) =0$, then there exists an exact solution
  of the differential equation, near $y$.
\end{abstract}

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

\section{Introduction and preliminaries}

In 1940, Ulam  \cite{Ulam} posed the following problem
concerning the stability of functional equations:
\begin{quote}
 Give conditions in order for a linear mapping near an approximately
 linear mapping to exist.
\end{quote}
The problem for  approximately additive
mappings, on Banach spaces, was solved by Hyers \cite{HYERS}.
The result by Hyers was generalized by
Rassias  \cite{RASSIAS}. Since then, the stability
problems of functional equations have been extensively investigated
by several mathematicians  \cite{JUN,PARK,RASSIAS}.

Alsina and Ger \cite{Alsina} were the first authors who investigated the
Hyers-Ulam stability of a differential equation. In
fact, they proved that if a differentiable function
$y: I \to \mathbb{R}$ satisfies $|y'(t) - y(t)| \leq \varepsilon$
for all $t \in I$, then there exists a differentiable function
$g: I \to \mathbb{R}$ satisfying $g'(t) = g(t)$ for any
$t\in I$ such that $|y(t) - g(t)| \leq 3\varepsilon$ for every $t \in I$.

The above result by Alsina and  Ger was generalized by
Miura,  Takahasi and  Choda \cite{Miura5}, by  Miura
\cite{Miura1}, also by  Takahasi,  Miura and Miyajima \cite{Takahasi1}.
Indeed, they dealt with the Hyers-Ulam
stability of the differential equation $y'(t) = \lambda y(t)$,
while Alsina and  Ger investigated the differential equation
$y'(t) = y(t)$.

 Miura et al \cite{Miura4} proved the Hyers-Ulam stability of the
first-order linear differential equations $y'(t) + g(t)y(t) =
0$, where g(t) is a continuous function, while Jung
\cite{SMJung1} proved the Hyers-Ulam stability of differential
equations of the form $\varphi (t)y' (t) = y(t)$.

Furthermore, the result of Hyers-Ulam stability for first-order
linear differential equations has been generalized
in  \cite{SMJung2,SMJung3,Miura4,Takahasi2,Yongjin,Wang1}.

Let us consider the Hyers-Ulam stability of the  $y''+ \beta (x) y
= 0$, it may be not stable for unbounded intervals. Indeed, for
$\beta (x)= 0$, $\varepsilon = 1/4$ and
$y(x) =x^2/16$ condition  $-\varepsilon < y'' < -\varepsilon$ is
fulfilled and the function $y_0(x) = C_1 x + C_2$, for which
$|y(x) - y_0(x)|=|\frac{x^2}{16} - C_1 x + C_2| $ is bounded, does
not exist.

 The aim of this paper is to investigate the Hyers-Ulam
stability of the second-order linear differential equation
 \begin{equation} \label{e1}
y'' + \beta (x) y = 0
\end{equation}
with boundary conditions
\begin{equation} \label{ebc}
y(a) = y(b) =0
\end{equation}
or with initial conditions
\begin{equation} \label{eic}
y(a) = y'(a) =0,
\end{equation}
where $y\in  C^2[a, b]$, $\beta (x) \in C[a, b]$,
$-\infty < a < b < +\infty$.

First of all, we give the definition of  Hyers-Ulam stability with
boundary conditions and with initial conditions.

\begin{definition} \label{def1.1} \rm
  We say that \eqref{e1} has the Hyers-Ulam
stability with boundary conditions \eqref{ebc}
if there exists a positive constant $K$ with the following property:
For every $\varepsilon > 0$, $y \in C^2[a, b]$, if
 \[
|y''+ \beta(x) y|\leq \varepsilon,
\]
and $y(a) = y(b) =0$, then there exists some $z \in C^2[a, b]$
satisfying
\[
z'' + \beta(x) z = 0
\]
and $z(a) = z(b) =0$, such that $|y(x) - z(x)| <K\varepsilon$.
\end{definition}

\begin{definition} \label{def1.2} \rm
We say that \eqref{e1} has the Hyers-Ulam stability with initial
conditions \eqref{eic} if there exists a positive constant $K$
 with the following property: For every $\varepsilon > 0$,
$y\in C^2[a, b]$, if
 \[
|y''+ \beta(x) y|\leq \varepsilon,
\]
and $y(a) = y'(a) =0$, then there exists some $z \in C^2[a, b]$
satisfying
\[
z'' + \beta(x) z = 0
\]
and $z(a) = z'(a) =0$, such that $|y(x) - z(x)| <K\varepsilon$.
\end{definition}


\section{Main Results}

In the following theorems, we will prove the Hyers-Ulam stability
with boundary conditions and with initial conditions.

Let $\beta (x) =1$, $a=0$, $b=1$, then it is easy to see that
for any $\varepsilon>0$, there exists
$ y(t) = \frac {\varepsilon x^2}{H} - \frac {\varepsilon x}{H}$,
with $H > 4$, such that
$|y''+ \beta (x) y| < \varepsilon$ with $y(0) = y(1) = 0$.

\begin{theorem} \label{thm2.1}
If $\max |\beta(x)| < 8/(b-a)^2$. Then \eqref{e1} has the
Hyers-Ulam stability with boundary conditions \eqref{ebc}.
\end{theorem}

\begin{proof}
For every $\varepsilon > 0$, $y \in C^2[a, b]$, if
$ |y''+ \beta(x) y|\leq \varepsilon$ and
$y(a) = y(b) =0$. Let $M= \max \{|y(x)|: x\in [a, b]\}$, since
$y(a) = y(b)= 0$, there exists $x_0\in (a, b)$ such that
$|y(x_0)| = M$. By Taylor formula, we have
\begin{gather*}
y(a) = y(x_0) + y'(x_0)(x_0 - a) + \frac {y''(\xi)}{2}(x_0 - a)^2,\\
y(b) = y(x_0) + y'(x_0)(b - x_0) + \frac {y''(\eta)}{2}(b - x_0)^2;
\end{gather*}
thus
\[
|y''(\xi)| = \frac {2M}{(x_0 - a)^2}, \quad
|y''(\eta)| = \frac{2M}{(x_0 - b)^2}
\]
On the case $x_0 \in (a, \frac{a+b}{2}]$, we have
\[
\frac {2M}{(x_0 - a)^2} \geq \frac {2M}{(b-a)^2/4}
= \frac {8M}{(b - a)^2}
\]
On the case $x_0 \in [\frac{a+b}{2}, b)$, we have
\[
\frac {2M}{(x_0 - b)^2} \geq \frac {2M}{(b-a)^2/4}
= \frac {8M}{(b - a)^2}.
\]
So
\[
\max |y''(x)| \geq \frac {8M}{(b -a )^2}
= \frac {8}{(b-a)^2} \max |y(x)|.
\]
Therefore,
\[
\max |y(x)| \leq \frac {(b - a)^2}{8} \max |y''(x)|.
\]
Thus
\begin{align*}
 \max |y(x)|
&\leq \frac {(b - a)^2}{8} [\max |y''(x) - \beta (x) y|
 + \max |\beta (x)| \max |y(x)|],\\
&\leq \frac {(b - a)^2}{8} \varepsilon
 + \frac {(b - a)^2}{8} \max |\beta (x)| \max |y(x)|].
\end{align*}
Let $ \eta = (b - a)^2\max |\beta(x)|/8$,
$K = (b -a)^2/\big(8(1 - \eta)\big)$.
Obviously, $z_0(x)=0$ is a solution of
$y'' - \beta (x) y = 0$ with the boundary conditions
$y(a) = y(b) = 0$.
\[
|y - z_0 | \leq K \varepsilon.
\]
Hence \eqref{e1} has the Hyers-Ulam stability with boundary conditions
\eqref{ebc}.
\end {proof}

Next,  we consider  the Hyers-Ulam
stability of $y''+ \beta (x) y =0$ in $[a, b]$ with initial
conditions \eqref{eic}. For example, let $\beta (x) =1$, $a=0$, $b=1$,
then for any $\varepsilon>0$, there exists
$ y(t) = \frac {\varepsilon x^2}{H}$ with $H > 3$, such that
$|y''+ \beta (x) y| < \varepsilon$ with $y(0) = y'(0) = 0$.

\begin{theorem} \label{thm2.2}
If $\max |\beta(x)| < 2/(b-a)^2$. Then \eqref{e1} has the
Hyers-Ulam stability with initial conditions \eqref{eic}.
\end{theorem}

\begin{proof} For every $\varepsilon > 0$,
$y \in C^2[a, b]$, if $ |y''+ \beta(x) y|\leq \varepsilon$ and
$y(a) = y'(a) =0$. By Taylor formula, we have
\[
y(x) = y(a) + y'(a)(x - a) + \frac {y''(\xi)}{2}(x - a)^2.
\]
Thus
\[
|y(x)| = |\frac {y''(\xi)}{2}(x - a)^2 |
\leq \max |y''(x)|\frac {(b - a)^2}{2};
\]
so, we obtain
\begin{align*}
\max |y(x)|
&\leq \frac {(b - a)^2}{2} [\max |y''(x) - \beta (x) y|
+ \max |\beta (x)| \max |y(x)|]\\
&\leq \frac {(b - a)^2}{2} \varepsilon
+ \frac {(b - a)^2}{2} \max |\beta (x)| \max |y(x)|].
\end{align*}
Let $ \eta = (b - a)^2 \max |\beta(x)|/2$,
$K = (b - a)^2/\big(2(1 - \eta)\big)$.
It is easy to see that $z_0(x)=0$ is a solution
of $y'' - \beta (x) y = 0$ with the initial conditions
$y(a) = y'(a)= 0$.
\[
|y - z_0 | \leq K \varepsilon.
\]
Hence \eqref{e1} has the Hyers-Ulam stability with initial conditions
\eqref{eic}.
\end {proof}

\subsection*{Acknowledgements}
This work  was supported by grant 10871213 from the
National Natural Science Foundation of China.

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\end{document}