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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 03, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/03\hfil Gaussian functions]
{An approximation property of Gaussian functions}

\author[S.-M. Jung, H. \c{S}evli, S. \c{S}evgin \hfil EJDE-2013/03\hfilneg]
{Soon-Mo Jung, Hamdullah \c{S}evli, Sebaheddin \c{S}evgin}  % in alphabetical order

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

\address{Hamdullah \c{S}evli \newline
Department of Mathematics,
Faculty of Sciences and Arts,
Istanbul Commerce University, 34672 Uskudar, Istanbul, Turkey}
\email{hsevli@yahoo.com}

\address{Sebaheddin \c{S}evgin \newline
Department of Mathematics,
Faculty of Art and Science,
Yuzuncu Yil University, 65080 Van, Turkey}
\email{ssevgin@yahoo.com}

\thanks{Submitted October 5, 2012. Published January 7, 2013.}
\subjclass[2000]{34A30, 34A40, 41A30, 39B82, 34A25}
\keywords{Linear first order differential equation;
power series method; \hfill\break\indent 
Gaussian function; approximation; Hyers-Ulam stability;
local Hyers-Ulam stability}

\begin{abstract}
 Using the power series method, we solve the inhomogeneous
 linear first order differential equation
 $$
 y'(x) + \lambda (x-\mu) y(x) = \sum_{m=0}^\infty a_m (x-\mu)^m,
 $$
 and prove an approximation property of Gaussian functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Let $Y$ and $I$ be a normed space and an open subinterval of
$\mathbb{R}$, respectively.
If for any function $f : I \to Y$ satisfying the differential
inequality
$$
\big\| a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots +
 a_1(x)y'(x) + a_0(x)y(x) + h(x) \big\| \leq \varepsilon
$$
for all $x \in I$ and for some $\varepsilon \geq 0$, there
exists a solution $f_0 : I \to Y$ of the differential equation
$$
a_n(x)y^{(n)}(x) + a_{n-1}(x)y^{(n-1)}(x) + \cdots +
a_1(x)y'(x) + a_0(x)y(x) + h(x) = 0
$$
such that $\| f(x) - f_0(x) \| \leq K(\varepsilon)$ for any
$x \in I$, where $K(\varepsilon)$ depends on $\varepsilon$ only,
then we say that the above differential equation satisfies the
Hyers-Ulam stability (or the local Hyers-Ulam stability if the
domain $I$ is not the whole space $\mathbb{R}$).
We may apply these terminologies for other differential
equations.
For a more detailed definition of the Hyers-Ulam stability,
refer to \cite{czerwik0,hir,jung2}.


Ob\a{l}oza seems to be the first author who investigated the
Hyers-Ulam stability of linear differential equations
(see \cite{ob1,ob2}).
Here, we introduce a result of Alsina and Ger (see \cite{ag}):
If a differentiable function $f : I \to \mathbb{R}$ is a
solution of the differential inequality
$| y'(x) - y(x) | \leq \varepsilon$, where $I$ is an open
subinterval of $\mathbb{R}$, then there exists a solution
$f_0 : I \to \mathbb{R}$ of the differential equation
$y'(x) = y(x)$ such that $| f(x) - f_0(x) | \leq 3\varepsilon$
for any $x \in I$.
This result of Alsina and Ger was generalized by Takahasi,
Miura and Miyajima:
They proved in \cite{tmm} that the Hyers-Ulam stability holds
for the Banach space valued differential equation
$y'(x) = \lambda y(x)$ (see also \cite{mjt,motn,popa}).

Using the conventional power series method, the first author
investigated the general solution of the inhomogeneous linear
first order differential equations of the form,
$$
y'(x) - \lambda y(x) = \sum_{m=0}^\infty a_m (x-c)^m,
$$
where $\lambda$ is a complex number and the convergence radius
of the power series is positive.
This result was applied for proving an approximation property
of exponential functions in a neighborhood of $c$
(see \cite{115}).

Throughout this paper, we assume that $\rho$ is a positive
real number or infinity.
In \S 2 of this paper, using an idea from \cite{115}, we will
investigate the general solution of the inhomogeneous linear
differential equation of the first order,
\begin{equation}
y'(x) + \lambda (x-\mu) y(x)
= \sum_{m=0}^{\infty} a_m (x-\mu)^m,
\label{eq:1.1}
\end{equation}
where the coefficients $a_m$ of the power series are given
such that the radius of convergence is at least $\rho$.
Moreover, we prove the (local) Hyers-Ulam stability of linear
first order differential equation \eqref{eq:2.1} in a class of
special analytic functions.

\section{General Solution of \eqref{eq:1.1}}

The linear first order differential equation
\begin{equation}
y'(x) + \lambda (x-\mu) y(x) = 0
\label{eq:2.1}
\end{equation}
has a general solution of the form
$y(x) = c \exp\big\{ -\frac{\lambda}{2}(x-\mu)^2 \}$, which is
called a Gaussian function.
We recall that $\rho$ is a positive real number or infinity.


\begin{theorem}\label{thm:2.1}
Let $\lambda \neq 0$ and $\mu$ be a complex number and a real
number, respectively.
Assume that the radius of convergence of power series
$\sum_{m=0}^\infty a_m (x-\mu)^m$ is at least $\rho$.
Every solution $y : (\mu-\rho, \mu+\rho) \to \mathbb{C}$ of
the inhomogeneous differential equation \eqref{eq:1.1} can
be expressed as
\begin{equation}
y(x) = y_h(x) + \sum_{m=0}^\infty c_m (x-\mu)^m,
\label{eq:2.2}
\end{equation}
where the coefficients $c_m$ are given by
\begin{gather}
c_{2m} = \sum_{i=0}^{m-1} (-1)^i \frac{a_{2m-1-2i}}{\lambda}
 \prod_{k=0}^{i} \frac{\lambda}{2m-2k} +
 (-1)^m c_0 \prod_{k=0}^{m-1} \frac{\lambda}{2m-2k},
 \label{eq:2.3a} \\
c_{2m+1} = \sum_{i=0}^{m-1} (-1)^i \frac{a_{2m-2i}}{\lambda}
 \prod_{k=0}^{i} \frac{\lambda}{2m+1-2k} +
 (-1)^m c_1 \prod_{k=0}^{m-1} \frac{\lambda}{2m+1-2k}
 \label{eq:2.3b}
\end{gather}
for each $m \in \mathbb{N}_0$, and $y_h(x)$ is a solution of
the corresponding homogeneous differential equation
\eqref{eq:2.1}.
\end{theorem}


\begin{proof}
Since each solution of  \eqref{eq:1.1} can be expressed as
a power series in $x-\mu$, we put
$y(x) = \sum_{m=0}^\infty c_m (x-\mu)^m$ in \eqref{eq:1.1}
to obtain
\begin{align*}
y'(x) + \lambda (x-\mu) y(x)
&= c_1 + \sum_{m=0}^\infty (m+2) c_{m+2} (x-\mu)^{m+1} +
 \sum_{m=0}^\infty \lambda c_m (x-\mu)^{m+1} \\
&= c_1 + \sum_{m=0}^\infty
 \big[ (m+2) c_{m+2} + \lambda c_m \big] (x-\mu)^{m+1} \\
&= a_0 + \sum_{m=0}^\infty a_{m+1} (x-\mu)^{m+1},
\end{align*}
from which we obtain the following recurrence formula
\begin{equation}
\begin{gathered}
c_1 = a_0, \\
(m+2) c_{m+2} + \lambda c_m = a_{m+1} \quad (m \in \mathbb{N}_0).
\end{gathered}
\label{eq:rec}
\end{equation}

We will now prove the formula \eqref{eq:2.3a} for any
$m \in \mathbb{N}_0$:
If we set $m = 0$ in \eqref{eq:2.3a}, then we get $c_0 = c_0$
which is true.
We assume that the formula \eqref{eq:2.3a} is true for some
$m \in \mathbb{N}_0$.
Then, it follows from \eqref{eq:rec} and the induction hypothesis
that
\begin{align*}
&c_{2m+2} \\
 &= \frac{a_{2m+1}}{2m+2} - \frac{\lambda}{2m+2} c_{2m} \\
 &= \frac{a_{2m+1}}{2m+2} - \frac{\lambda}{2m+2}
 \Big[ \sum_{i=0}^{m-1} (-1)^i \frac{a_{2m-1-2i}}{\lambda}
 \prod_{k=0}^{i} \frac{\lambda}{2m-2k} +
 (-1)^m c_0 \prod_{k=0}^{m-1}
 \frac{\lambda}{2m-2k}
 \Big] \\
 &= \frac{a_{2m+1}}{2m+2} + \sum_{i=0}^{m-1} (-1)^{i+1}
 \frac{a_{2m-1-2i}}{\lambda} \prod_{k=-1}^{i}
 \frac{\lambda}{2m-2k} + (-1)^{m+1} c_0
 \prod_{k=-1}^{m-1} \frac{\lambda}{2m-2k} \\
 &= \frac{a_{2m+1}}{2m+2} + \sum_{i=0}^{m-1} (-1)^{i+1}
 \frac{a_{2m-1-2i}}{\lambda} \prod_{k=0}^{i+1}
 \frac{\lambda}{2m+2-2k} + (-1)^{m+1} c_0
 \prod_{k=0}^{m} \frac{\lambda}{2m+2-2k} \\
 &= \frac{a_{2m+1}}{2m+2} + \sum_{i=1}^{m} (-1)^i
 \frac{a_{2m+1-2i}}{\lambda} \prod_{k=0}^{i}
 \frac{\lambda}{2(m+1)-2k} + (-1)^{m+1} c_0
 \prod_{k=0}^{m} \frac{\lambda}{2(m+1)-2k} \\
 &= \sum_{i=0}^{m} (-1)^i \frac{a_{2m+1-2i}}{\lambda}
 \prod_{k=0}^{i} \frac{\lambda}{2(m+1)-2k} +
 (-1)^{m+1} c_0 \prod_{k=0}^{m} \frac{\lambda}{2(m+1)-2k},
\end{align*}
which can be obtained provided we replace $m$ in \eqref{eq:2.3a}
with $m+1$.
Hence, we conclude that the formula \eqref{eq:2.3a} is true for
all $m \in \mathbb{N}_0$.
Similarly, we can also prove the validity of \eqref{eq:2.3b}
for all $m \in \mathbb{N}_0$.

Indeed, in view of \eqref{eq:rec},
$y_p(x) = \sum_{m=0}^\infty c_m (x-\mu)^m$ is a solution of the
inhomogeneous linear differential equation \eqref{eq:1.1}.
Since every solution of Eq. \eqref{eq:1.1} is a sum of a
solution $y_h(x)$ of the corresponding homogeneous equation
and a particular solution $y_p(x)$ of the inhomogeneous
equation, it can be expressed by \eqref{eq:2.2}.

The formulas \eqref{eq:2.3a} and \eqref{eq:2.3b} can be merged
in a new one:
\begin{equation}
c_m = \sum_{i=0}^{[m/2]-1} (-1)^i \frac{a_{m-1-2i}}{\lambda}
 \prod_{k=0}^{i} \frac{\lambda}{m-2k} +
 (-1)^{[m/2]} c_{0,1} \prod_{k=0}^{[m/2]-1}
 \frac{\lambda}{m-2k}
\label{eq:20120902-1}
\end{equation}
for all $m \in \mathbb{N}_0$, where $c_{0,1} = c_0$ for $m$
even, $c_{0,1} = c_1$ for $m$ odd, and $[m/2]$ denotes the
largest integer not exceeding $m/2$.
Let us define
$$
C := \max \Big\{ \frac{1}{|\lambda|} \prod_{k=0}^i
 \frac{| \lambda |}{m-2k} \;|\; m \in \mathbb{N}_0;\;
 i \in \{ 0, 1, \ldots, [m/2]-1 \} \Big\}.
$$

For any $\varepsilon > 0$, we can choose an (sufficiently large)
integer $m_\varepsilon$ such that
$$
\prod_{k=0}^{[m/2]-1} \frac{|\lambda|}{m-2k} \leq \varepsilon
$$
for all integers $m \geq m_\varepsilon$.
Thus, in view of \eqref{eq:20120902-1}, there exists a constant
$D > 0$ such that
\begin{equation}
| c_m | \leq (C+D) \sum_{i=0}^{m-1} | a_i |
\label{eq:20120830}
\end{equation}
for all sufficiently large integers $m$.
(Since the inhomogeneous term $\sum_{m=0}^\infty a_m (x-\mu)^m$
has to be nonzero for some $x \in (\mu-\rho, \mu+\rho)$, there
exists an $m_0 \in \mathbb{N}_0$ such that $a_{m_0} \neq 0$
and hence, $\sum_{i=0}^{m-1} | a_i | > 0$ for all sufficiently
large integer $m$.)

Finally, it follows from \eqref{eq:20120830} and
\cite[Problem 8.8.1 (p)]{kosmala} that
\begin{align*}
\limsup_{m \to \infty} | c_m |^{1/m}
&=  \limsup_{m \to \infty} \Big(\frac{1}{m} | c_m |\Big)^{1/m} \\
&\leq  \limsup_{m \to \infty}
 \Big(\frac{C+D}{m} \sum_{i=0}^{m-1} | a_i |\Big)^{1/m} \\
&\leq  \limsup_{m \to \infty} | a_m |^{1/m}.
\end{align*}
By use of the Cauchy-Hadamard theorem
(see \cite[Theorem 8.8.2]{kosmala}), the radius of convergence
of the power series for $y_p(x)$ is at least $\rho$.
Therefore, $y(x)$ in Eq. \eqref{eq:2.2} is well defined on
$(\mu-\rho, \mu+\rho)$.
\end{proof}

\begin{remark}\label{rem:2.1} \rm
We notice that Theorem \ref{thm:2.1} is true if we set
$c_0 = 0$.
\end{remark}

\section{Local Hyers-Ulam stability of \eqref{eq:2.1}}


Let $\rho$ be a positive real number or the infinity.
We denote by $\widetilde{C}$ the set of all functions
$f : (\mu-\rho, \mu+\rho ) \to \mathbb{C}$ with the following
properties:
\begin{itemize}
\item[(a)] $f(x)$ is expressible by a power series
 $\sum_{m=0}^\infty b_m (x-\mu)^m$ whose radius
 of convergence is at least $\rho$;

\item[(b)] There exists a constant $K \geq 0$ such that
\[
\sum_{m=0}^\infty | a_m (x-\mu)^m | \leq K
 \big| \sum_{m=0}^\infty a_m (x-\mu)^m \big|
 \]
 for all  $x \in (\mu-\rho, \mu+\rho)$, where
 $a_0 = b_1$ and
 $a_m = (m+1) b_{m+1} + \lambda b_{m-1}$ for
 any $m \in \mathbb{N}$.
\end{itemize}
If we define
$$
( y_1 + y_2 )(x) = y_1(x) + y_2(x) \quad\text{and}\quad
( \lambda y_1 )(x) = \lambda y_1(x)
$$
for all $y_1, y_2 \in \widetilde{C}$ and $\lambda \in \mathbb{C}$,
then $\widetilde{C}$ is a vector space over complex numbers.
We remark that the set $\widetilde{C}$ is large enough to be a
vector space.

We investigate an approximation property of Gaussian functions.
More precisely, we prove the (local) Hyers-Ulam stability of
the linear first order differential equation \eqref{eq:2.1} for
the functions in $\widetilde{C}$.

\begin{theorem}\label{thm:3.1}
Let $\lambda \neq 0$ and $\mu$ be a complex number and a real
number, respectively.
If a function $y \in \widetilde{C}$ satisfies the differential
inequality
\begin{equation}
\big| y'(x) + \lambda (x-\mu) y(x) \big| \leq \varepsilon
\label{eq:3.1}
\end{equation}
for all $x \in (\mu-\rho, \mu+\rho)$ and for some
$\varepsilon \geq 0$, then there exists a solution
$y_h : (\mu-\rho, \mu+\rho) \to \mathbb{C}$ of the differential
equation $\eqref{eq:2.1}$ such that
$$
\big| y(x) - y_h(x) \big|
\leq \Big( | b_1 |
 \exp \big\{ \frac{|\lambda|}{2} (x-\mu)^2
 \big\} +
 \frac{K \varepsilon}{2}
 \frac{\exp \big\{ \frac{|\lambda|}{2} (x-\mu)^2
 \big\} - 1}
 {\frac{|\lambda|}{2} (x-\mu)^2}
 \Big) | x-\mu |
$$
for any $x \in (\mu-\rho, \mu+\rho)$.
In particular, it holds that $y_h \in \widetilde{C}$.
\end{theorem}


\begin{proof}
Since $y$ belongs to $\widetilde{C}$, $y(x)$ can be expressed by
$y(x) = \sum_{m=0}^\infty b_m (x-\mu)^m$ and it follows from
(a) and (b) that
\begin{equation}
\begin{aligned}
&y'(x) + \lambda (x-\mu) y(x)\\
&=  b_1 + \sum_{m=0}^\infty (m+2) b_{m+2} (x-\mu)^{m+1} +
 \sum_{m=0}^\infty \lambda b_m (x-\mu)^{m+1} \\
&=  b_1 + \sum_{m=0}^\infty
 \big[ (m+2) b_{m+2} + \lambda b_m \big] (x-\mu)^{m+1}
 \\
&=  \sum_{m=0}^\infty a_m (x-\mu)^m
\end{aligned} \label{eq:3.3}
\end{equation}
for all $x \in (\mu-\rho, \mu+\rho)$.
By considering \eqref{eq:3.1} and \eqref{eq:3.3}, we have
$$
\Big| \sum_{m=0}^\infty a_m (x-\mu)^m \Big| \leq \varepsilon
$$
for any $x \in (\mu-\rho, \mu+\rho)$.
This inequality, together with (b), yields
\begin{equation}
\sum_{m=0}^\infty \big| a_m (x-\mu)^m \big|
\leq K \Big| \sum_{m=0}^\infty a_m (x-\mu)^m \Big|
\leq K \varepsilon
\label{eq:condition1}
\end{equation}
for all $x \in (\mu-\rho, \mu+\rho)$.

Now, it follows from Theorem \ref{thm:2.1}, \eqref{eq:20120902-1},
\eqref{eq:3.3}, and \eqref{eq:condition1} that there exists a
solution $y_h : (\mu-\rho, \mu+\rho) \to \mathbb{C}$ of the
differential equation \eqref{eq:2.1} such that
\begin{align*}
&\big| y(x) - y_h(x) \big| \\
&\leq  \sum_{m=0}^\infty | c_m | | x-\mu |^m
\text{ }\leq\text{ } | c_0 | + | c_1 | | x-\mu | +
 \sum_{m=2}^\infty | c_m | | x-\mu |^m \\
&\leq  | c_0 | + | c_1 | | x-\mu | + \sum_{m=2}^\infty
 \sum_{i=0}^{[m/2]-1}
 \frac{| a_{m-2i-1} (x-\mu)^{m-2i-1} |}{| \lambda (x-\mu) |}
 \prod_{k=0}^i \frac{| \lambda (x-\mu)^2 |}{m-2k} \\
&\quad + \sum_{m=2}^\infty | c_{0,1} | | x-\mu |^{m-2[m/2]}
 \prod_{k=0}^{[m/2]-1} \frac{| \lambda (x-\mu)^2 |}{m-2k} \\
&\leq  | c_0 | + | c_1 | | x-\mu | + \sum_{m=2}^\infty
 \frac{| a_{m-1} (x-\mu)^{m-1} |}{| \lambda (x-\mu) |}
 \frac{| \lambda (x-\mu)^2 |}{m} \\
&\quad + \sum_{m=4}^\infty
 \frac{| a_{m-3} (x-\mu)^{m-3} |}{| \lambda (x-\mu) |}
 \frac{| \lambda (x-\mu)^2 |}{m}
 \frac{| \lambda (x-\mu)^2 |}{m-2} \\
&\quad + \sum_{m=6}^\infty
 \frac{| a_{m-5} (x-\mu)^{m-5} |}{| \lambda (x-\mu) |}
 \frac{| \lambda (x-\mu)^2 |}{m}
 \frac{| \lambda (x-\mu)^2 |}{m-2}
 \frac{| \lambda (x-\mu)^2 |}{m-4} 
 + \dots  \\
&\quad + | c_0 | \frac{| \lambda (x-\mu)^2 |}{2} +
 | c_1 | | x-\mu | \frac{| \lambda (x-\mu)^2 |}{3} +
 | c_0 | \frac{| \lambda (x-\mu)^2 |}{4}
 \frac{| \lambda (x-\mu)^2 |}{2} \\
&\quad + | c_1 | | x-\mu | \frac{| \lambda (x-\mu)^2 |}{5}
 \frac{| \lambda (x-\mu)^2 |}{3} +
 | c_0 | \frac{| \lambda (x-\mu)^2 |}{6}
 \frac{| \lambda (x-\mu)^2 |}{4}
 \frac{| \lambda (x-\mu)^2 |}{2} \\
&\quad + | c_1 | | x-\mu | \frac{| \lambda (x-\mu)^2 |}{7}
 \frac{| \lambda (x-\mu)^2 |}{5}
 \frac{| \lambda (x-\mu)^2 |}{3} + \cdots \\
&\leq  K \varepsilon
 \Big( \frac{| x-\mu |}{2} +
 \frac{| \lambda (x-\mu)^3 |}{4 \cdot 2} +
 \frac{| \lambda^2 (x-\mu)^5 |}{6 \cdot 4 \cdot 2} +
 \cdots
 \Big) \\
&\quad + | c_0 | \Big( 1 + \frac{| \lambda (x-\mu)^2 |}{2} +
 \frac{| \lambda (x-\mu)^2 |^2}{4 \cdot 2} +
 \frac{| \lambda (x-\mu)^2 |^3}{6 \cdot 4 \cdot 2} +
 \cdots
 \Big) \\
&\quad + | c_1 | | x-\mu |
 \Big( 1 + \frac{| \lambda (x-\mu)^2 |}{3} +
 \frac{| \lambda (x-\mu)^2 |^2}{5 \cdot 3} +
 \frac{| \lambda (x-\mu)^2 |^3}{7 \cdot 5 \cdot 3} +
 \dots
 \Big)
\end{align*}
for all $x \in (\mu-\rho, \mu+\rho)$, where $c_{0,1} = c_0$
for $m$ even, $c_{0,1} = c_1$ for $m$ odd.

In view of \eqref{eq:rec}, Remark \ref{rem:2.1}, and (b), we know that
$y_p(x) = b_1 (x-\mu) + \sum_{m=2}^\infty c_m (x-\mu)^m$
is a particular solution of the inhomogeneous differential equation
\eqref{eq:1.1}, i.e., we can set $c_0 = 0$ and $c_1 = b_1$ in
Theorem \ref{thm:2.1}.
Hence, we obtain
\begin{align*}
&\big| y(x) - y_h(x) \big| \\
&\leq  | c_0 | + | c_1 | | x-\mu | +
 \Big( \frac{K \varepsilon}{| \lambda (x-\mu) |} +
 | c_0 | + | c_1 | | x-\mu |
 \Big) \sum_{i=1}^\infty
 \frac{| \lambda (x-\mu)^2 |^i}{2^i i!} \\
&=  | b_1 | | x-\mu | +
 \Big( \frac{K \varepsilon}{| \lambda (x-\mu) |} +
 | b_1 | | x-\mu |
 \Big) \sum_{i=1}^\infty \frac{1}{i!}
 \Big| \frac{\lambda}{2} (x-\mu)^2 \Big|^i \\
&=  \Big( | b_1 |
 \exp \Big\{ \frac{|\lambda|}{2} (x-\mu)^2
 \Big\} +
 \frac{K \varepsilon}{2}
 \frac{\exp \Big\{ \frac{|\lambda|}{2} (x-\mu)^2
 \Big\} - 1}
 {\frac{|\lambda|}{2} (x-\mu)^2}
 \Big) | x-\mu |
\end{align*}
for any $x \in (\mu-\rho, \mu+\rho)$.

As we already remarked, there exists a real number $c$ such
that
$$
y_h(x) = c\exp \big\{ -\frac{\lambda}{2} (x-\mu)^2 \big\}.
$$
Hence, $y_h(x)$ has a power series expansion in $x-\mu$, namely,
\begin{equation}
y_h(x) = \sum_{m=0}^\infty b_m^\ast (x-\mu)^m,
\label{eq:20120829}
\end{equation}
where
$$
b_{2m}^\ast = (-1)^m \frac{c}{m!}
 \Big( \frac{\lambda}{2} \Big)^m
\quad\text{and}\quad
b_{2m+1}^\ast = 0
$$
for all $m \in \mathbb{N}_0$.
The radius of convergence of the power series \eqref{eq:20120829}
is infinity.

It follows from (b) that $a_0^\ast = b_1^\ast = 0$ and
$$
a_{2m}^\ast = (2m+1) b_{2m+1}^\ast + \lambda b_{2m-1}^\ast = 0
$$
for every $m \in \mathbb{N}$.
Moreover, we have
\begin{align*}
a_{2m+1}^\ast 
&= (2m+2) b_{2m+2}^\ast + \lambda b_{2m}^\ast \\
 &= (2m+2) (-1)^{m+1} \frac{c}{(m+1)!}
 \Big( \frac{\lambda}{2} \Big)^{m+1} +
 \lambda (-1)^m \frac{c}{m!}
 \Big( \frac{\lambda}{2} \Big)^m 
 = 0
\end{align*}
for all $m \in \mathbb{N}_0$, i.e., $a_m^\ast = 0$ for all
$m \in \mathbb{N}_0$.
Therefore,
$y_h(x) = c\exp \big\{ -\frac{\lambda}{2} (x-\mu)^2 \big\}$
satisfies both conditions (a) and (b).
That is, $y_h$ belongs to $\widetilde{C}$.
\end{proof}

According to the previous theorem, each approximate solution
of the differential equation \eqref{eq:2.1} can be well
approximated by a Gaussian function in a (small) neighborhood
of $\mu$.
More precisely, by applying l'Hospital's rule, we can easily
prove the following corollary.

\begin{corollary}\label{cor:3.2}
Let $\lambda \neq 0$ and $\mu$ be a complex number and a real
number, respectively.
If a function $y \in \widetilde{C}$ satisfies the differential
inequality $\eqref{eq:3.1}$ for all $x \in (\mu-\rho, \mu+\rho)$
and for some $\varepsilon \geq 0$, then there exists a complex
number $c$ such that
$$
\Big|
y(x) - c \exp \big\{ -\frac{\lambda}{2} (x-\mu)^2 \big\}
\Big| = O \big( | x-\mu | \big) \quad\text{as}\quad x \to \mu,
$$
where $O( \cdot )$ denotes the Landau symbol $($big-O$)$.
\end{corollary}

\subsection*{Acknowledgments}
This research was completed with the support of the Scientific
and Technological Research Council of Turkey while the first
author was a visiting scholar at Istanbul Commerce University,
Istanbul, Turkey.

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\end{document}