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

\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
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\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 $-\infty<a<b<+\infty$. 
Assume that $g$ is an arbitrarily complex number, $h:I\to C$ is continuous 
and integrable on $I$. Moreover, suppose $\varphi:I \to [0,\infty)$ 
is an integrable function on $I$. If a continuously differentiable 
function $y:I \to X$ satisfies the differential inequality
\begin{equation}
\|y'(t)+gy(t)+h(t)\|\le \varphi(t)\label{e3}
\end{equation}
 for all $t\in I$, then there exists a unique $x\in X$ such that
\[
\|y(t)-e^{-\int_{a}^{t}gdu}(x-\int_{a}^{t}e^{\int_{a}^{v}gdu}h(v)dv)\|
\le e^{-\Re(\int_{a}^{t}gdu)}\int_{t}^{b}\varphi(v)e^{\Re(\int_{a}^{v}gdu)}dv
\]
\end{lemma}

\begin{proof} 
For simplicity, we use the  notation
\[ 
z(t):=e^{\int_{a}^{t}gdu}y(t)+\int_{a}^{t}e^{\int_{a}^{v}gdu}h(v)dv   
 \]
for each $t\in I$. By making use of this notation and by \eqref{e3}, we obtain
\begin{align*}
\|z(t)-z(s)\|
&= \|e^{\int_{a}^{t}gdu}y(t)-e^{\int_{a}^{s}gdu}y(s)
 +\int_{s}^{t}e^{\int_{a}^{v}gdu}h(v)dv\|\\
&= \|\int_{s}^{t}\frac{d}{dv}(e^{\int_{a}^{v}gdu}y(v))dv
 +\int_{s}^{t}e^{\int_{a}^{v}gdu}h(v)dv\|\\
&= \|\int_{s}^{t}e^{\int_{a}^{v}gdu}(y'(v)+gy(v)+h(v))dv\|\\
&\le |\int_{s}^{t}e^{\Re(\int_{a}^{v}gdu)}\varphi(v)dv|
\end{align*}
for any $s,t\in I$.

Since $g$ is a constant number, we know that $e^{\Re(\int_{a}^{v}gdu)}$ is
boundary and continuous. What is more, $\varphi(v)$ is integrable and hence
$e^{\Re(\int_{a}^{v}gdu)}\varphi(v)$ is integrable. Since $X$ is
completed, there exists an $x\in X$ such that $z(s)$
converges to $x$ as $s\to b$.

Thus, it follows from the above argument that for any $t\in I$,
\begin{align*}
&\|y(t)-e^{-\int_{a}^{t}gdu}(x-\int_{a}^{t}e^{\int_{a}^{v}gdu}h(v)dv)\|\\
&= \|e^{-\int_{a}^{t}gdu}(z(t)-x)\|\\
&\le e^{-\Re(\int_{a}^{t}gdu)}\|z(t)-z(s)\|+e^{-\Re(\int_{a}^{t}gdu)}\|z(s)-x\|\\
&\le e^{-\Re(\int_{a}^{t}gdu)}|\int_{s}^{t}\varphi(v)e^{\Re(\int_{a}^{v}gdu)}dv|
 +e^{-\Re(\int_{a}^{t}gdu)}\|z(s)-x\|\\
&\to e^{-\Re(\int_{a}^{t}gdu)}\int_{t}^{b}\varphi(v)
 e^{\Re(\int_{a}^{v}gdu)}dv
\end{align*}
as $s \to b$, since $z(s) \to x$ as $ s\to b$.
Obviously, $y_0(t)=e^{-\int_{a}^{t}gdu}(x-\int_{a}^{t}e^{\int_{a}^{v}gdu}h(v)dv)$
 is a solution of $y'(t)+gy(t)+h(t)=0 $.

It now remains to prove the uniqueness of $x$.
Assume that $x_1\in X$ also satisfies \eqref{e3} in place of $x$.
Then, we have
\[
\|e^{-\int_{a}^{t}gdu}(x-x_1)\|\le2e^{-\Re(\int_{a}^{t}gdu)}
\int_{t}^{b}\varphi(v)e^{\Re(\int_{a}^{v}gdu)}dv
\]
for any $t\in I$. It follows from the integrability hypotheses that
\begin{align*}
\|x-x_1\|\le2\int_{t}^{b}e^{\Re{(\int_{a}^{v}gdu)}}\varphi(v)dv\to 0
\end{align*}
as $t\to b$. This implies the uniqueness of $x$.
\end{proof}



\begin{corollary} \label{coro2.2}
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 $-\infty<a<b<+\infty$.
 Assume that $g$ is an arbitrarily complex number, $h:I\to C$ is 
continuous and integrable on $ I$. Moreover, suppose 
$\varphi:I \to [0,\infty)$ is an integrable function on $I$. 
If a continuously differentiable function $y:I \to X$ satisfies 
the differential inequality
\begin{equation}
\|y'(t)+gy(t)+h(t)\|\le \varphi(t)\label{e4}
\end{equation}
 for all $t\in I$, then there exists a unique $x\in X$ such that
\[
\|y(t)-e^{-\int_{b}^{t}gdu}(x-\int_{b}^{t}e^{\int_{b}^{v}gdu}h(v)dv)\|
\le e^{-\Re(\int_{b}^{t}gdu)}\int_{a}^{t}\varphi(v)e^{\Re(\int_{b}^{v}gdu)}dv
\]
\end{corollary}

\begin{proof}
Let $J=(-b,-a)$ and define $h_1(t)=h(-t),y_1(t)=y(-t)$ and
$\varphi_1(t)=\varphi(-t)$, respectively.
Using these definitions, we may transform the inequality \eqref{e4} into
$$
\|y_1'(t)-gy_1(t)-h_1(t)\|\le \varphi_1(t) 
$$
for each $t\in J$.

 Hence, we can now use Lemma \ref{lem2.1} to conclude that there exists a 
unique $x\in X$ such that
 \begin{align*}
&\|y_1(t)-e^{\int_{-b}^{t}gdu}(x+\int_{-b}^{t}e^{-\int_{-b}^{v}gdu}h_1(v)dv)\|\\
&\le e^{\Re(\int_{-b}^{t}gdu)}\int_{t}^{-a}\varphi_1(v)e^{-\Re(\int_{-b}^{v}gdu)}dv
\end{align*}
for any $t\in J$.
Indeed, we can transform the above inequality into
 \begin{align*}
\|y(t)-e^{-\int_{b}^{t}gdu}(x-\int_{b}^{t}e^{\int_{b}^{v}gdu}h(v)dv)\|
\le e^{-\Re(\int_{b}^{t}gdu)}\int_{a}^{t}\varphi(v)e^{\Re(\int_{b}^{v}gdu)}dv
\end{align*}
by some tedious substitutions.
\end{proof}

In the following theorems, we investigate the Hyers-Ulam-Rassias of 
\eqref{e1} and \eqref{e2}.

\begin{theorem} \label{thm2.3}
Let $\varphi:I \to [0,\infty)$ be an integrable function on $ I$. 
Assume that $\alpha, \beta $ are complex numbers. If a twice 
continuously differentiable function $y(t)$ satisfies the inequality
\begin{equation}
\|y''(t) + \alpha y'(t)+\beta y(t)\|\le \varphi(t) \label{e5}
\end{equation}
Then \eqref{e1} has the Hyers-Ulam-Rassias stability.
\end{theorem}

\begin{proof}
Let $\lambda_1$ and $ \lambda_2$ be the roots of the characteristic 
equation $\lambda^2+\alpha \lambda+\beta=0$.
Define $g(t)=y'(t)-\lambda_1y(t)$, thus
\begin{align*}
|g'(t)-\lambda_2g(t)|
&=|y''(t)-\lambda_1y'(t)-\lambda_2(y'(t)-\lambda_1y(t))|\\
&=|y''(t)-\alpha y'(t)+\beta y(t)|
\end{align*}
Hence, we have
$|g'(t)-\lambda_2g(t)|\le\varphi(t)$.
By using Lemma \ref{lem2.1}, there exists a unique $x_1\in X$ such
that
$$
\|g(t)-x_1e^{t\lambda_2-a\lambda_2}\|\le
e^{\Re{(t\lambda_2-a\lambda_2)}}\int_{t}^{b}e^{-\Re{(\int_{a}^{v}\lambda_2du)}}
\varphi(v)dv
$$
 where $ x_1=\lim_{t\to b}g(t)e^{-\lambda_2t+\lambda_2a} $ and  
$x_1e^{t\lambda_2-a\lambda_2}$ satisfies $g'(t)-\lambda_2g(t)=0$.

Since $g(t)=y'(t)-\lambda_1y(t)$, we have
$$
\|y'(t)-\lambda_1y(t)-x_1e^{t\lambda_2-a\lambda_2}\|
\le e^{\Re{(t\lambda_2-a\lambda_2)}}\int_{t}^{b}
e^{-\Re{(\int_{a}^{v}\lambda_2du)}}\varphi(v)dv 
$$
For simplicity, we define 
$\psi(t)=e^{\Re{(t\lambda_2-a\lambda_2)}}\int_{t}^{b}e^{-\Re{(\int_{a}^{v}
\lambda_2du)}}\varphi(v)dv$, thus
$$
\|y'(t)-\lambda_1y(t)-x_1e^{t\lambda_2-a\lambda_2}\|\le\psi(t)
$$
By using Lemma \ref{lem2.1} again, there exists  a unique $x_2 \in X$ such
that
\begin{align*}
&\|y(t)-e^{\int_{a}^{t}\lambda_1du}(x_2+\int_{a}^{t}
e^{-\int_{a}^{v}\lambda_1du}x_1e^{v\lambda_2-a\lambda_2}dv)\|\\
&\le e^{\Re(\int_{a}^{t}\lambda_1du)}\int_{t}^{b}\psi(v)
e^{\Re(\int_{a}^{v}-\lambda_1du)}dv
\end{align*}
where $x_2=\lim_{t\to b}(e^{-\int_{a}^{t}\lambda_1du}y(t)
-\int_{a}^{t}e^{-\int_{a}^{v}\lambda_1du}\cdot
x_1\cdot e^{\int_{a}^{v}\lambda_2du}dv)$.
Furthermore, it is easy to prove that
$e^{\int_{a}^{t}\lambda_1du}(x_2+\int_{a}^{t}
e^{-\int_{a}^{v}\lambda_1du}x_1e^{v\lambda_2-a\lambda_2}dv)$
is a solution of \eqref{e1}.
\end{proof}


\begin{theorem} \label{thm2.4}
Let $\varphi:I \to [0,\infty)$ is an integrable function on $ I$. 
Assume that $\alpha, \beta $ are complex numbers, and $f:I \to X$ 
is continuous and integrable on $I$. If a twice continuously differentiable 
function $y(t)$ satisfies the inequality
\begin{equation}
\|y''(t) + \alpha y'(t)+\beta y(t)-f(t)\|\le \varphi(t) \label{e6}
\end{equation}
Then \eqref{e2} has the Hyers-Ulam-Rassias stability.
\end{theorem}

\begin{proof}
Let $\lambda_1$ and $ \lambda_2$ be the roots of the characteristic
 equation $\lambda^2+\alpha \lambda+\beta=0$.
Define $g(t)=y'(t)-\lambda_1y(t)$, thus
$$
|g'(t)-\lambda_2g(t)|=|y''(t)-\lambda_1y'(t)-\lambda_2(y'(t)
-\lambda_1y(t))|=|y''(t)-\alpha y'(t)+\beta y(t)|.
$$
Hence, we have
$|g'(t)-\lambda_2g(t)-f(t)|\le\varphi(t)$.
By using  Lemma \ref{lem2.1}, there exists a unique $x_1\in X$ such
that
$$
\|g(t)-e^{t\lambda_2-a\lambda_2}(x_1+\int_{a}^{t}
e^{-\int_{a}^{v}\lambda_2du}f(v)dv)\|
\le e^{\Re{(t\lambda_2-a\lambda_2)}}\int_{t}^{b
}e^{-\Re{(\int_{a}^{v}\lambda_2du)}}\varphi(v)dv
$$
where $ x_1=\lim_{t\to b}(g(t)e^{-\lambda_2t+\lambda_2a}
-\int_{a}^{t}e^{-\int_{a}^{v}\lambda_2du}f(v)dv )$.

For simplicity, we define
\begin{gather*}
\phi(t)=e^{t\lambda_2-a\lambda_2}(x_1+\int_{a}^{t}
e^{-\int_{a}^{v}\lambda_2du}f(v)dv),\\
\psi(t)=e^{\Re{(t\lambda_2-a\lambda_2)}}\int_{t}^{b}
e^{-\Re{(\int_{a}^{v}\lambda_2du)}}\varphi(v)dv
\end{gather*}
thus,
$$
\|y'(t)-\lambda_1y(t)-\phi(t)\|\le\psi(t)
$$
By using Lemma \ref{lem2.1} again, there exists  a unique $x_2 \in X$ such
that
$$
\|y(t)-e^{\int_{a}^{t}\lambda_1du}(x_2+\int_{a}^{t}
e^{-\int_{a}^{v}\lambda_1du}\cdot \phi(v)dv)\|
\le e^{\Re(\int_{a}^{t}\lambda_1du)}\int_{t}^{b}\psi(v)
e^{\Re(\int_{a}^{v}-\lambda_1du)}dv
$$
where 
$$
x_2=\lim_{t\to b}(e^{-\int_{a}^{t}\lambda_1du}y(t)
-\int_{a}^{t}e^{-\int_{a}^{v}\lambda_1du}\phi(v)dv).
$$
Furthermore, it is easy to show that
$e^{\int_{a}^{t}\lambda_1du}(x_2+\int_{a}^{t}e^{-\int_{a}^{v}\lambda_1du}\cdot
\phi(v)dv)$ is a solution of \eqref{e2}.
\end{proof}

If  $\alpha$ and $\beta$ are real numbers, the approximating function 
will be a real function even if the roots of the characteristic equation 
are complex numbers.

\begin{corollary} \label{coro2.5}
Let $\varphi:I \to [0,\infty)$ be an integrable function on $ I$. 
Assume that $\alpha, \beta $ are real numbers, $y(t)$ satisfies the inequality
$$
\|y''(t) + \alpha y'(t)+\beta y(t)\|\le \varphi(t) 
$$
where $y:I\to X$ is a twice continuously differentiable function, 
$X$ is a real Banach space.
Then \eqref{e1} has the Hyers-Ulam-Rassias stability.
Moreover, the approximating function is a real function.
\end{corollary}

\begin{proof}
What we have to do is just to verify that if the approximate function is real.
It is easy to know that when the roots are real, the corollary holds.
Therefore, we suppose that the roots of the characteristic equation 
are complex numbers. Let $r_1=p_1+ip_2$ and $r_2=p_1-ip_2$ be the 
roots of the characteristic equation, and 
$\lim_{t\to b}y(t)=d_1,\lim_{t\to b}y'(t)=d_2$, so 
$\lim_{t\to b}g(t)=d_2-r_1*d_1$ ($p_1,p_2,d_1,d_2 $ are real numbers).

By some tedious calculations, we can know that the approximating function is
$$
\frac{1}{p_2}[d_1p_2\cos(p_2(b-t))+(-d_2+d_1p_1)\sin(p_2(b-t))]
(\cosh(p_1(b-t))-\sinh(p_1(b-t))).
$$
which is a real function. This completes the proof of our corollary.
\end{proof}

\begin{corollary} \label{coro2.6}
Let $\varphi:I \to [0,\infty)$ be an integrable function on $ I$. 
Assume that $\alpha, \beta $ are real numbers, $y(t)$ satisfies the inequality
$$
\|y''(t) + \alpha y'(t)+\beta y(t)-f(t)\|\le \varphi(t) 
$$
where $y:I\to X$ is a twice continuously differentiable function and 
$f:I\to X$ is continuous and integrable on $I$, $X$ is a real Banach space. 
Then Eq\eqref{e2} has the Hyers-Ulam-Rassais stability.
Furthermore, the approximating function is a real function.
\end{corollary}

\begin{remark} \label{rmk2.7} \rm
By using the corollary \ref{coro2.2}, we can get similar results with 
Theorem \ref{thm2.3}, 
Theorem \ref{thm2.4}, Corollary \ref{coro2.5} and Corollary \ref{coro2.6}.
\end{remark}


\begin{example} \label{examp2.8} \rm
Given $y\in C^2(1,2)$, $\lim_{t\to 2}y(t)=2$, $\lim_{t\to 2}y'(t)=1$, 
and $y$ satisfies the inequality $|y''(t)-3y'(t)+2y(t)|<t$.
By using  Theorem \ref{thm2.3}, we have
\begin{align*}
|y(t)-(3e^{t-2}-e^{2t-4})|\le\frac{3}{4}-3e^{t-2}+\frac{5}{4}e^{2t-4}
+\frac{1}{2}t;
\end{align*}
moreover, $y_0(t)=3e^{t-2}-e^{2t-4}$ satisfies 
$\lim_{t\to 2}y_0(t)=2,\lim_{t\to 2}y_0'(t)=1$ and
$y_0''-3y_0'+2y_0=0$.
\end{example}


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

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