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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 184, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2013/184\hfil Hyers-Ulam stability]
{Hyers-Ulam stability of linear second-order differential equations
in complex Banach spaces}
\author[Y. Li, J. Huang \hfil EJDE-2013/184\hfilneg]
{Yongjin Li, Jinghao Huang} % in alphabetical order
\address{Yongjin Li \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou 510275, China}
\email{stslyj@mail.sysu.edu.cn}
\address{Jinghao Huang \newline
Department of Mathematics, Sun Yat-Sen University,
Guangzhou 510275, China}
\email{hjinghao@mail2.sysu.edu.cn}
\thanks{Submitted May 1, 2013. Published August 10, 2013.}
\subjclass[2000]{34K20, 26D10}
\keywords{Hyers-Ulam stability; differential equation}
\begin{abstract}
We prove the Hyers-Ulam stability of linear second-order differential equations
in complex Banach spaces. That is, if $y$ is an approximate solution of the
differential equation $y''+ \alpha y'(t) +\beta y = 0$ or
$y''+ \alpha y'(t) +\beta y = f(t)$, then there exists an exact
solution of the differential equation near to $y$.
\end{abstract}
\maketitle
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\section{Introduction and preliminaries}
In 1940, Ulam \cite {Ulam} gave a wide-ranging talk about a
series of important unsolved problems. Among those was the
question concerning the stability of group homomorphisms.
Hyers \cite {HYERS} solved the problem for the case of
approximately additive mappings between Banach spaces.
Since then, the stability problems of
functional equations have been extensively investigated by several
mathematicians \cite{JUN,PARK,RASSIAS}.
Assume that $Y$ is a normed space and $I$ is an open subset of $\mathbb{R}$.
Suppose that $a_i:I\to \mathbb{K}$ and $h:I\to Y$ are continuous functions
and $\mathbb{K}$ is either $\mathbb{R}$ of $\mathbb{C}$, for any function
$f:I \to Y$ satisfying the differential inequality
$$
\|a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\dots+a_1(x)y'(x)+a_0(x)y(x)+h(x)\|
\le\varepsilon
$$
for all $x\in I$ and for some $\varepsilon\ge0$.
We say that
$$
a_n(x)y^{(n)}(x)+a_{n-1}(x)y^{(n-1)}(x)+\dots+a_1(x)y'(x)+a_0(x)y(x)+h(x)=0
$$
satisfies the Hyers-Ulam stability, if there exists a solution
$f_0:I\to Y$ of the above differential equation and
$\|f(x)-f_0(x)\|\le K(\varepsilon)$ for any $x\in I$, where
$K(\varepsilon)$ is an expression of $\varepsilon$ only.
If the above statement is also true when we replace
$\varepsilon$ and $K(\varepsilon)$ by $\varphi(t)$ and $\Phi(\varepsilon)$,
where $\varphi,\Phi:I\to[0,\infty)$ are functions not depending
on $f$ and $f_0$ explicitly, then we say that the corresponding
differential equation has the Hyers-Ulam-Rassias stability
(or generalized Hyers-Ulam stability).
Ob{\l}oza may be the first author to investigate the Hyers-Ulam
stability of differential equations (see \cite{M.Oboza1,M.Oboza2}).
Then, Alsina and Ger prove the Hyers-Ulam
stability of $y'(t) - y(t)=0$ \cite {Alsina}.
The above result of Alsina and Get has been generalized by Miura,
Takahasi and Choda \cite {Miura5}, by Miura \cite {Miura1},
and also by Takahasi, Miura and Miyajima \cite{Takahasi1}.
While \cite {Miura4}, Miura et al \cite{S.-M.Jung1} also proved
the Hyers-Ulam stability of linear differential equations of first
order $y'(t) + g(t)y(t) = 0$ and $\varphi (t)y' (t) = y(t)$.
Furthermore, the result of Hyers-Ulam stability for first-order
linear differential equations has been generalized in
(see \cite{S.-M.Jung2,S.-M.Jung3, Miura4,Takahasi2,Wang1}).
In the meantime, Yongjin Li et al \cite {Yongjin} do some work in
linear differential equations of second order in the form of
$y''(t) + \alpha y'(t)+\beta y(t) = 0$ and $y''(t) + \alpha
y'(t)+\beta y(t) = f(t)$ under the assumption that the
characteristic equation $\lambda^2 + \alpha \lambda+\beta =
0$ has two different positive roots. And the Hyers-Ulam stability
for second-order linear differential equations in the form of
$y''(t)+\beta(x)y=0$ with boundary conditions was investigated in
\cite{Yongjin2}.
The aim of this article is to study the Hyers-Ulam-Rassias
stability of the following linear differential equations of second
order in complex Banach spaces:
\begin{equation}
y''(t) + \alpha y'(t)+\beta y(t) = 0 \label{e1}
\end{equation}
and
\begin{equation}
y''(t) + \alpha y'(t)+\beta y(t)= f(t) \label{e2}
\end{equation}
\section{Main Results}
In the following theorems, we will prove the Hyers-Ulam-Rassias stability
of linear differential equations of second order.
Before stating the main theorem, we need the following lemma. For
the sake of convenience, all the integrals and derivations will be
viewed as existing and $\Re (\omega)$ denotes the real part of
complex number $\omega$.
\begin{lemma} \label{lem2.1}
Let $X$ be a complex Banach space and let $I=(a,b)$ be an open interval,
where $a,b\in R$ are arbitrarily given with $-\infty