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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 215, 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 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/215\hfil Superstability of differential equations]
{Superstability of differential equations with boundary conditions}

\author[J. Huang, Q. H. Alqifiary, Y. Li \hfil EJDE-2014/215\hfilneg]
{Jinghao Huang, Qusuay H. Alqifiary, Yongjin Li} 

\address{Jinghao Huang \newline
Department of Mathematics, Sun Yat-Sen
University, Guangzhou, China}
\email{hjinghao@mail2.sysu.edu.cn}

\address{Qusuay H. Alqifiary \newline
Department of Mathematics,
University of Belgrade,
Belgrade, Serbia.\newline
University of Al-Qadisiyah, Al-Diwaniya, Iraq}
\email{qhaq2010@gmail.com}

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

\thanks{Submitted January 28, 2014. Published October 14, 2014.}
\subjclass[2000]{44A10, 39B82, 34A40, 26D10}
\keywords{Hyers-Ulam stability; superstability; linear differential equations;
\hfill\break\indent boundary conditions; initial conditions}

\begin{abstract}
 In this article, we establish the superstability of differential equations
 of second order with boundary conditions or with initial conditions
 as well as the superstability of differential equations of higher order
 with initial conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

In 1940, Ulam \cite{Ul} posed a problem concerning the
stability of functional equations:
``Give conditions in order for a linear function near an
approximately linear function to exist.''

A year later, Hyers \cite{Hy1} gave an answer to the problem of
Ulam for additive functions defined on Banach spaces:
Let $X_1$ and $X_2$ be real Banach spaces and $\varepsilon > 0$.
Then for every function $f : X_1 \to X_2$ satisfying
$$
\| f(x+y) - f(x) - f(y) \| \leq \varepsilon  \quad (x, y \in X_1),
$$
there exists a unique additive function $A : X_1 \to X_2$ with
the property
$$
\| f(x) - A(x) \| \leq \varepsilon  \quad (x \in X_1).
$$

After Hyers's result, many mathematicians have extended Ulam's
problem to other functional equations and generalized Hyers's
result in various directions (see \cite{Cz,Hy2,Ju4,Ra1}).
A generalization of Ulam's problem was recently proposed by
replacing functional equations with differential equations:
The differential equation
$\varphi \big( f, y, y', \dots, y^{(n)} \big) = 0$ has the
Hyers-Ulam stability if for given $\varepsilon > 0$ and a
function $y$ such that
$$
\big| \varphi\big( f, y, y', \dots, y^{(n)} \big) \big|
\leq \varepsilon,
$$
there exists a solution $y_0$ of the differential equation such
that $| y(t) - y_0(t) | \leq K(\varepsilon)$ and
$\lim_{\varepsilon \to 0} K(\varepsilon) = 0$.

Ob\a{l}oza seems to be the first author who has investigated
the Hyers-Ulam stability of linear differential equations
(see \cite{ob1,ob2}).
Thereafter, Alsina and Ger published their paper \cite{AlGe},
which handles the Hyers-Ulam stability of the linear differential
equation $y'(t) = y(t)$:
If a differentiable function $y(t)$ is a solution of the
inequality $| y'(t) - y(t) | \leq \varepsilon$ for any
$t \in (a, \infty)$, then there exists a constant $c$ such that
$| y(t) - ce^t | \leq 3\varepsilon$ for all $t \in (a, \infty)$.

Those previous results were extended to the Hyers-Ulam stability
of linear differential equations of first order and higher
order with constant coefficients in \cite{Mi2,Ta1,Ta2} and
in \cite{Mi1}, respectively.
Furthermore, Jung has also proved the Hyers-Ulam stability of
linear differential equations (see \cite{Ju1,Ju2,Ju3}).
Rus investigated the Hyers-Ulam stability of differential and
integral equations using the Gronwall lemma and the technique
of weakly Picard operators (see \cite{Ru1,Ru2}).
Recently, the Hyers-Ulam stability problems of linear
differential equations of first order and second order with
constant coefficients were studied by using the method of
integral factors (see \cite{LiSh,Wa}).
The results given in \cite{Ju2,LiSh,Mi2} have been generalized
by Cimpean and Popa \cite{CiPo} and by Popa and Ra\c{s}a
\cite{popa1,popa2} for the linear differential equations of
$n$th order with constant coefficients.
Furthermore, the Laplace transform method was recently applied
to the proof of the Hyers-Ulam stability of linear differential
equations (see \cite{Re}).

In 1979, Baker,  Lawrence and  Zorzitto \cite{JJF} proved a new type
of stability of the exponential equation $f(x+y)=f(x)f(y)$. More precisely,
they proved that if a complex-valued mapping $f$ defined on a normed vector
space satisfies the inequality $|f(x+y)-f(x)f(y)|\le\delta$ for some given
 $\delta > 0$ and for all $x,y$, then either $f$ is bounded or $f$ is exponential.
Such a phenomenon is called the superstability of the exponential equation,
which is a special kind of Hyers-Ulam stability. It seems that the results of
 G\v{a}vru\c{t}a,  Jung and  Li \cite{Ga} are the earliest one concerning
the superstability of differential equations.

In this paper,  we prove the superstability of the linear
differential equations of second order with initial and boundary conditions
as well as linear differential equations of higher order in the form of
\eqref{eq:e5.0.1} with initial conditions.

First of all, we give the definition of superstability with initial and
boundary conditions.

\begin{definition} \label{def1.1} \rm
Assume that for any function $y\in C^n[a,b]$, if $y$ satisfies the differential
inequality
\[
|\varphi \big( f, y, y', \dots, y^{(n)} \big) |\le \epsilon
\]
for all $x\in[a,b]$ and for some $\epsilon\ge 0$ with initial(or boundary)
conditions, then either $y$ is a solution of the differential equation
\begin{equation}
\varphi \big( f, y, y', \dots, y^{(n)} \big) = 0\label{eq:linequ}
\end{equation}
 or $|y(x)|\le K\epsilon$ for any $x\in[a,b]$, where $K$ is a constant not
depending on $y$ explicitly. Then, we say that \eqref{eq:linequ}
has superstability with initial (or boundary) conditions.
\end{definition}


\section{Preliminaries}

\begin{lemma}\label{lem2.1}
Let $y \in C^2[a,b]$ and $y(a)=0=y(b)$, then
\[
\max|y(x)|\leq \frac{(b-a)^2}{8}\max|y''(x)|.
\]
\end{lemma}

\begin{proof}
Let $M=\max \{|y(x)|:x \in [a,b] \}$. Since $y(a)=0=y(b)$, there exists
 $x_0 \in (a,b)$ such that $|y(x_0)|=M$. By Taylor's 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
$$
|y''(\xi)|=\frac{2M}{(x_0-a)^2},\quad
|y''(\eta)|=\frac{2M}{(b-x_0)^2}.
$$
In the case $x_0 \in (a,\frac{a+b}{2}]$, we have
$$
\frac{2M}{(x_0-a)^2}\geq \frac{2M}{(b-a)^{2}/4}=\frac{8M}{(b-a)^2};
$$
In 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,
$$
\max|y(x)| \leq \frac{(b-a)^2}{8}\max|y''(x)| .
$$
\end{proof}

\begin{lemma}\label{lem2.2}
Let $y \in C^2[a,b]$ and $y(a)=0=y'(a)$, then
\[
\max|y(x)|\leq \frac{(b-a)^2}{2}\max|y''(x)|.
\]
\end{lemma}

\begin{proof}
By Taylor formula, we have
$$
y(x)=y(a)+y'(a)(x-a)+\frac{y''(\xi)}{2}(x-a)^2 .
$$
We have $(x-a)^2 \leq (b-a)^2$. Therefore,
$$
y(x) \leq \frac{y''(\xi)}{2}(b-a)^2 .
$$
Thus
$$
\max|y(x)| \leq \frac{(b-a)^2}{2}\max|y''(x)| .
$$
\end{proof}

\begin{theorem}[\cite{Ga}]\label{thm2.3}
Consider the differential equation
\begin{equation} \label{eq:e2.1}
y''(x)+\beta (x)y(x)=0
\end{equation}
with boundary conditions
\begin{equation}\label{eq:e2.2}
y(a)=0=y(b),
\end{equation}
where $y \in C^2[a,b]$, $\beta(x)\in C[a,b]$, $-\infty<a<b<+\infty$.
If $\max |\beta(x)|<8/(b-a)^2$. Then \eqref{eq:e2.1} has the superstability
with boundary conditions \eqref{eq:e2.2}.
\end{theorem}

\begin{theorem}[\cite{Ga}]\label{thm2.4}
Consider the differential equation \eqref{eq:e2.1}
with initial conditions
\begin{equation}\label{eq:e2.3}
y(a)=0=y'(a),
\end{equation}
where $y \in C^2[a,b]$, $\beta(x)\in C[a,b],\ -\infty<a<b<+\infty$.
If $\max |\beta(x)|<2/(b-a)^2$. Then \eqref{eq:e2.1} has the superstability
with initial conditions \eqref{eq:e2.3}.
\end{theorem}


\section{Main results}

In the following theorems, we investigate the superstability of the differential equation
\begin{equation}\label{eq:e3.0.1}
y''(x)+p(x)y'(x)+q(x)y(x)=0
\end{equation}
with boundary conditions
\begin{equation}\label{eq:e3.0.2}
y(a)=0=y(b)
\end{equation}
or initial conditions
\begin{equation}\label{eq:e3.0.3}
y(a)=0=y'(a),
\end{equation}
where $y \in C^2[a,b]$, $p \in C^1[a,b]$, $q \in C^{0}[a,b]$,
$-\infty<a<b<+\infty$.

\begin{theorem}\label{Th_3.1}
If
\begin{equation} \label{ast}
\max \{ |q(x)- \frac{1}{2}p'(x)-\frac{p^{2}(x)}{4} | \} < 8/(b-a)^2.
\end{equation}
Then \eqref{eq:e3.0.1} has the superstability with boundary conditions
\eqref{eq:e3.0.2}.
\end{theorem}

\begin{proof}
Suppose that $y \in C^2[a,b] $ satisfies the inequality
\begin{equation}\label{eq:e3.1.1}
|y''(x)+p(x)y'(x)+q(x)y(x)|\leq \epsilon
\end{equation}
for some $\epsilon >0$.
Let
\begin{equation}\label{eq:e3.0.0}
u(x)=y''(x)+p(x)y'(x)+q(x)y(x),
\end{equation}
for all $x \in [a,b]$, and define $z(x)$ by
\begin{equation}\label{eq:e3.1.2}
y(x)=z(x)\exp\Big(-\frac{1}{2} \int_a^x { p(\tau)d\tau} \Big).
\end{equation}
By a substitution \eqref{eq:e3.1.2} in \eqref{eq:e3.0.0}, we obtain
\[
z''(x)+\big( q(x)- \frac{1}{2}p'(x)-\frac{p^{2}(x)}{4}\big)z(x)
=u(x)\exp\Big(\frac{1}{2} \int_a^x { p(\tau)d\tau} \Big).
\]
 Then it follows from inequality \eqref{eq:e3.1.1} that
\begin{align*}
\big|z''(x)+\big( q(x)- \frac{1}{2}p'(x)
-\frac{p^{2}(x)}{4}\big)z(x)\big|
& =  |u(x)exp(\frac{1}{2} \int_a^x { p(\tau)d\tau} )| \\
&\leq  \exp\Big(\frac{1}{2} \int_a^x { p(\tau)d\tau}\Big)  \epsilon.
\end{align*}
From \eqref{eq:e3.0.2} and \eqref{eq:e3.1.2} we have
\begin{equation}\label{eq:e3.1.3}
z(a)=0=z(b).
\end{equation}
Define $\beta(x)=q(x)- \frac{1}{2}p'(x)-\frac{p^{2}(x)}{4}$,
then it follows from \eqref{ast}  and by Lemma \ref{lem2.1},
\begin{align*}
&\max|z(x)| \\
&\le \frac{(b-a)^2}{8}\max|z''(x)| \\
&\le \frac{(b-a)^2}{8}[\max|z''(x)+\beta(x)z(x) |+\max|\beta(x)|\max|z(x)|]\\
&\le \frac{(b-a)^2}{8}\max\Big\{\exp\Big(\frac{1}{2} \int_a^x { p(\tau)d\tau}\Big)
\Big\}  \epsilon + \frac{(b-a)^2}{8}\max| \beta(x) | \max | z(x) |.
\end{align*}
Obviously, $\max\{exp(\frac{1}{2} \int_a^x { p(\tau)d\tau})\}<\infty$  on $[a,b]$. Hence, there exists a constant $K>0$  such that $|z(x)|\leq K \epsilon$ for all $x\in [a,b]$.

Moreover, $\max\{\exp(-\frac{1}{2} \int_a^x { p(\tau)d\tau})\}<\infty$ on
 $[a,b]$ which implies that there exists a constant $\tilde{K}>0$ such that
\begin{align*}
|y(x)|& =  \big|z(x)\exp\Big(-\frac{1}{2} \int_a^x { p(\tau)d\tau} \Big)|\\
& \leq  \max\Big\{\exp\Big(-\frac{1}{2} \int_a^x { p(\tau)d\tau}\Big)\Big\} K
 \epsilon \\
& \leq  \tilde{K} \epsilon.
 \end{align*}
Thus \eqref{eq:e3.0.1} has superstability stability with boundary conditions
\eqref{eq:e3.0.2}.
\end{proof}

As in Theorem \ref{thm2.4}, we can prove  the following theorem.

\begin{theorem}\label{thm3.2}
If
\[
\max \{ q(x)- \frac{1}{2}p'(x)-\frac{p^{2}(x)}{4}\} < 2/(b-a)^2.
\]
Then \eqref{eq:e3.0.1} has superstability stability with initial
conditions \eqref{eq:e3.0.3}.
\end{theorem}

Now, as examples, we investigate the superstability of the differential equation
\begin{equation}\label{eq:e4.0.1}
\alpha(x)y''(x)+\beta(x)y'(x)+\gamma(x)y(x)=0
\end{equation}
with boundary conditions
\begin{equation}\label{eq:e4.0.2}
y(a)=0=y(b)
\end{equation}
and initial conditions
\begin{equation}\label{eq:e4.0.3}
y(a)=0=y'(a),
\end{equation}
where $y \in C^2[a,b]$, $\alpha, \beta, \gamma \in C^1[a,b]$,
$-\infty<a<b<+\infty$  and $\alpha(x) \neq 0 $ on $[a,b]$.

\begin{theorem} \label{Th_3.3}
(1) If
\[
\max \{ \frac{\gamma(x)}{\alpha(x)}
- \frac{1}{2}(\frac{\beta(x)}{\alpha(x)})'
-\frac{1}{4}(\frac{\beta(x)}{\alpha(x)})^2 \} <  8/(b-a)^2,
\]
then \eqref{eq:e4.0.1} has superstability with boundary conditions \eqref{eq:e4.0.2}.

(2) If
\[
\max \{ \frac{\gamma(x)}{\alpha(x)}- \frac{1}{2}(\frac{\beta(x)}{\alpha(x)})'
-\frac{1}{4}(\frac{\beta(x)}{\alpha(x)})^2 \} < 2/(b-a)^2,
\]
 then \eqref{eq:e4.0.1} has superstability with initial conditions \eqref{eq:e4.0.3}.
\end{theorem}

\begin{corollary} \label{coro3.4}
(1) If
\[
\max \{\frac{l(x)}{k(x)} - \frac{1}{2}\frac{d}{dx}\frac{k'(x)}{k(x)}
-\frac{(k'(x)/k(x))^{2}}{4}\} < 8/(b-a)^2,
\]
then
\begin{equation} \label{eq:e4.0.4}
\frac{d}{dx}[k(x)y'(x)]+l(x)y(x)=0
\end{equation}
has superstability with boundary conditions \eqref{eq:e4.0.2}, where
$k\in C^1[a,b]$,  $k(x)\neq 0$ on $[a,b]$ and $l\in C^0[a,b]$.

(2) If 
\[
\max \{\frac{l(x)}{k(x)} - \frac{1}{2}\frac{d}{dx}\frac{k'(x)}{k(x)}
-\frac{(k'(x)/k(x))^{2}}{4}\} < 2/(b-a)^2,\]
then \eqref{eq:e4.0.4}
has superstability with initial conditions \eqref{eq:e4.0.3}.
\end{corollary}

\begin{example} \rm
The differential equation
\begin{equation}
y''(x)+ 2y'(x)+ y(x)=0
\end{equation}
has the superstability with boundary conditions \eqref{eq:e4.0.2}
on any closed bounded interval $[a,b]$ and the superstability with initial
conditions \eqref{eq:e4.0.3} on any closed bounded interval $[a,b]$.
\end{example}

In the following theorem, we investigate the stability of  differential
equation of higher order of the form
\begin{equation}\label{eq:e5.0.1}
y^{(n)}(x)+\beta (x)y(x)=0
\end{equation}
with  initial conditions
\begin{equation}
\label{eq:e5.0.2}
y(a)=y'(a)=\dots=y^{(n-1)}(a)=0,
\end{equation}
where $n\in\mathbb{N^+}$, $y \in C^n[a,b]$, $\beta \in C^{0}[a,b]$,
$-\infty<a<b<+\infty$.

\begin{theorem} \label{thm3.6}
If $\max |\beta(x)|<\frac{n!}{(b-a)^n}$. Then \eqref{eq:e5.0.1}
has the superstability with initial conditions \eqref{eq:e5.0.2}.
\end{theorem}

\begin{proof}
For every $\epsilon >0$, $y \in C^2[a,b]$, if
$|y^{(n)}(x)+\beta (x)y(x)|\leq \epsilon$ and
$y(a)=y'(a)=\dots=y^{(n-1)}(a)=0$. Similarly to the proof of Lemma \ref{lem2.2},
$$
y(x)=y(a)+y'(a)(x-a)+\dots+\frac{y^{(n-1)}(a)}{(n-1)!}(x-a)^{n-1}
+\frac{y^{(n)}(\xi)}{n!}(x-a)^n.
$$
Thus
$$
|y(x)|=|\frac{y^{(n)}(\xi)}{n!}(x-a)^n |
\le\max{|y^{(n)}(x)|}\frac{(b-a)^n}{n!}
$$
for every $x\in[a,b]$; so, we obtain
\begin{align*}
\max{|y(x)|}
& \le  \frac{(b-a)^n }{n!} [\max{ | y^{(n)}(x)
+ \beta(x)y(x) |}]+\frac{(b-a)^n}{n!} \max{ |\beta(x)y(x) |}\\
& \le \frac{(b-a)^n }{n!} \epsilon +\frac{(b-a)^n }{n!}
\max{ |\beta(x) |}\max{ |y(x) |}.
\end{align*}
Let $ \eta=\frac{(b-a)^n }{n!} \max{ |\beta(x) |}$,
$K=\frac{(b-a)^n }{n!(1-\eta)}$. It is easy to see that
$$
|y(x)|\le K\epsilon.
$$
Hence \eqref{eq:e5.0.1} has superstability with initial conditions \eqref{eq:e5.0.2}.
\end{proof}

\subsection*{Acknowledgements}
The authors would like to thank the anonymous referee for his or her
 corrections and suggestions.
Yongjin Li was supported by the National Natural Science Foundation 
of China (10871213). 

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