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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 80, pp. 1--11.\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-2014/80\hfil Hyers-Ulam stability]
{Laplace transform and generalized Hyers-Ulam
 stability of linear differential equations}

\author[Q. H. Alqifiary, S.-M. Jung \hfil EJDE-2014/80\hfilneg]
{Qusuay H. Alqifiary, Soon-Mo Jung}  % in alphabetical order

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

\thanks{Submitted March 5, 2014. Published March 21, 2014.}
\subjclass[2000]{44A10, 39B82, 34A40, 26D10}
\keywords{Laplace transform method;  differential equations;
 \hfill\break\indent generalized Hyers-Ulam stability}

\begin{abstract}
 By applying the Laplace transform method, we
 prove that the linear differential equation
 $$
 y^{(n)}(t)+\sum_{k=0}^{n-1}{\alpha_k y^{(k)}(t)}=f(t)
 $$
 has the generalized Hyers-Ulam stability, where $\alpha_k$ is
 a scalar, $y$ and $f$ are $n$ times continuously differentiable
 and of exponential order.
\end{abstract}

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

\section{Introduction}

In 1940, Ulam \cite{Ka} 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{r3} 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,HyIsRa,Jud,Kc}).
A generalization of Ulam's problem was recently proposed by
replacing functional equations with differential equations:
The differential equation $\varphi(f,y,y', \dots,y^{(n)}) = 0$
has Hyers-Ulam stability if for a given $\varepsilon > 0$ and a
function $y$ such that
$| \varphi(f, y, y', \dots, y^{(n)}) |$ $\leq \varepsilon$,
there exists a solution $y_a$ of the differential equation
such that $| y(t) - y_a(t) | \leq K(\varepsilon)$ and
$\lim_{\varepsilon \to 0} K(\varepsilon) = 0$.
If the preceding statement is also true when we replace
$\varepsilon$ and $K(\varepsilon)$ by $\varphi(t)$ and
$\Phi(t)$, where $\varphi,\, \Phi$ are appropriate functions
not depending on $y$ and $y_a$ explicitly, then we say that
the corresponding differential equation has the generalized
Hyers-Ulam stability (or Hyers-Ulam-Rassias stability).

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{MMT1,TMM,TTMM} and
in \cite{MMT2}, respectively.
Furthermore, Jung has also proved the Hyers-Ulam stability of
linear differential equations (see \cite{r19,r21,r20}).
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,WZS}).
The results given in \cite{r21,LiSh,MMT1} 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.



Recently, Rezaei, Jung and Rassias have proved the Hyers-Ulam
stability of linear differential equations by using the Laplace
transform method (see \cite{Ke}).

In this paper, by using the Laplace transform method, we prove
that the linear differential equation of the $n$th order
$$
y^{(n)}(t) + \sum_{k=0}^{n-1} {\alpha_k y^{(k)}(t)} = f(t)
$$
has the generalized Hyers-Ulam stability, where $\alpha_k$ is
a scalar, $y$ and $f$ are $n$ times continuously differentiable
and of exponential order, respectively.


\section{Preliminaries}

Throughout this paper, $\mathbb{F}$ will denote either the real
field $\mathbb{R}$ or the complex field $\mathbb{C}$.
A function $f : (0, \infty) \to \mathbb{F}$ is said to be of
exponential order if there are constants $A, B \in \mathbb{R}$
such that
$$
| f(t) | \leq Ae^{tB}
$$
for all $t > 0$.
For each function $f : (0, \infty) \to \mathbb{F}$ of exponential
order, we define the Laplace transform of $f$ by
$$
F(s) = \int_{0}^{\infty} f(t) e^{-st}dt.
$$
There exists a unique number $-\infty \leq \sigma < \infty$
such that this integral converges if $\Re(s) > \sigma$ and
diverges if $\Re(s) < \sigma$, where $\Re(s)$ denotes the real
part of the (complex) number $s$.
The number $\sigma$ is called the abscissa of convergence and
denoted by $\sigma_{f}$.
It is well known that $| F(s) | \to 0$ as $\Re(s) \to \infty$.
Furthermore, $f$ is analytic on the open right half plane
$\{ s \in \mathbb{C}: \Re(s)> \sigma \}$ and we have
$$
\frac{d}{ds} F(s) = -\int_{0}^{\infty} te^{-st} f(t) dt \quad
(\Re(s)> \sigma).
$$
The Laplace transform of $f$ is sometimes denoted by
$\mathcal{L}(f)$.
It is well known that $\mathcal{L}$ is linear and one-to-one.

Conversely, let $f(t)$ be a continuous function whose Laplace
transform $F(s)$ has the abscissa of convergence $\sigma_{f}$,
then the formula for the inverse Laplace transforms yields
$$
f(t) = \frac{1}{2\pi i} \lim_{T \to \infty}
 \int_{\alpha-iT}^{\alpha+iT}
 F(s) e^{st} ds
 = \frac{1}{2\pi} \int_{-\infty}^{\infty}
 e^{(\alpha+iy)t} F(\alpha+iy) dy
$$
for any real constant $\alpha > \sigma_{f}$, where the first
integral is taken along the vertical line $\Re(s) = \alpha$
and converges as an improper Riemann integral and the second
integral is used as an alternative notation for the first
integral (see \cite{Da}).
Hence, we have
\begin{gather*}
\mathcal{L}(f)(s) = \int_{0}^{\infty} f(t) e^{-st} dt
 \quad (\Re(s) > \sigma_{f}) \\
\mathcal{L}^{-1}(F)(t)
 = \frac{1}{2\pi} \int_{-\infty}^{\infty}
 e^{(\alpha+iy)t} F(\alpha+iy) dy
 \quad (\alpha > \sigma_{f}).
\end{gather*}

The convolution of two integrable functions
$f, g : (0,\infty) \to \mathbb{F}$ is defined by
$$
(f*g)(t) := \int_0^t {f(t - x) g(x) dx}.
$$
Then $\mathcal{L}(f*g) = \mathcal{L}(f) \mathcal{L}(g)$.


\begin{lemma}[\cite{Ke}] \label{lem1}
Let $P(s) = \sum_{k=0}^n {\alpha_k s^k}$ and
$Q(s) = \sum_{k=0}^m {\beta_k s^k}$, where $m, n$ are
nonnegative integers with $m < n$ and $\alpha_k, \beta_k$ are
scalars.
Then there exists an infinitely differentiable function
$g : (0,\infty) \to \mathbb{F}$ such that
$$
\mathcal{L}(g) = \frac{Q(s)}{P(s)} \quad ( \Re(s) > \sigma_{{}_P} )
$$
and
$$
g^{(i)}(0)
= \begin{cases}
 0 & \text{for } i \in \{ 0, 1, \dots, n-m-2 \},\\
 \beta_m / \alpha_n
 & \text{for } i = n-m-1
 \end{cases}
$$
where $\sigma_{{}_P} = \max \{ \Re(s) : P(s) = 0 \}$.
\end{lemma}


\begin{lemma}[\cite{Ke}] \label{lem2}
Given an integer $n > 1$, let $f : (0,\infty) \to \mathbb{F}$
be a continuous function and let $P(s)$ be a complex polynomial
of degree $n$.
Then there exists an $n$ times continuously differentiable
function $h : (0,\infty) \to \mathbb{F}$ such that
$$
\mathcal{L}(h) = \frac{\mathcal{L}(f)}{P(s)} \quad
(\Re(s) > \max\{ \sigma_{{}_P}, \sigma_{f} \} ),
$$
where $\sigma_{{}_P} = \max\{ \Re(s) : P(s) = 0 \}$ and
$\sigma_{f}$ is the abscissa of convergence for $f$.
In particular, it holds that $h^{(i)}(0) = 0$ for every
$i \in \{ 0, 1, \dots, n-1 \}$.
\end{lemma}

\section{Main Results}

Let $\mathbb{F}$ denote either $\mathbb{R}$ or $\mathbb{C}$.
In the following theorem, using the Laplace transform method,
we investigate the generalized Hyers-Ulam stability of the
linear differential equation of first order
\begin{equation}
\label{eq:1}
y'(t) + \alpha y(t) = f(t).
\end{equation}



\begin{theorem}\label{thm:3.1}
Let $\alpha$ be a constant in $\mathbb{F}$ and let
$\varphi : (0,\infty) \to (0,\infty)$ be an integrable function.
If a continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
\begin{equation}
\label{eq:2}
| y'(t) + \alpha y(t) - f(t) | \leq \varphi(t)
\end{equation}
for all $t > 0$, then there exists a solution
$y_\alpha : (0,\infty) \to \mathbb{F}$ of the differential
equation $\eqref{eq:1}$ such that
$$
| y(t) - y_\alpha(t) |
\leq e^{-\Re(\alpha)t} \int_0^t e^{\Re(\alpha)x} \varphi(x) dx
$$
for any $t > 0$.
\end{theorem}

\begin{proof}
If we define a function $z : (0,\infty) \to \mathbb{F}$ by
$z(t) := y'(t) + \alpha y(t) - f(t)$ for each $t > 0$, then
\begin{equation}
\mathcal{L}(y) - \frac{y(0) + \mathcal{L}(f)}{s + \alpha}
= \frac{\mathcal{L}(z)}{s + \alpha}.
\label{eq:20131113-1}
\end{equation}
If we set $y_\alpha(t) := y(0) e^{-\alpha t} + (E_{-\alpha}*f)(t)$,
where $E_{-\alpha}(t) = e^{-\alpha t}$, then $y_\alpha(0) = y(0)$
and
\begin{equation}
\mathcal{L} (y_\alpha)
= \frac{y(0) + \mathcal{L}(f)}{s + \alpha}
= \frac{{y_\alpha(0) + \mathcal{L}(f)}}{s + \alpha}.
\label{eq:20131113-2}
\end{equation}
Hence, we get
$$
\mathcal{L}\big( y'_\alpha(t) + \alpha y_\alpha(t) \big)
= s\mathcal{L}(y_\alpha) - y_\alpha(0) + \alpha \mathcal{L}(y_\alpha)
= \mathcal{L}(f).
$$
Since $\mathcal{L}$ is a one-to-one operator, it holds that
$$
y'_\alpha(t) + \alpha y_\alpha(t) = f(t).
$$
Thus, $y_\alpha$ is a solution of \eqref{eq:1}.

Moreover, by \eqref{eq:20131113-1} and \eqref{eq:20131113-2},
we obtain
$\mathcal{L}(y) - \mathcal{L}(y_\alpha) = \mathcal{L}(E_{-\alpha}*z)$.
Therefore, we have
\begin{equation}
y(t) - y_\alpha(t) = (E_{-\alpha}*z)(t).
\label{eq:20131113-3}
\end{equation}
In view of \eqref{eq:2}, it holds that
\begin{equation}
| z(t) | \leq \varphi(t)
\label{eq:20131113-4}
\end{equation}
for all $t > 0$, and it follows from the definition of
convolution, \eqref{eq:20131113-3}, and \eqref{eq:20131113-4}
that
\begin{align*}
| y(t) - y_\alpha(t) |
& = | (E_{-\alpha}*z)(t) | \\
& = \big| \int_0^t E_{-\alpha}(t-x) z(x) dx \big| \\
& \leq \int_0^t \big| e^{-\alpha(t-x)} \big| \varphi(x) dx  \\
& \leq e^{-\Re(\alpha)t}  \int_0^t e^{\Re(\alpha)x} \varphi(x) dx
\end{align*}
for all $t > 0$.
(We remark that $\int_0^t e^{\Re(\alpha)x} \varphi(x) dx$
exists for each $t > 0$ provided $\varphi$ is an integrable
function.)
\end{proof}


\begin{corollary}\label{cor:3.1}
Let $\alpha$ be a constant in $\mathbb{F}$ and let
$\varphi : (0,\infty) \to (0,\infty)$ be an integrable function
such that
\begin{equation}
\int_0^t e^{\Re(\alpha)(x-t)} \varphi(x) dx \leq K \varphi(t)
\label{eq:20131121-1}
\end{equation}
for all $t > 0$ and for some positive real constant $K$.
If a continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
$\eqref{eq:2}$ for all $t > 0$, then there exists a solution
$y_\alpha : (0,\infty) \to \mathbb{F}$ of the differential
equation $\eqref{eq:1}$ such that
$$
| y(t) - y_\alpha(t) | \leq K \varphi(t)
$$
for any $t > 0$.
\end{corollary}

In the following remark, we show that there exists an integrable
function $\varphi : (0,\infty) \to (0,\infty)$ satisfying the
condition \eqref{eq:20131121-1}.

\begin{remark}\rm
Let $\alpha$ be a constant in $\mathbb{F}$ with $\Re(\alpha) > -1$.
If we define $\varphi(t) = Ae^t$ for all $t > 0$ and for some
$A > 0$, then we have
\begin{align*}
\int_0^t e^{\Re(\alpha)(x-t)} \varphi(x) dx
&= \int_0^t e^{\Re(\alpha)(x-t)} A e^x dx \\
&= \frac{1}{1 + \Re(\alpha)}
 \left( Ae^t - Ae^{-\Re(\alpha)t} \right) \\
&\leq \frac{1}{1 + \Re(\alpha)} \varphi(t)
\end{align*}
for each $t > 0$.
\end{remark}


Now, we apply the Laplace transform method to the proof of the
generalized Hyers-Ulam stability of the linear differential
equation of second order
\begin{equation}
\label{eq:4}
y''(t) + \beta y'(t) + \alpha y(t) = f(t).
\end{equation}

\begin{theorem}\label{thm:3.2}
Let $\alpha$ and $\beta$ be constants in $\mathbb{F}$ such that
there exist $a, b \in \mathbb{F}$ with $a + b = -\beta$,
$ab = \alpha$, and $a \neq b$.
Assume that $\varphi : (0,\infty) \to (0,\infty)$ is an
integrable function.
If a twice continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
\begin{equation}
\label{eq:22}
| y''(t) + \beta y'(t) + \alpha y(t) - f(t) | \leq \varphi(t)
\end{equation}
for all $t > 0$, then there exists a solution
$y_c : (0,\infty) \to \mathbb{F}$ of the differential equation
$\eqref{eq:4}$ such that
$$
| y(t) - y_c(t) |
\leq \frac{e^{\Re(a)t}}{| a - b |}
 \int_0^t e^{-\Re(a)x} \varphi(x) dx +
 \frac{e^{\Re(b)t}}{| a - b |}
 \int_0^t e^{-\Re(b)x} \varphi(x) dx
$$
for all $t > 0$.
\end{theorem}

\begin{proof}
If we define a function $z : (0,\infty) \to \mathbb{F}$ by
$z(t) := y''(t) + \beta y'(t) + \alpha y(t) - f(t)$ for each
$t > 0$, then we have
\begin{equation} \label{eq:5}
\mathcal{L}(z) = \big( s^2 + \beta s + \alpha \big) \mathcal{L}(y) -
 [ sy(0) + \beta y(0) + y'(0) ] - \mathcal{L}(f).
\end{equation}
In view of \eqref{eq:5}, a function
$y_0 : (0,\infty) \to \mathbb{F}$ is a solution of \eqref{eq:4}
if and only if
\begin{equation} \label{eq:6}
\big( s^2 + \beta s + \alpha \big) \mathcal{L}(y_0) - s y_0(0) -
[ \beta y_0(0) + y'_0(0) ] = \mathcal{L}(f).
\end{equation}
Now, since $s^2 + \beta s + \alpha = (s-a)(s-b)$, \eqref{eq:5}
implies that
\begin{equation} \label{eq:7}
\mathcal{L}(y) - \frac{sy(0) + [\beta y(0) + y'(0)] + \mathcal{L}(f)}
{(s - a)(s - b)} = \frac{\mathcal{L}(z)}{(s - a)(s - b)}.
\end{equation}

If we set
\begin{equation}
y_c(t) := y(0) \frac{ae^{at} - be^{bt}}{a - b} +
 [\beta y(0) + y'(0)] E_{a,b}(t) + (E_{a,b}*f)(t),
\label{eq:20131119-1}
\end{equation}
where $E_{a,b}(t) := \frac{e^{at} - e^{bt}}{a-b}$, then
$y_c(0) = y(0)$.
Moreover, since
\begin{align*}
&y'_c(t) = y(0) \frac{a^2 e^{at} - b^2 e^{bt}}{a-b} +
 [\beta y(0) + y'(0)] \frac{ae^{at} - be^{bt}}{a-b} +
 \frac{d}{dt}(E_{a,b}*f)(t), \\
&(E_{a,b}*f)(t) = \frac{e^{at}}{a-b} \int_0^t e^{-ax} f(x) dx -
 \frac{e^{bt}}{a-b} \int_0^t e^{-bx} f(x) dx,
\end{align*}
we have
\begin{align*}
y'_c(0) & = y(0) \frac{a^2 - b^2}{a-b} +
 [\beta y(0) + y'(0)] \frac{a-b}{a-b} \\
 & = (a+b) y(0) + \beta y(0) + y'(0) \\
% & = -\beta y(0) + \beta y(0) + y'(0) \\
 & = y'(0).
\end{align*}
It follows from \eqref{eq:20131119-1} that
\begin{equation} \label{eq:8}
\mathcal{L}(y_c)
= \frac{sy_c(0) + [ \beta y_c(0) + y'_c(0) ] + \mathcal{L}(f)}
 {(s - a)(s - b)}.
\end{equation}
Now, \eqref{eq:6} and \eqref{eq:8} imply that $y_c$ is a
solution of \eqref{eq:4}.
Applying \eqref{eq:7} and \eqref{eq:8} and considering the
facts that $y_c(0) = y(0)$, $y'_c(0) = y'(0)$, and
$\mathcal{L}(E_{a,b}*z) = \frac{\mathcal{L}(z)}{(s - a)(s - b)}$, we
obtain $\mathcal{L}(y) - \mathcal{L}(y_c) = \mathcal{L}(E_{a,b}*z)$ or
equivalently, $y(t) - y_c(t) = (E_{a,b}*z)(t)$.

In view of \eqref{eq:22}, it holds that
$| z(t) | \leq \varphi(t)$, and it follows from the definition
of the convolution that
\begin{align*}
| y(t) - y_{c}(t) |
& = | (E_{a,b}*z)(t) | \\
& \leq \frac{e^{\Re(a)t}}{| a - b |}
 \int_0^t {e^{-\Re(a)x} \varphi(x) dx} +
 \frac{e^{\Re(b)t}}{| a - b |}
 \int_0^t {e^{-\Re(b)x} \varphi(x) dx}
\end{align*}
for any $t > 0$.
We remark that $\int_0^t e^{-\Re(a)x} \varphi(x) dx$ and
$\int_0^t e^{-\Re(b)x} \varphi(x) dx$ exist for any $t > 0$
provided $\varphi$ is an integrable function.
\end{proof}

\begin{corollary}\label{cor:3.2}
Let $\alpha$ and $\beta$ be constants in $\mathbb{F}$ such that
there exist $a, b \in \mathbb{F}$ with $a + b = -\beta$,
$ab = \alpha$, and $a \neq b$.
Assume that $\varphi : (0,\infty) \to (0,\infty)$ is an
integrable function for which there exists a positive real
constant $K$ with
\begin{equation}
\int_0^t \left( e^{\Re(a)(t-x)} + e^{\Re(b)(t-x)} \right)
 \varphi(x) dx \leq K \varphi(t)
\label{eq:20131121-2}
\end{equation}
for all $t > 0$.
If a twice continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
$\eqref{eq:22}$ for all $t > 0$, then there exists a solution
$y_c : (0,\infty) \to \mathbb{F}$ of the differential equation
$\eqref{eq:4}$ such that
$$
| y(t) - y_c(t) | \leq \frac{K}{| a - b |} \varphi(t)
$$
for all $t > 0$.
\end{corollary}

We now show that there exists an integrable function
$\varphi : (0,\infty) \to (0,\infty)$ which satisfies the
condition \eqref{eq:20131121-2}.


\begin{remark} \rm
Let $\alpha$ and $\beta$ be constants in $\mathbb{F}$ such that
there exist $a, b \in \mathbb{F}$ with $a + b = -\beta$,
$ab = \alpha$, $\Re(a) < 1$, $\Re(b) < 1$, and $a \neq b$.
If we define $\varphi(t) = Ae^t$ for all $t > 0$ and for some
$A > 0$, then we get
\begin{align*}
&\int_0^t \Big( e^{\Re(a)(t-x)} + e^{\Re(b)(t-x)} \Big)
 \varphi(x) dx\\
&= \int_0^t
 \Big( e^{\Re(a)(t-x)} + e^{\Re(b)(t-x)} \Big)
 Ae^x dx \\
&= \frac{A}{1 - \Re(a)} \Big( e^t - e^{\Re(a)t} \Big)
 + \frac{A}{1 - \Re(b)} \Big( e^t - e^{\Re(b)t} \Big)
 \\
&\leq \Big( \frac{1}{1 - \Re(a)} + \frac{1}{1 - \Re(b)}
 \Big) \varphi(t)
\end{align*}
for all $t > 0$.
\end{remark}

Similarly, we apply the Laplace transform method to investigate
the generalized Hyers-Ulam stability of the linear differential
equation of $n$th order
\begin{equation}
\label{eq:10}
y^{(n)}(t) + \sum_{k=0}^{n-1} \alpha_k y^{(k)}(t) = f(t)
\end{equation}


\begin{theorem}\label{thm:3.3}
Let $\alpha_0, \alpha_1, \dots, \alpha_{n}$ be scalars in
$\mathbb{F}$ with $\alpha_n = 1$, where $n$ is an integer larger
than $1$.
Assume that $\varphi : (0,\infty) \to (0,\infty)$ is an
integrable function of exponential order.
If an $n$ times continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
\begin{equation} \label{eq:11}
\big| y^{(n)}(t) + \sum_{k=0}^{n-1} \alpha_k y^{(k)}(t) -
 f(t) \big| \leq \varphi(t)
\end{equation}
for all $t > 0$, then there exist real constants $M > 0$ and
$\sigma_g$ and a solution $y_c : (0,\infty) \to \mathbb{F}$ of
the differential equation $\eqref{eq:10}$ such that
$$
| y(t) - y_c(t) |
\leq M \int_0^t e^{\alpha (t-x)} \varphi(x) dx
$$
for all $t > 0$ and $\alpha > \sigma_g$.
\end{theorem}

\begin{proof}
Applying  integration by parts repeatedly, we derive
$$
\mathcal{L}\big( y^{(k)} \big)
= s^k \mathcal{L}(y) - \sum_{j=1}^k s^{k-j} y^{(j-1)}(0)
$$
for any integer $k > 0$.
Using this formula, we can prove that a function
$y_0 : (0,\infty) \to \mathbb{F}$ is a solution of \eqref{eq:10}
if and only if
\begin{equation} \label{eq:12}
\begin{aligned}
\mathcal{L}(f)
& = \sum_{k=0}^n \alpha_k s^k \mathcal{L}(y_0) -
 \sum_{k=1}^n \alpha_k \sum_{j=1}^k s^{k-j} y_0^{(j-1)}(0)
 \\
& = \sum_{k=0}^n \alpha_k s^k \mathcal{L}(y_0) -
 \sum_{j=1}^n \sum_{k=j}^n \alpha_k s^{k-j} y_0^{(j-1)}(0)
 \\
& = P_{n,0}(s) \mathcal{L}(y_0) -
 \sum_{j=1}^n P_{n,j}(s) y_0^{(j-1)}(0),
\end{aligned}
\end{equation}
where $P_{n,j}(s) := \sum_{k=j}^n \alpha_k s^{k-j}$ for
$j \in \{ 0, 1, \dots, n \}$.

Let us define a function $z : (0,\infty) \to \mathbb{F}$ by
\begin{equation}
z(t) := y^{(n)}(t) + \sum_{k=0}^{n-1} \alpha_k y^{(k)}(t) - f(t)
\label{eq:20131118-1}
\end{equation}
for all $t > 0$.
Then, similarly as in \eqref{eq:12}, we obtain
$$
\mathcal{L}(z) = P_{n,0}(s) \mathcal{L}(y) -
 \sum_{j=1}^n P_{n,j}(s) y^{(j-1)}(0) - \mathcal{L}(f).
$$
Hence, we get
\begin{equation} \label{eq:13}
\mathcal{L}(y) - \frac{1}{P_{n,0}(s)}
\Big( \sum_{j=1}^n P_{n,j}(s) y^{(j-1)}(0) + \mathcal{L}(f) \Big)
= \frac{\mathcal{L}(z)}{P_{n,0}(s)}.
\end{equation}

Let $\sigma_{f}$ be the abscissa of convergence for $f$, let
$s_1, s_2, \dots, s_n$ be the roots of the polynomial
$P_{n,0}(s)$, and let $\sigma_{{}_P} = \max\{ \Re(s_k) :
 k \in \{ 1, 2, \dots, n \} \}$.
For any $s$ with
$\Re(s) > \max\{ \sigma_{f}, \sigma_{{}_P} \}$, we set
\begin{equation}
\label{eq:14}
G(s) := \frac{1}{P_{n,0}(s)}
 \Big( \sum_{j=1}^n P_{n,j}(s) y^{(j-1)}(0) + \mathcal{L}(f)
 \Big).
\end{equation}
By Lemma \ref{lem2}, there exists an $n$ times continuously
differentiable function $f_0$ such that
\begin{equation} \label{eq:15}
\mathcal{L}(f_0) = \frac{\mathcal{L}(f)}{P_{n,0}(s)}
\end{equation}
for all $s$ with
$\Re(s) > \max\{ \sigma_{f}, \sigma_{{}_P} \}$ and
\begin{equation}
f_0^{(i)}(0) = 0
\label{eq:20131116-1}
\end{equation}
for any $i \in \{ 0, 1, \dots, n-1 \}$.

For $j \in \{ 1, 2, \dots, n \}$, we note that
\begin{equation}
\label{eq:16}
\frac{P_{n,j}(s)}{P_{n,0}(s)} = \frac{1}{s^j} -
\frac{\sum_{k=0}^{j-1} \alpha_k s^k}{s^j P_{n,0}(s)}
\end{equation}
for every $s$ with $\Re(s) > \max\{ 0, \sigma_{{}_P} \}$.
Applying Lemma \ref{lem1} for the case of
$Q(s) = \sum_{k=0}^{j-1} \alpha_k s^k$ and
$P(s) = s^j P_{n,0}(s)$, we can find an infinitely
differentiable function $g_j$ such that
\begin{equation}
\label{eq:17}
\mathcal{L}(g_j) = \frac{\sum_{k=0}^{j-1} \alpha_k s^k}
 {s^j P_{n,0}(s)}
\end{equation}
and $g_j^{(k)}(0) = 0$ for $k \in \{ 0, 1, \dots, n-1 \}$.

Let
\begin{equation} \label{eq:18}
f_j (t) := \frac{t^{j-1}}{(j-1)!} - g_j (t)
\end{equation}
for $j \in \{ 1, 2, \dots, n \}$.
Then we have
\begin{equation}
f_j^{(i)}(0) = \begin{cases}
0 & \text{for } i \in \{ 0, 1, \dots, j-2, j, j+1, \dots,
 n-1 \}, \\
1 & \text{for } i = j-1.
\end{cases}
\label{eq:20131116-2}
\end{equation}
If we define
$$
y_c (t) := \sum_{j=1}^n y^{(j-1)}(0) f_j(t) + f_0(t),
$$
then the conditions \eqref{eq:20131116-1} and
\eqref{eq:20131116-2} imply that
\begin{equation}
y_c^{(i)}(0) = y^{(i)}(0)
\label{eq:20131116-3}
\end{equation}
for every $i \in \{ 0, 1, \dots, n-1 \}$.
Moreover, it follows from \eqref{eq:14}--\eqref{eq:20131116-3}
that
\begin{equation}  \label{eq:19}
\begin{aligned}
\mathcal{L}(y_c)
& = \sum_{j=1}^n y^{(j-1)}(0) \mathcal{L}(f_j) + \mathcal{L}(f_0)
 \\
& = \sum_{j=1}^n y^{(j-1)}(0)
 \Big( \frac{1}{s^j} - \mathcal{L}(g_j) \Big) +
 \frac{\mathcal{L}(f)}{P_{n,0}(s)} \\
& = \frac{1}{P_{n,0}(s)}
 \Big( \sum_{j=1}^n P_{n,j}(s) y^{(j-1)}(0) + \mathcal{L}(f)
 \Big)
\end{aligned}
\end{equation}
for each $s$ with
$\Re(s) > \max\{ 0, \sigma_{f}, \sigma_{{}_P} \}$.

Now, \eqref{eq:12} implies that $y_c$ is a solution of
\eqref{eq:10}.
Moreover, by \eqref{eq:13} and \eqref{eq:19}, we have
\begin{equation}
\label{eq:20}
\mathcal{L}(y) - \mathcal{L}(y_c) = \frac{\mathcal{L}(z)}{P_{n,0}(s)}.
\end{equation}

Applying Lemma \ref{lem1} for the case of $Q(s) = 1$ and
$P(s) = P_{n,0}(s)$, we find an infinitely differentiable
function $g : (0,\infty) \to \mathbb{F}$ such that
\begin{equation} \label{eq:21}
\mathcal{L}(g) = \frac{1}{P_{n,0}(s)}
\end{equation}
which implies that
$$
g(t) = \mathcal{L}^{-1} \bigg( \frac{1}{P_{n,0}(s)} \bigg)
 = \frac{1}{2\pi} \int_{-\infty }^\infty e^{(\alpha + iy)t}
 \frac{1}{P_{n,0} (\alpha + iy)} dy
$$
for any real constant $\alpha > \sigma_g$.
Moreover, it holds that
\begin{equation} \label{eq:23}
\begin{aligned}
| g(t-x) |
& \leq \frac{1}{2\pi} \int_{-\infty}^\infty
 \big| e^{(\alpha + iy)(t-x)} \big|
 \frac{1}{| P_{n,0}(\alpha + iy) |} dy \\
& \leq \frac{1}{2\pi} \int_{-\infty}^\infty e^{\alpha(t-x)}
 \frac{1}{| P_{n,0}(\alpha + iy) |} dy \\
& \leq \frac{1}{2\pi} e^{\alpha(t-x)} \int_{-\infty}^\infty
 \frac{1}{| P_{n,0}(\alpha + iy) |} dy \\
& \leq M e^{\alpha(t-x)}
\end{aligned}
\end{equation}
for all $\alpha > \sigma_g$, where
$$
M = \frac{1}{2\pi} \int_{-\infty}^\infty
 \frac{1}{| P_{n,0}(\alpha + iy) |} dy < \infty,
$$
because $n$ is an integer larger than $1$.
By \eqref{eq:11} and \eqref{eq:20131118-1}, it also holds
that $| z(t) | \leq \varphi(t)$ for all $t > 0$.

In view of \eqref{eq:20}, \eqref{eq:21}, and \eqref{eq:23},
we obtain
$$
\mathcal{L}(y) - \mathcal{L}(y_c) = \mathcal{L}(g) \mathcal{L}(z)
= \mathcal{L}(g*z).
$$
Consequently, we have $y(t) - y_c(t) = (g*z)(t)$ for any
$t > 0$.
Hence, it follows from \eqref{eq:11}, \eqref{eq:20131118-1},
and \eqref{eq:23} that
$$
| y(t) - y_c(t) |
= | (g*z)(t) |
\leq \int_0^t | g(t-x) | | z(x) | dx
\leq M \int_0^t e^{\alpha(t-x)} \varphi(x) dx
$$
for all $t > 0$ and for any real constant $\alpha > \sigma_g$,
which completes the proof.
\end{proof}


\begin{corollary}\label{cor:3.3}
Let $\alpha_0, \alpha_1, \dots, \alpha_{n}$ be scalars in
$\mathbb{F}$ with $\alpha_n = 1$, where $n$ is an integer larger
than $1$.
Assume that there exist real constants $\alpha$ and $K > 0$
such that a function $\varphi : (0,\infty) \to (0,\infty)$
satisfies
$$
\int_0^t e^{\alpha (t-x)} \varphi(x) dx \leq K \varphi(t)
$$
for all $t > 0$.
Moreover, assume that the constant $\sigma_g$ given in Theorem
\ref{thm:3.3} is less than $\alpha$.
If an $n$ times continuously differentiable function
$y : (0,\infty) \to \mathbb{F}$ satisfies the inequality
$\eqref{eq:11}$ for all $t > 0$, then there exist a real
constants $M > 0$ and a solution $y_c : (0,\infty) \to \mathbb{F}$
of the differential equation $\eqref{eq:10}$ such that
$$
| y(t) - y_c(t) | \leq K M \varphi(t)
$$
for all $t > 0$.
\end{corollary}



\begin{remark} \rm
Assume that $\alpha < 1$.
If we define $\varphi(t) = Ae^t$ for all $t > 0$ and for some
$A > 0$, then we get
\begin{align*}
\int_0^t e^{\alpha (t-x)} \varphi(x) dx
= \int_0^t e^{\alpha (t-x)} A e^x dx
= \frac{A}{1-\alpha} \big( e^t - e^{\alpha t} \big)
\leq \frac{1}{1-\alpha} \varphi(t)
\end{align*}
for all $t > 0$.
\end{remark}

\subsection*{Acknowledgements}
This research was supported by Basic Science Research Program
through the  National Research Foundation of Korea
(NRF) funded by the Ministry of Education
(No. 2013R1A1A2005557).

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\end{document}