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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 172, 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-2013/172\hfil Ulam-Hyers-Rassias stability]
{Ulam-Hyers-Rassias stability of semilinear differential
equations with impulses}

\author[Xuezhu Li, Jinrong Wang \hfil EJDE-2013/172\hfilneg]
{Xuezhu Li, Jinrong Wang}  % in alphabetical order

\address{Xuezhu Li \newline
Department of Mathematics,
Guizhou University, Guiyang, Guizhou 550025, China}
\email{xzleemath@126.com}

\address{Jinrong Wang \newline
Department of Mathematics,
Guizhou University, Guiyang, Guizhou 550025, China}
\email{wjr9668@126.com}

\thanks{Submitted March 20, 2013. Published July 26, 2013.}
\subjclass[2000]{34G20, 34D10, 45N05}
\keywords{Semilinear differential equations; impulses;
\hfill\break\indent Ulam-Hyers-Rassias stability}

\begin{abstract}
 In this article, we present Ulam-Hyers-Rassias and  Ulam-Hyers
 stability results for semilinear differential equations with
 impulses on a compact interval. An example is also provided to
 illustrate our results.
\end{abstract}

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


\section{Introduction}

Many researchers paid  attention to the stability
properties of all kinds of equations since Ulam \cite{Ulam} raised the famous
stability problem of functional equations (Ulam problem) in 1940.
Such problems have been taken up by Hyers \cite{Hyers41}, Rassias
\cite{Rassias} and other mathematicians. Recently, the study of this
area has the grown to be one of the most important subjects in the
mathematical analysis area. For the advanced contribution on Ulam
problem, we refer the reader to Andr\'{a}s and Kolumb\'{a}n
\cite{Andras-NATMA}, Andr\'{a}s and M\'{e}sz\'{a}ros
\cite{Andras-AMC}, Burger et al \cite{Burger}, C\u{a}dariu
\cite{Cadariu1}, Cimpean and Popa \cite{Cimpean}, Hyers
\cite{Hyers}, Hegyi and Jung \cite{Hegyi}, Jung
\cite{Jung1,Jung-aml1}, Lungu and Popa \cite{Lungu}, Miura et al
\cite{Miura1,Miura2}, Ob{\l}oza \cite{Obloza1,Obloza2}, Rassias
\cite{Rassias98,Rassias00}, Rus \cite{Rus2009,Rus2010}, Takahasi et
al \cite{Takahasi}  and Wang et al \cite{Wang-JMAA}.


However, Ulam-Hyers-Rassias stability of semilinear differential
equations with impulses have not been studied. Motivated by recent
works \cite{Rus2010,Wang-JMAA}, we investigate Ulam-Hyers-Rassias
stability of the following semilinear differential equations with
impulses
\begin{equation}\label{sy.1-im-no}
\begin{gathered}
x'(t)=\lambda x(t)+f(t,x(t)),\quad t\in J':=J\setminus
\{t_{1},\dots,t_{m}\},\; J:=[0,T],\\
\Delta x(t_{k})=I_{k}(x(t^{-}_k)),\quad k=1,2,\dots,m,
\end{gathered}
\end{equation}
where $0<T<+\infty$, $\lambda>0$, $f:J \times \mathbb{R}\to
\mathbb{R}$ is continuous, $I_{k}: \mathbb{R}\to \mathbb{R}$ and
$t_{k}$ satisfy $0=t_{0}<t_{1}< \dots <t_{m}<t_{m+1}=T$,
$\Delta x(t_k):=x(t_k^+)-x(t_k^-)$,
$x(t_{k}^{+})=\lim_{\epsilon \to 0^{+}}x(t_{k}+\epsilon)$ and
$x(t_{k}^{-})=\lim_{\epsilon\to 0^{-}} x(t_{k}+\epsilon)$
represent the right and left limits of
$x(t)$ at $t=t_{k}$.

We will adopt the concepts in Wang et al \cite{Wang-JMAA} and
introduce four types of Ulam stabilities (see Definitions
\ref{def1}--\ref{def4}): Ulam-Hyers stability, generalized
Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized
Ulam-Hyers-Rassias stability for the equation \eqref{sy.1-im-no}.
Next, we present the Ulam-Hyers-Rassias stability results for the
equation \eqref{sy.1-im-no} on a compact interval by virtue of an
integral inequality of Gronwall type for piecewise continuous
functions  (see Lemma \ref{Gronwall-class}).

\section{Preliminaries}

In this section, we introduce some notation, and
preliminary facts. Throughout this paper, let $C(J,\mathbb{R})$ be
the Banach space of all continuous functions from $J$ into
$\mathbb{R}$ with the norm $\|x\|_{C}:= \sup\{|x(t)|:t\in J\}$ for
$x\in C(J,\mathbb{R})$. We also introduce the Banach space
$PC(J,\mathbb{R}):=\{x:J \to \mathbb{R}:x\in
C((t_{k},t_{k+1}],\mathbb{R})$, $k=0,\dots,m$ and there exist
 $x(t_{k}^{-})$ and $x(t_{k}^{+}),k=1,\dots,m, \text{ with }
x(t_{k}^{-})=x(t_{k})\}$ with the norm $\|x\|_{PC}:=
\sup\{|x(t)|:t\in J\}$.  Denote $PC^{1}(J,\mathbb{R}):= \{x\in
PC(J,\mathbb{R}) : x'\in PC(J,\mathbb{R})\}$. Set
$\|x\|_{PC^1}:=\max\{\|x\|_{PC},\|x'\|_{PC}\}$. It can be seen that
endowed with the norm $\|\cdot\|_{PC^{1}}$, $PC^{1}(J,\mathbb{R})$
is also a Banach space.

\begin{definition} \label{def2.1} \rm
By a $PC^1$-solution of the  impulsive Cauchy problem
\begin{equation}\label{impulsive-Cauchy}
\begin{gathered}
x'(t)=\lambda x(t)+f(t,x(t)),\quad t\in J',\\
\Delta x(t_{k})=I_{k}(x(t^{-}_k)),\quad k=1,2,\dots,m,\\
x(0)=x_{0},\quad x_0\in \mathbb{R},
\end{gathered}
\end{equation}
we mean the function $x\in PC^1(J,\mathbb{R})$ that satisfies
\[
x(t)=e^{\lambda t}x_{0}+\int_{0}^{t}e^{\lambda(t-s)}f(s,x(s))ds
+\sum_{0<t_{k}<t}e^{\lambda(t-t_{k})}I_{k}(x(t^-_{k})),\quad t\in J.
\]
\end{definition}

Bainov and Hristova \cite{Bainov} studied the following integral
inequality of Gronwall type for piecewise continuous functions which
can be used in the sequel.

\begin{lemma}\label{Gronwall-class}
Let for $t\geq t_{0}\geq0$ the following inequality hold
\begin{equation}
 x( t)\leq a(t)+\int_{t_{0}}^{t} g(t,s)x(s)
ds +\sum_{t_{0}<t_{k}<t}\beta_{k}(t)x(t_{k}),
\end{equation}
where $\beta_{k}(t)(k\in\mathbb{N})$ are nondecreasing functions for
$t\geq t_{0}$, $a\in PC([t_{0},\infty),\mathbb{R}_{+})$, $a$ is
nondecreasing and $g(t,s)$ is a continuous nonnegative function for
$t,s\geq t_{0}$ and nondecreasing with respect to $t$ for any fixed
$s\geq t_{0}$.
Then, for $t\geq t_{0}$, the following inequality is valid:
\[
x(t)\leq  a(t)\prod_{t_{0}<t_{k}<t}(1+\beta_{k}(t))\exp
\Big(\int_{t_{0}}^{t}g(t,s)ds\Big).
\]
\end{lemma}

\section{Basic concepts and remarks}

Here, we adopt the concepts in Wang et al \cite{Wang-JMAA}  and
introduce Ulam's type stability concepts for the equation
\eqref{sy.1-im-no}.

Let $\epsilon>0$, $\psi\ge0$ and $\varphi\in PC(J,\mathbb{R}_+)$ is
nondecreasing. We consider the set of inequalities
\begin{equation}\label{U-H stable}
\begin{gathered}
|y'(t)-\lambda y(t)-f(t,y(t))|\leq\epsilon,\quad t\in J',\\
|\Delta y(t_{k})-I_{k}(y(t^{-}_k))|\leq \epsilon, \quad k=1,2,\dots,m;
\end{gathered}
\end{equation}
the set of inequalities
\begin{equation}\label{generalized U-H-R stable}
\begin{gathered}
|y'(t)-\lambda y(t)-f(t,y(t))|\leq\varphi(t), \quad t\in J',\\
|\Delta y(t_{k})-I_{k}(y(t^{-}_k))|\leq \psi,\quad k=1,2,\dots,m;
\end{gathered}
\end{equation}
and the set of inequalities
\begin{equation}\label{U-H-R stable}
\begin{gathered}
|y'(t)-\lambda y(t)-f(t,y(t))|\leq\epsilon\varphi(t), \quad t\in J',\\
|\Delta y(t_{k})-I_{k}(y(t^{-}_k))|\leq\epsilon
\psi,\quad k=1,2,\dots,m.
\end{gathered}
\end{equation}

\begin{definition}\label{def1} \rm
 Equation \eqref{sy.1-im-no} is Ulam-Hyers stable if there exists a
real number $c_{f,m}>0$ such that for each $\epsilon>0$ and for each
solution $y\in PC^1(J,\mathbb{R})$ of the inequality \eqref{U-H
stable} there exists a solution $x\in PC^1(J,R)$ of the equation
\eqref{sy.1-im-no} with
\[
|y(t)-x(t)|\leq c_{f,m}\epsilon ,\quad t\in J.
\]
\end{definition}

\begin{definition}\label{def2} \rm
Equation \eqref{sy.1-im-no} is generalized Ulam-Hyers stable if
there exists $\theta_{f,m}\in C(\mathbb{R}_+,\mathbb{R}_+)$,
$\theta_{f,m}(0)=0$ such that for each solution $y\in
PC^1(J,\mathbb{R})$  of the inequality \eqref{U-H stable} there
exists a solution $x\in PC^1(J,\mathbb{R})$  of the equation
\eqref{sy.1-im-no} with
\[
|y(t)-x(t)|\leq\theta_{f,m}(\epsilon),\quad t\in J.
\]
\end{definition}

\begin{definition}\label{def3}\rm
Equation \eqref{sy.1-im-no} is Ulam-Hyers-Rassias stable with
respect to $(\varphi,\psi)$ if there exists $c_{f,m,\varphi}>0$ such
that for each $\epsilon>0$ and for each solution $y\in
PC^1(J,\mathbb{R})$ of the inequality \eqref{U-H-R stable} there
exists a solution $x\in PC^1(J,\mathbb{R})$ of the equation
\eqref{sy.1-im-no} with
\[
|y(t)-x(t)|\leq c_{f,m,\varphi}\epsilon (\varphi(t)+\psi),\quad t\in J.
\]
\end{definition}

\begin{definition}\label{def4}\rm
The equation \eqref{sy.1-im-no} is generalized Ulam-Hyers-Rassias
stable with respect to $(\varphi,\psi)$ if there exists
$c_{f,m,\varphi}>0$ such that for each solution $y\in
PC^1(J,\mathbb{R})$  of the inequality \eqref{generalized U-H-R
stable} there exists a solution $x\in PC^1(J,\mathbb{R})$ of the
equation \eqref{sy.1-im-no} with
\begin{align*}
|y(t)-x(t)|\leq c_{f,m,\varphi}(\varphi(t)+\psi),~t\in J.
\end{align*}
\end{definition}

\begin{remark}\label{def1-4-remark} \rm
It is clear that: (i) Definition \ref{def1} implies
Definition \ref{def2}; (ii) Definition \ref{def3} implies
Definition \ref{def4}; (iii) Definition \ref{def3} for $\varphi
(t)=\psi=1$ implies Definition \ref{def1}.
\end{remark}

\begin{remark}\label{remark1} \rm
A function $y\in PC^{1}(J,\mathbb{R})$ is a solution of the
inequality
\eqref{U-H stable} if and only if there is $g\in PC(J,\mathbb{R})$
and a sequence  $g_k$, $k=1,2,\dots,m$ (which depend on $y$) such that
\begin{itemize}
\item[(i)] $|g(t)|\leq\epsilon \varphi(t),~t\in J$ and $|g_k|\le\epsilon \psi$,
$k=1,2,\dots,m$;
\item[(ii)] $y'(t)=f(t,y(t))+g(t)$, $t\in J'$;
\item[(iii)] $\Delta y(t_{k})=I_{k}(y(t^{-}_k))+g_k$, $k=1,2,\dots,m$.
\end{itemize}
\end{remark}
One can have similar remarks for  inequalities \eqref{generalized
U-H-R stable}  and \eqref{U-H stable}.


\begin{remark} \rm
If $y\in PC^{1}(J,\mathbb{R})$ is a solution of the inequality
\eqref{U-H-R stable} then $y$ is a solution of the
integral inequality
\begin{equation}\label{est1}
\begin{aligned}
&\Big|y(t)-e^{\lambda t}y(0)-\sum_{i=1}^{k}e^{\lambda(t-t_{i})}I_i(y(t_i^-))
-\int_0^{t}e^{\lambda(t-s)}f(s,y(s))ds\Big| \\
&\leq e^{\lambda t}m\epsilon
\psi+\epsilon\int_0^{t}e^{\lambda(t-s)}\varphi(s)ds,\quad t\in J.
\end{aligned}
\end{equation}
\end{remark}

In fact, by Remark \ref{remark1} we have
\begin{equation}\label{est1-eq}
\begin{gathered}
y'(t)=f(t,y(t))+g(t),\quad t\in J',\\
\Delta y(t_{k})=I_{k}(y(t^{-}_k))+g_k,\quad k=1,2,\dots,m.
\end{gathered}
\end{equation}
Clearly, the solution of  \eqref{est1-eq} is given by
\begin{align*}
y(t)&= e^{\lambda
t}y(0)+\sum_{i=1}^{k}e^{\lambda(t-t_{i})}I_i(y(t_i^-))
+\sum_{i=1}^{k}e^{\lambda(t-t_{i})}g_i\\
&\quad +\int_0^{t}e^{\lambda(t-s)}f(s,y(s))ds
+\int_0^{t}e^{\lambda(t-s)}g(s)ds,~t\in (t_k,t_{k+1}].
\end{align*}
From this it follows that
\begin{align*}
&\Big|y(t)-e^{\lambda
t}y(0)-\sum_{i=1}^{k}e^{\lambda(t-t_{i})}I_i(y(t_i^-))
-\int_0^{t}e^{\lambda(t-s)}f(s,y(s))ds\Big|\\
&\leq \sum_{i=1}^{m}e^{\lambda(t-t_{i})}|g_i|
 +\int_0^{t}e^{\lambda(t-s)}|g(s)|ds\\
&\leq e^{\lambda t}m\epsilon \psi
 +\epsilon\int_0^{t}e^{\lambda(t-s)}\varphi(s)ds.
\end{align*}
Clearly, one can give similar remarks for the solutions of the
inequalities \eqref{generalized U-H-R stable}  and \eqref{U-H
stable}.


\section{Ulam-Hyers-Rassias stability results}

We use the following assumptions:
\begin{itemize}
\item[(H1)]  $f\in C(J\times\mathbb{R},\mathbb{R})$.

\item[(H2)] There exists  $L_{f}(\cdot)\in C(J,\mathbb{R}_+)$ such
that
\[
 |f(t,u_{1})-f(t,u_{2})|\leq L_{f}(t)|u_{1}-u_{2}|,\quad
\text{for each $t\in J$ and all }u_{1},u_{2} \in \mathbb{R}.
\]

\item[(H3)] There exists $\rho_{k}>0$ such that
$ |I_{k}(u_1)-I_{k}(u_2)|\leq \rho_{k}|u_1-u_2|$ for each
$u_1,u_2\in \mathbb{R}$ and $k=1,2,\dots,m$.

\item[(H4)]  Let $\varphi\in C(J,\mathbb{R_{+}})$ be a nondecreasing
function. There exists $c_{\varphi}>0$ such that
\[
\int_{0}^{t}\varphi(s)ds\leq c_{\varphi}\varphi(t),\quad\text{for
each }t\in J.
\]
\end{itemize}
Now, we are ready to state the following Ulam-Hyers-Rassias stable
result.

\begin{theorem}\label{theorem-U-H-R}
Assume that {\rm (H1)--(H4)} are satisfied. Then
\eqref{sy.1-im-no} is Ulam-Hyers-Rassias stable with respect to
$(\varphi,\psi)$.
\end{theorem}

\begin{proof} Let $y\in PC^1(J,\mathbb{R})$ be a solution of the
inequality \eqref{U-H-R stable}. Denote by $x$ the unique solution
of the impulsive Cauchy problem
\begin{equation}\label{sy.1-ref}
\begin{gathered}
x'(t)=\lambda x(t)+f(t,x(t)),\quad t\in J',\\
\Delta x(t_k)=I_k(x(t_k^-)),\quad k=1,2,\dots,m,\\
x(0)=y(0).
\end{gathered}
\end{equation}
Then we have
\[
x(t)=\begin{cases}
 e^{\lambda t}y(0)+\int_{0}^{t}
e^{\lambda(t-s)}f(s,x(s))ds, &\text{for }t\in[0,t_1],\\
e^{\lambda t}y(0)+e^{\lambda(t-t_{1})}I_1(x(t_1^-))
+\int_{0}^{t}e^{\lambda(t-s)}f(s,x(s))ds, &\text{for }t\in(t_1,t_2],\\
\dots\\
e^{\lambda t}y(0)+\sum_{k=1}^{m}e^{\lambda(t-t_{k})}I_k(x(t_k^-))+
\int_{0}^{t}e^{\lambda(t-s)}f(s,x(s))ds, &\text{for }t\in(t_m,T].
\end{cases}
\]
By \eqref{est1}, for each $t\in(t_k,t_{k+1}]$,  we have
\begin{align*}
&\Big|y(t)-e^{\lambda t}y(0)-\sum_{i=1}^{k}
e^{\lambda(t-t_{i})}I_i(y(t_i^-))-\int_0^{t}e^{\lambda(t-s)}f(s,y(s))ds\Big|\\
&\leq e^{\lambda t}m\epsilon \psi
 +\epsilon\int_0^{t}e^{\lambda(t-s)}\varphi(s)ds\\
&\leq \varepsilon e^{\lambda T}(m+c_{\varphi})[\psi+\varphi(t)].
\end{align*}
Hence for each $t\in(t_k,t_{k+1}]$, it follows that
\begin{align*}
&|y(t)-x(t)|\\
&\leq \Big|y(t)-e^{\lambda
t}y(0)-\sum_{i=1}^{k}e^{\lambda(t-t_{i})}I_i(y(t_i^-))
-\int_0^{t}e^{\lambda(t-s)}f(s,y(s))ds\Big|\\
&\quad +\sum_{i=1}^{k}e^{\lambda(t-t_{i})}|I_i(y(t_i^-))-I_i(x(t_i^-))|
 +\int_{0}^{t}e^{\lambda(t-s)}|f(s,y(s))-f(s,x(s))|ds\\
&\leq \varepsilon e^{\lambda T}(m+c_{\varphi})[\psi+\varphi(t)]
+\int_{0}^{t}e^{\lambda (t-s)}L_f(s)|y(s)-x(s)|ds\\
&\quad +e^{\lambda T}\sum_{i=1}^{k}\rho_{i}|y(t_i^-)-x(t_i^-)|.
\end{align*}
By Lemma \ref{Gronwall-class}, we obtain
\begin{align*}
|y(t)-x(t)|
&\leq \varepsilon e^{\lambda T}(m+c_{\varphi})[\psi+\varphi(t)]
\Big(\prod_{0<t_{k}<t}(1+\rho_{k}e^{\lambda t})
 e^{\int_{0}^{t}e^{\lambda (t-s)}L_{f}(s)ds}\Big)\\
&\leq c_{f,m,\varphi}\varepsilon(\varphi(t)+\psi),\quad t\in J,
\end{align*}
where
\[
c_{f,m,\varphi}:= e^{\lambda T}
(m+c_{\varphi})\prod_{k=1}^{m}(1+\rho_{k}e^{\lambda T})e^{e^{\lambda
T}\int_{0}^{T}L_{f}(s)ds}>0.
\]
Thus,  \eqref{sy.1-im-no} is Ulam-Hyers-Rassias stable
with respect to $(\varphi,\psi)$. The proof is complete.
\end{proof}


Next, we replace (H3) by
\begin{itemize}
\item[(H3*)] There exist nondecreasing functions
$\rho_{k}\in C(\mathbb{R}_+,\mathbb{R}_+)$, with
$\rho_k(0)=0$ such that
\[
 |I_{k}(u_1)-I_{k}(u_2)|\leq \rho_{k}(|u_1-u_2|),
\]
for each $u_1,u_2\in \mathbb{R}$ and $k=1,2,\dots,m$.
\end{itemize}

Next, we present the following Ulam-Hyers stable result.

\begin{theorem}\label{theorem-g-U-H}
Assume that {\rm (H1), (H2)} and {\rm (H3*)} are satisfied. Then
\eqref{sy.1-im-no} is generalized Ulam-Hyers stable.
\end{theorem}

\begin{proof}
  From the proof in Theorem
\ref{theorem-U-H-R}, we are led to the inequality
\begin{equation}\label{ineq1}
\begin{aligned}
 |v(t)|&\leq \varepsilon e^{\lambda T}(m+T)
 + e^{\lambda T}\int_{0}^{t}L_f(s)|v(s)|ds\\
&\quad +e^{\lambda T}\sum_{i=1}^{k}\rho_{i}(|v(t_i^-)|),\quad
t\in(t_k,t_{k+1}],
\end{aligned}
\end{equation}
where $v(t):=y(t)-x(t)$.

Let $M_k:=\sup_{t\in[t_k,t_{k+1}]}\{|v(t)|\}$ for $k=0,\dots,m$.
Then the inequality \eqref{ineq1} implies
$$
|v(t)|\leq (m+T)e^{\lambda T}\epsilon+ e^{\lambda
T}\int_{0}^{t}L_f(s)\left|v(s)\right|ds +e^{\lambda
T}\sum_{i=1}^{k}\rho_{i}(M_{i-1})
$$
for $t\in(t_k,t_{k+1}]$.
Using the standard Gronwall inequality we obtain
\begin{equation}\label{ineq2}
M_{k}\leq  e^{\lambda
T}\Big((m+T)\epsilon+\sum_{i=1}^{k}\rho_{i}(M_{i-1})\Big)
e^{e^{\lambda T}\int_{0}^{T}L_f(s)ds}.
\end{equation}
Setting
\begin{gather*}
\theta_0(\epsilon)= (m+T)\epsilon e^{e^{\lambda
T}\int_{0}^{T}L_f(s)ds},\\
\theta_k(\epsilon)= \Big((m+T)\epsilon+\sum_{i=1}^{k}\rho_{i}(e^{\lambda
T}\theta_{i-1}(\epsilon))\Big)e^{e^{\lambda
T}\int_{0}^{T}L_f(s)ds},\quad k=1,\dots,m.
\end{gather*}
Obviously, the inequality \eqref{ineq2} implies
\[
M_k\le e^{\lambda T}\theta_k(\epsilon),\quad  k=0,\dots,m.
\]
Let $\theta_{f,m}(\epsilon)=\max\{e^{\lambda T}\theta_k(\epsilon):
k=0,\dots,m\}$. Hence
\[
|v(t)|\leq \theta_{f,m}(\epsilon).
\]
Clearly $\theta_{f,m}\in C(\mathbb{R}_+,\mathbb{R}_+)$ and
$\theta_{f,m}(0)=0$. Thus, the equation \eqref{sy.1-im-no} is
generalized Ulam-Hyers stable. The proof is complete.
\end{proof}

\section{Example}

Let $\lambda=1$, $\varphi(t)=t$, $\psi=1$. We consider the linear
impulsive ordinary differential equation
\begin{equation}\label{E4.1}
\begin{gathered}
x'(t)=x(t),\quad t\in [0,2]\setminus \{1\},\\
\Delta x(1)=\frac{|x(1^{-})|}{1+|x(1^{-})|},
\end{gathered}
\end{equation}
and the inequalities
\begin{equation}\label{E4.2}
\begin{gathered}
|y'(t)-y(t)|\leq\epsilon t,\quad t\in ( [0,2]\setminus\{1\},\\
\big|\Delta y(1)-\frac{|y(1^{-})|}{1+|y(1^{-})|}\big|
\leq\epsilon,\quad \epsilon>0.
\end{gathered}
\end{equation}
Let $y\in PC^{1}([0,2],\mathbb{R})$ be a solution of inequality
\eqref{E4.2}. Then there exist $g\in PC^{1}([0,2],\mathbb{R})$ and
$g_1\in \mathbb{R}$ such that:
\begin{gather}\label{E4.2-add}
|g(t)|\leq \epsilon t, \quad t\in [0,2],\; |g_1|\le\epsilon,\\
\label{E4.3}
y'(t)= y(t)+g(t),\quad t\in [0,2]\setminus\{1\}, \\
\label{E4.3-im}
\Delta y(1)= \frac{|y(1^{-})|} {1+|y(1^{-})|}+g_1.
\end{gather}
Integrating  \eqref{E4.3} from $0$ to $t$ via \eqref{E4.3-im}, we
have
\[
y(t)= e^{t}y(0)+\chi_{(1,2]}(t)e^{t-1}\Big(\frac{|y(1^{-})|}{1+|y(1^{-})|}
+g_1\Big)+\int_{0}^{t}e^{t-s}g(s)ds,
\]
for the characteristic function $\chi_{(1,2]}(t)$ on $(1,2]$.

Let us consider the solution $x$ of \eqref{E4.1} given by
\[
x(t)=
e^{t}y(0)+\chi_{(1,2]}(t)e^{t-1}\frac{|x(1^{-})|}{1+|x(1^{-})|}.
\]
We have
\begin{align*}
|y(t)-x(t)|
&= \Big|\chi_{(1,2]}(t)e^{t-1}\Big(\frac{|y(1^{-})|}{1+|y(1^{-})|}
-\frac{|x(1^{-})|}{1+|x(1^{-})|}
+g_1\Big)+\int_0^{t}e^{t-s}g(s)ds\Big|\\
&\leq e^{t}|y(1^{-})-x(1^{-})|+e^{t}|g_1|+e^{t}\int_0^{t}|g(s)|ds\\
&\leq e^{t}|y(1^{-})-x(1^{-})|
+e^{t}\epsilon+\epsilon e^{t}\int_0^{t} s ds\\
&\leq e^{t}|y(1^{-})-x(1^{-})|
+e^{t}\epsilon+\epsilon e^{t}\frac{1}{2}t^2\\
&\leq e^{t}|y(1^{-})-x(1^{-})| +e^{t}\epsilon(1+t),~t\in [0,2],
\end{align*}
which gives
\[
|y(t)-x(t)|\leq  e^{2}(1+e^2)\epsilon (t+1),\quad t\in [0,2].
\]
Thus, \eqref{E4.1} is Ulam-Hyers-Rassias stable with
respect to $(t,1)$.

\subsection*{Acknowledgments}
This work is supported by grant 11201091 from the National Natural
Science Foundation of China, and by the Key Support Subject
(Applied Mathematics) of Guizhou Normal College.

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