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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 83, pp. 1--10.\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-2014/83\hfil Existence and stability of solutions]
{Existence and stability of solutions to \\ 
nonlinear impulsive differential equations \\ in $\beta$-normed spaces}

\author[J. Wang, Y. Zhang \hfil EJDE-2014/83\hfilneg]
{Jinrong Wang, Yuruo Zhang}  % in alphabetical order

\address{Jinrong Wang \newline
Department of Mathematics,
Guizhou University, Guiyang, Guizhou 550025, China.\newline
School of Mathematics and Computer Science, Guizhou Normal
College, Guiyang, Guizhou 550018, China}
\email{sci.jrwang@gzu.edu.cn}

\address{Yuruo Zhang \newline
Department of Mathematics,
Guizhou University, Guiyang, Guizhou 550025, China}
\email{yrzhangmath@126.com}


\thanks{Submitted February 3, 2014. Published March 26, 2014.}
\subjclass[2000]{34A37, 34D10}
\keywords{Nonlinear impulsive differential equations; existence;
stability}

\begin{abstract}
 In this article, we  consider nonlinear impulsive differential
 equations in $\beta$-normed spaces. We give new concepts of
 $\beta$-Ulam's type stability. Also we present sufficient conditions
 for the existence  of solutions for impulsive Cauchy problems.
 Then we obtain generalized  $\beta$-Ulam-Hyers-Rassias stability results
 for the impulsive problems on a compact interval.
 An example illustrates our main 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}

In the past decades, many researchers  studied 
differential equations with instantaneous impulses of the type
\begin{equation}\label{sy.0}
\begin{gathered}
x'(t)=f(t,x(t)),\quad t\in J':=J\setminus
\{t_1,\dots,t_m\},\; J:=[0,T],\\
x(t^{+}_k)=x(t_k^-)+I_k(x(t^{-}_k)),\quad k=1,2,\dots,m.
\end{gathered}
\end{equation}
where $f: J\times  \mathbb{R}\to  \mathbb{R}$ and 
$I_k: \mathbb{R}\to \mathbb{R}$ and $t_k$ satisfy
$0=t_0<t_1< \dots <t_m<t_{m+1}=T$, $x(t_k^{+})=\lim_{\epsilon \to
0^{+}}x(t_k+\epsilon)$ and $x(t_k^{-})=\lim_{\epsilon\to
0^{-}} x(t_k+\epsilon)$ represent the right and left limits of
$x(t)$ at $t=t_k$  respectively. Here, $I_k$ is a sequence of
instantaneously impulse operators and have been used to describe
abrupt changes such as shocks, harvesting, and natural disasters.
For more existence, stability and periodic solutions on \eqref{sy.0}
and other impulsive models, one can read the monographs of
\cite{Bain,Benchohra,Samo}.

In pharmacotherapy,  the above instantaneous impulses can not
describe the certain dynamics of evolution processes. For example,
one considers the hemodynamic equilibrium of a person, the
introduction of the drugs in the bloodstream and the consequent
absorption for the body are gradual and continuous process. So we do
not expect to use \eqref{sy.0} to describe such process. In fact,
the above situation should be shown by a new case of impulsive
action, which starts at an arbitrary fixed point and stays active on
a finite time interval. From the viewpoint of general theories,
Hern\'andez and O'Regan \cite{Hernandez} initially offered to
study a new class of abstract semilinear impulsive differential
equations with not instantaneous impulses in a $PC$-normed Banach
space. Meanwhile, Pierri et al. \cite{Pierri} continue the work in a
$PC_{\alpha}$-normed Banach space and develop the results in
\cite{Hernandez}.


Motivated by \cite{Hernandez,Pierri,Rus2009,Rus2010,Wang-JMAA}, we
continue to study existence and uniqueness of solutions to
differential equations with not instantaneous impulses in a
$P\beta$-normed Banach space (see Section 2) of the form
\begin{equation}\label{sy.1-im-no}
\begin{gathered}
x'(t)=f(t,x(t)),\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
x(t)=g_i(t,x(t)),\quad t\in (t_i,s_{i}],\; i=1,2,\dots,m,
\end{gathered}
\end{equation}
where $t_i$, $s_i$ are pre-fixed numbers satisfying $0=s_0<t_1\leq
s_1\leq t_2<\dots<s_{m-1}\leq t_m\leq s_m\leq t_{m+1}=T$, $f:[0,T]
\times \mathbb{R}\to \mathbb{R}$ is continuous and $g_i:
[t_i,s_{i}] \times \mathbb{R}\to \mathbb{R}$ is continuous
for all $i=1,2,\dots,m$. An improved existence and uniqueness
result is obtained.

It is remarkable that Ulam type stability problems \cite{Ulam} have
attracted many famous researchers. The readers can refer to
 monographs of C\u{a}dariu \cite{Cadariu1}, Hyers
\cite{Hyers41,Hyers}, Jung \cite{Jung2,Jung2-add},  Rassias
\cite{Rassias} and other recent works
\cite{Andras-NATMA,Andras-AMC,Cimpean,Fadd1,Hegyi,Jun,Jung,Fadd3,Fadd2,Fadd4,Lungu,Park,Popa}
in standard normed spaces and \cite{Cieplinski,XuTZ} in
$\beta$-normed spaces.

We introduce some auxiliary facts and offer four new concepts of
$\beta$-Ulam's type stability for \eqref{sy.1-im-no} (see
Definitions \ref{def1}--\ref{def4}). This is our main original
contribution of this paper. It is quite useful in many applications
such as numerical analysis, optimization, biology and economics,
where finding the exact solution is quite difficult. As a result,
existence and uniqueness and a generalized $\beta$-Ulam's type
stability result on a compact interval are established. An example
is given to illustrate our main results.

\section{Preliminaries}


\begin{definition}\label{beta norm}\rm
(see Jung et al. \cite{Jung} or Balachandran \cite{Balachandran})
Suppose $E$ is a vector space over $\mathbb{K}$. A function
$\|\cdot\|_{\beta}~(0<\beta \leq 1):E\to [0,\infty)$ is
called a $\beta$-norm if and only if it satisfies
(i) $\|x\|_{\beta}=0$ if and only if $x=0$; 
(ii) $\|\lambda x\|_{\beta}=|\lambda|^{\beta}\| x\|_{\beta}$ for all 
 $\lambda \in\mathbb{K}$ and all $x\in E$; 
(iii) $\|x+y\|_{\beta}\leq \|x\|_{\beta}+\|y\|_{\beta}$. 
The pair $(E,\|\cdot\|_{\beta})$ is
called a $\beta$-normed space. A $\beta$-Banach space is a complete
$\beta$-normed space.
\end{definition}


Throughout this paper, let $J=[0,T]$, $\beta\in(0,1)$ be a fixed
constant and $C(J,\mathbb{R})$ be the Banach space of all continuous
functions from $J$ into $\mathbb{R}$ with the new norm
$\|x\|_{\beta}:= \max\{|x(t)|^\beta:t\in J\}$ for $x\in
C(J,\mathbb{R})$. For example, $\|z\|_{\frac{1}{2}}=\sqrt{e}$ for
$z=t$, $t\in [0,e]$. We need the $P\beta$-Banach space
$PC(J,\mathbb{R}):=\{x:J \to \mathbb{R}:x\in
C((t_k,t_{k+1}],\mathbb{R}),~k=0,1,\dots,m$ and there exist
 $x(t_k^{-})$ and $x(t_k^{+})$, $k=1,\dots,m$,  with 
$x(t_k^{-})=x(t_k)\}$ with the norm 
$\|x\|_{P\beta}:= \sup\{|x(t)|^\beta:t\in J\}$. For example,
 $\|z\|_{P\frac{1}{2}}=e$
for $z=t$, $t\in [0,1]$ and $z=e^t$, $t\in(1,2]$. Meanwhile, we set
$PC^{1}(J,\mathbb{R}):= \{x\in PC(J,\mathbb{R}) : x'\in
PC(J,\mathbb{R})\}$ with
$\|x\|_{P\beta^1}:=\max\{\|x\|_{\beta},\|x'\|_{\beta}\}$. Clearly,
$PC^{1}(J,\mathbb{R})$  endowed with the norm
$\|\cdot\|_{P\beta^{1}}$ is a $P\beta$-Banach space.


\begin{definition}[\cite{Hernandez}] \rm
A function $x\in PC^{1}(J,\mathbb{R})$ is called a solution of the
problem
\begin{equation}\label{sy.1-im-Cauchy}
\begin{gathered}
x'(t)=f(t,x(t)),\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
x(t)=g_i(t,x(t)),\quad t\in (t_i,s_{i}],\; i=1,2,\dots,m,\\
x(0)=x_0, \quad x_0\in \mathbb{R},
\end{gathered}
\end{equation}
 if $x$ satisfies
\begin{gather*}
x(0)=x_0;\\
x(t)=g_i(t,x(t)),\quad t\in(t_i,s_i],\; i=1,2,\dots,m;\\
x(t)=x_0+\int_0^{t}f(s,x(s))ds,\quad t\in [0,t_1];\\
x(t)=g_i(s_i,x(s_i))+\int_{s_i}^{t}f(s,x(s))ds,\quad 
t\in (s_i,t_{i+1}],\; i=1,2,\dots,m.
\end{gather*}
\end{definition}

In general, we do not expect to get a precise solution of
\eqref{sy.1-im-Cauchy}. However, we can try to get a function which
satisfies some suitable approximation inequalities.

Let $0<\beta<1$, $\epsilon>0$, $\psi\ge0$ and $\varphi\in
PC(J,\mathbb{R}_+)$. We consider the following inequalities:
\begin{equation}\label{U-H stable}
\begin{gathered}
|y'(t)-f(t,y(t))|\leq\epsilon,\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
|y(t)-g_i(t,y(t))|\leq \epsilon,\quad t\in (t_i,s_{i}],\; i=1,2,\dots,m,
\end{gathered}
\end{equation}
and
\begin{equation}\label{generalized U-H-R stable}
\begin{gathered}
|y'(t)-f(t,y(t))|\leq\varphi(t),\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
|y(t)-g_i(t,y(t))|\leq \psi,\quad t\in (t_i,s_{i}],\; i=1,2,\dots,m,
\end{gathered}
\end{equation}
and
\begin{equation}\label{U-H-R stable}
\begin{gathered}
|y'(t)-f(t,y(t))|\leq\epsilon\varphi(t),\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
|y(t)-g_i(t,y(t))|\leq\epsilon \psi,\quad t\in (t_i,s_{i}],~i=1,2,\dots,m.
\end{gathered}
\end{equation}

Next, our aim is to find a solution $y(\cdot)$ close to the measured
output $x(\cdot)$ and whose closeness is defined in the sense of
$\beta$-Ulam's type stabilities.

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

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

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

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

Obviously, 
(i) Definition \ref{def1} implies  Definition \ref{def2}; 
(ii) Definition \ref{def3} implies Definition \ref{def4}; 
(iii) Definition \ref{def3} for $\varphi (\cdot)=\psi=1$
implies Definition \ref{def1}; 
(iv) Definitions \ref{def1}-\ref{def4} become to Ulam's stability 
concepts in Wang et al.~\cite{Wang-JMAA} when $\beta=1$ and $s_i=t_i$.


\begin{remark}\label{remark1} \rm
A function $y\in PC^{1}(J,\mathbb{R})$ is a solution of 
\eqref{generalized U-H-R stable} if and only if there is 
$G\in PC(J,\mathbb{R})$ and a sequence  $G_i$, $i=1,2,\dots,m$ 
(which depend on $y$) such that
\begin{itemize}
\item[(i)] $|G(t)|\leq \varphi(t)$, $t\in J$ and $|G_i|\le \psi$, $i=1,2,\dots,m$;
\item[(ii)] $y'(t)=f(t,y(t))+G(t)$, $t\in (s_i,t_{i+1}]$, $i=0,1,2,\dots,m$;
\item[(iii)] $y(t)=g_i(t,y(t))+G_i$, $t\in (t_i,s_{i}]$, $i=1,2,\dots,m$.
\end{itemize}
\end{remark}

By Remark \ref{remark1} we get the following results.

\begin{remark} \rm
If $y\in PC^{1}(J,\mathbb{R})$ is a solution of 
\eqref{generalized U-H-R stable} then $y$ is a solution of the
integral inequality
\begin{equation}\label{est1}
\begin{gathered}
|y(t)-g_i(t,y(t))|\leq \psi,\quad t\in(t_i,s_i],\; i=1,2,\dots,m;\\
\big|y(t)-y(0)-\int_0^{t}f(s,y(s))ds\big|
\leq \int_0^{t}\varphi(s)ds,\quad t\in[0,t_1];\\
\big|y(t)-g_i(s_i,y(s_i))-\int_{s_i}^{t}f(s,y(s))ds\big| 
\leq \psi+\int_{s_i}^{t}\varphi(s)ds,\\
 t\in [s_i,t_{i+1}],\; i=1,2,\dots,m.
\end{gathered}
\end{equation}
\end{remark}

We can give similar remarks for the solutions of the
inequalities \eqref{U-H stable} and \eqref{U-H-R stable}.
To study Ulam's type stability, we need the following
integral inequality results (see \cite[Theorem 16.4]{Bainov92}).

\begin{lemma}\label{Gronwall-class} 
(i) Let the following
inequality holds
\[
 u(t)\leq a(t)+\int_0^{t}b(s)u(s) ds,\quad t\geq 0,
\]
where $u$, $a,\in PC(\mathbb{R}_{+},\mathbb{R}_{+})$, $a$ is
nondecreasing and $b(t)> 0$.
Then, for $t\in \mathbb{R}_{+}$, 
\[
u(t)\leq  a(t)\exp\Big(\int_0^{t}b(s) ds\Big).
\]
(ii) Assume 
\[ % \label{im-in-in}
 u(t)\leq a(t)+\int_0^{t}b(s)u(s)
ds +\sum_{0<t_k<t}\beta_ku(t^{-}_k),\quad t\geq 0,
\]
where $u$, $a,b\in PC(\mathbb{R}_{+},\mathbb{R}_{+})$, $a$ is
nondecreasing and $b(t)> 0$, $\beta_k>0$, $k\in \{1,\dots,m\}$.
Then, for $t\in \mathbb{R}_{+}$, 
\[ %\label{inqq}
u(t)\leq  a(t)(1+\beta)^{k}\exp\Big(\int_0^{t}b(s)
ds\Big),\quad t\in (t_k,t_{k+1}],\; k\in\{1,\dots,m\},
\]
where $\beta=\sup_{k\in \{1,\dots,m\}}\{\beta_k\}$.
\end{lemma}


\section{Main results}

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

\item[(H2)] There exists a  positive constant $L_f$ such that
\[
 |f(t,u_1)-f(t,u_2)|\leq L_f|u_1-u_2|,
\]
for each $t\in J$ and  all $u_1,u_2 \in \mathbb{R}$.


\item[(H3)] $g_i\in C([t_i,s_{i}]\times\mathbb{R},\mathbb{R})$ and
there are positive constants $L_{g_i}$, $i=1,2,\dots,m$ such that
\[
|g_i(t,u_1)-g_i(t,u_2)|\leq L_{g_i}|u_1-u_2|,
\]
for each $t\in [t_i,s_{i}]$ and  all $u_1,u_2 \in \mathbb{R}$.

\item[(H4)]:  Let $\varphi\in C(J,\mathbb{R_{+}})$ be a nondecreasing
function. There exists $c_{\varphi}>0$ such that
\[
\int_0^{t}\varphi(s)ds\leq c_{\varphi}\varphi(t),
\]
for each $t\in J$.
\end{itemize}

Concerning the existence and uniqueness result for the solutions to
 \eqref{sy.1-im-Cauchy}, we give the following theorem.



\begin{theorem}\label{theorem-existence}
Assume that {\rm (H1)--(H3)} are satisfied. Then 
 \eqref{sy.1-im-Cauchy} has a unique solution $x$ provided
that
\begin{equation}\label{contract-}
\varrho:=\max\{L_{g_i}^\beta+L_f^\beta(t_{i+1}-s_i)^\beta,L_f^\beta
t_1^\beta: i=1,2,\dots,m\}<1.
\end{equation}
\end{theorem}

\begin{proof}
 Consider a mapping $F: PC(J,\mathbb{R})\to PC(J,\mathbb{R})$ defined by
\begin{gather*}
(Fx)(0)= x_0;\\
(Fx)(t)= g_i(t,x(t)),\quad t\in(t_i,s_i],~i=1,2,\dots,m;\\
(Fx)(t)= x_0+\int_0^{t}f(s,x(s))ds,\quad t\in [0,t_1];\\
(Fx)(t)= g_i(s_i,x(s_i))+\int_{s_i}^{t}f(s,x(s))ds,\quad t\in (s_i,t_{i+1}],\;
i=1,2,\dots,m.
\end{gather*}
Obviously, $F$ is well defined.

For any $x,y\in PC(J,\mathbb{R})$ and 
$t\in (s_i,t_{i+1}]$, $i=1,2,\dots,m$, we have
\begin{align*}
|(Fx)(t)-(Fy)(t)|
&\leq L_{g_i}|x(s_i)-y(s_i)|+L_f\int_{s_i}^{t}|x(s)-y(s)|ds\\
&\leq  L_{g_i}\|x-y\|_{C}+L_f\int_{s_i}^{t}\max_{t\in
[s_i,t_{i+1}]}|x(s)-y(s)|ds\\
&\leq  L_{g_i}\|x-y\|_{C}+L_f(t_{i+1}-s_i)\|x-y\|_{PC},
\end{align*}
which implies 
\[
|(Fx)(t)-(Fy)(t)|^\beta 
\leq L^\beta_{g_i}\|x-y\|_{P\beta}+L^\beta_f(t_{i+1}-s_i)^\beta\|x-y\|_{P\beta}.
\]
This reduces to
\[
\|Fx-Fy\|_{P\beta}
\leq \big(L^\beta_{g_i}+L^\beta_f(t_{i+1}-s_i)^\beta\big)\|x-y\|_{P\beta},
\quad t\in (s_i,t_{i+1}].
\]
Proceeding as above, we obtain that
\begin{gather*}
\|Fx-Fy\|_{P\beta}
\leq L^\beta_ft_1^{\beta}\|x-y\|_{P\beta},\quad t\in [0,t_1],\\
\|Fx-Fy\|_{P\beta}\leq L^\beta_{g_i}\|x-y\|_{P\beta},\quad
t\in(t_i,s_i],\; i=1,2,\dots,m.
\end{gather*}
From the above facts, we have
\[
\|Fx-Fy\|_{P\beta}\leq \varrho\|x-y\|_{P\beta},
\]
where $\varrho$ is defined in \eqref{contract-}. Finally, we can
deduce that $F$ is a contraction mapping. Then, one can derive the
result immediately.
\end{proof}

Next, we discuss hte stability of 
\eqref{sy.1-im-no} by using the concept of generalized
$\beta$-Ulam-Hyers-Rassias in the above section.



\begin{theorem}\label{theorem-U-H-R}
Assume that {\rm (H1)-(H4)} and \eqref{contract-} are satisfied. 
Then \eqref{sy.1-im-no} is generalized $\beta$-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 
\eqref{generalized 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)=f(t,x(t)),\quad t\in (s_i,t_{i+1}],\; i=0,1,2,\dots,m,\\
x(t)=g_i(t,x(t)),\quad t\in (t_i,s_{i}],\; i=1,2,\dots,m,\\
x(0)=y(0).
\end{gathered}
\end{equation}
Then we obtain
\[
x(t)=\begin{cases}
g_i(t,x(t)), & t\in(t_i,s_i],\; i=1,2,\dots,m;\\
y(0)+\int_0^{t}f(s,x(s))ds, &t\in [0,t_1];\\
g_i(s_i,x(s_i))+\int_{s_i}^{t}f(s,x(s))ds, & t\in (s_i,t_{i+1}],\; i=1,2,\dots,m.
\end{cases}
\]
Keeping in mind \eqref{est1}, for each $t\in (s_i,t_{i+1}]$,
$i=1,2,\dots,m$, we have
\[
\big|y(t)-g_i(s_i,y(s_i))-\int_{s_i}^{t}f(s,y(s))ds\big| 
\leq \psi+\int_{s_i}^{t}\varphi(s)ds 
\leq \psi+ c_{\varphi}\varphi(t),
\]
and for $t\in(t_i,s_i]$, $i=1,2,\dots,m$, we have
\[
|y(t)-g_i(t,y(t))|\leq  \psi,
\]
and for $t\in [0,t_1]$, we have
\[
\big|y(t)-y(0)-\int_0^{t}f(s,y(s))ds\big| \leq c_{\varphi}\varphi(t).
\]
Hence,  for each $t\in (s_i,t_{i+1}]$, $i=1,2,\dots,m$, we have
\begin{align*}
&|y(t)-x(t)|\\
&= \big|y(t)-g_i(s_i,x(s_i))-\int_{s_i}^{t}f(s,x(s))ds\big|\\
&\leq \big|y(t)-g_i(s_i,y(s_i))-\int_{s_i}^{t}f(s,y(s))ds\Big|\\
&\quad +\big|g_i(s_i,y(s_i))-g_i(s_i,x(s_i))\Big|
 +\Big(\int_{s_i}^{t}|f(s,y(s))-f(s,x(s))|ds\Big)\\
&\leq (1+c_{\varphi})[\psi+\varphi(t)]+L_{g_i}|y(s_i)-x(s_i)|
 +\int_{s_i}^{t}L_f|y(s)-x(s)|ds\\
&\leq (1+c_{\varphi})[\psi+\varphi(t)]+\sum_{0< s_i<
t}L_{g_i}|y(s_i)-x(s_i)|+\int_0^{t}L_f|y(s)-x(s)|ds.
\end{align*}

Clearly, $ a(t):=(1+c_{\varphi})[\psi+\varphi(t)]$,
$t\in (s_i,t_{i+1}], $ is nondecreasing and
$a \in PC(\mathbb{R}_{+},\mathbb{R}_{+})$. By Lemma \ref{Gronwall-class}
(ii), we obtain
\begin{align*}
|y(t)-x(t)| 
&\leq (1+c_{\varphi})[\psi+\varphi(t)](1+L_g)^i\exp\Big(\int_0^{t}L_f
ds\Big)\\
&\leq (1+c_{\varphi})[\psi+\varphi(t)](1+L_g)^i\exp\big(L_ft_{i+1}\big)
\end{align*}
where $L_g=\max\{L_{g_1},L_{g_2},\dots,L_{g_m}\}$.
Thus,
\begin{equation}\label{UHR-E1}
\begin{aligned}
|y(t)-x(t)|^\beta
&\leq \big[(1+c_{\varphi})[\psi+\varphi(t)](1+L_g)^i\exp\big(L_ft_{i+1}\big)
 \big]^\beta\\
&\leq \big[(1+c_{\varphi})(1+L_g)^i\exp\big(L_ft_{i+1}\big)\big]^\beta
[\psi+\varphi(t)]^\beta\\
&\leq \big[(1+c_{\varphi})(1+L_g)^i\exp\big(L_ft_{i+1}\big)\big]^\beta
(\psi^\beta+\varphi(t)^\beta),
\end{aligned}
\end{equation}
for $t\in (s_i,t_{i+1}]$, $i=1,2,\dots,m$.


Further, for $t\in(t_i,s_i]$, $i=1,2,\dots,m$, we have
\begin{align*}
|y(t)-x(t)|^\beta
&\leq |y(t)-g_i(t,x(t))|^\beta\\
&\leq |y(t)-g_i(t,y(t))|^\beta+
|g_i(t,y(t))-g_i(t,x(t))|^\beta \\
&\leq \psi^\beta +L_{g_i}^\beta |y(t)-x(t)|^\beta,
\end{align*}
which yields 
\begin{equation}\label{UHR-E2}
 |y(t)-x(t)|^\beta \leq \frac{1}{1-L_{g_i}^\beta}  \psi^\beta.\quad 
\text{(\eqref{contract-} implies $L_{g_i}^\beta<1)$}
\end{equation}
Moreover, for $t\in [0,t_1]$, we have
\begin{align*}
|y(t)-x(t)|
&= \big|y(t)-y(0)-\int_0^tf(s,x(s))ds\big|\\
&\leq \big|y(t)-y(0)-\int_0^tf(s,y(s))ds\big|
+\Big(\int_0^t|f(s,y(s))-f(s,x(s))|ds\Big)\\
&\leq c_{\varphi}\varphi(t)+ \int_0^{t}L_f|y(s)-x(s)|ds.
\end{align*}
By Lemma \ref{Gronwall-class} (i), we obtain
\begin{align*}
|y(t)-x(t)| 
&\leq c_{\varphi}\varphi(t)\exp\Big(\int_0^{t}L_f ds\Big)\\
&\leq c_{\varphi}\varphi(t)\exp\big(L_ft_1\big).
\end{align*}
Thus, we obtain
\begin{equation}\label{UHR-E3}
\begin{aligned}
|y(t)-x(t)|^\beta 
&\leq \big[c_{\varphi}\varphi(t)\exp\big(L_ft_1\big)\big]^\beta\\
&\leq \big[c_{\varphi}\exp\big(L_ft_1\big)\big]^\beta\varphi(t)^\beta, \quad
t\in [0,t_1].
\end{aligned}
\end{equation}
Summarizing, we combine \eqref{UHR-E1}, \eqref{UHR-E2} and
\eqref{UHR-E3} and derive that
\begin{align*}
|y(t)-x(t)|^\beta 
&\leq \Big(\big[(1+c_{\varphi})(1+L_g)^i\exp\big(L_ft_{i+1}\big)\big]^\beta\\
&\quad +\frac{1}{1-L_{g_i}^\beta}
 +\big[c_{\varphi}\exp\big(L_ft_1\big)\big]^\beta\Big)(\psi^\beta+\varphi^\beta(t))\\
&:= c_{f,\beta,g_i,\varphi}(\psi^\beta+\varphi^\beta(t)),\quad t\in J,
\end{align*}
which implies that  \eqref{sy.1-im-no} is generalized
$\beta$-Ulam-Hyers-Rassias stable with respect to $(\varphi,\psi)$.
The proof is complete.
\end{proof}


\section{An example}

Consider the  nonlinear differential equation, without
instantaneous impulses,
\begin{equation}\label{E4.1}
\begin{gathered}
x'(t)=\frac{1}{1+15e^{t}}\arctan (t^2+x(t)), \quad t\in (0,1],\\
x(t)=\frac{1}{15+t^2}\ln( x(t)+1),\quad t\in (1,2],
\end{gathered}
\end{equation}
and inequalities
\begin{equation}\label{E4.2}
\begin{gathered}
\big|y'(t)-\frac{1}{1+15e^{t}}\arctan (t^2+y(t))\big|
\leq e^t, \quad t\in (0,1],\\
\big|y(t)-\frac{1}{15+t^2}\ln(y(t)+1)\big|\leq 1,\quad
 t\in (1,2].
\end{gathered}
\end{equation}

Set $J=[0,2]$, $0=s_0<t_1=1<s_1=2$ and $\beta=\frac{1}{2}$. Denote
$f(t,x(t))=\frac{1}{(1+15e^{t})}\arctan (t^2+x(t))$ with
$L_f=1/16$ for $t\in (0,1]$ and
$g_1(t,x(t))=\frac{1}{15+t^2}\ln( x(t)+1)$ with
$L_{g_1}=1/16$ for $t\in (1,2]$. We put $\varphi(t)= e^{t}$
and $\psi=1$.

Let $y\in PC^{1}([0,2],\mathbb{R})$ be a solution of the inequality
\eqref{E4.2}. Then there exist 
$G(\cdot)\in PC^{1}([0,2],\mathbb{R})$ and $G_1\in \mathbb{R}$ such that 
$|G(t)|\leq e^t$, $t\in (0,1]$, $|G_1|\leq 1$, $t\in (1,2]$, and
\begin{equation}\label{E4.3}
\begin{gathered}
y'(t)=\frac{1}{1+15e^{t}}\arctan (t^2+y(t))+G(t),\quad t\in (0,1],\\
y(t)=\frac{1}{15+t^2}\ln(y(t)+1)+G_1,\quad t\in (1,2].
\end{gathered}
\end{equation}
For $t\in [0,1]$, integrating  \eqref{E4.3} from $0$ to $t$, we have
\[
y(t)= y(0)+\int_0^{t}\big(\frac{1}{1+15e^{s}}\arctan
(s^2+y(s))+G(s)\big)ds.
\]
For $t\in (1,2]$, we have
\[
y(t)= \frac{1}{15+t^2}\ln(y(t)+1)+G_1.
\]
For
\begin{equation}\label{E4.1-cauchy}
\begin{gathered}
x'(t)=\frac{1}{1+15e^{t}}\arctan (t^2+x(t)),~t\in (0,1],\\
x(t)=\frac{1}{15+t^2}\ln( x(t)+1),~t\in (1,2],\\
x(0)=y(0),
\end{gathered}
\end{equation}
all the conditions in Theorem \ref{theorem-existence} are satisified.
Thus, \eqref{E4.1-cauchy} has a unique solution.

Let us take the solution $x$ of  \eqref{E4.1-cauchy}
given by
\begin{gather*}
x(t)= y(0)+\int_0^{t}\frac{1}{1+15e^{s}}\arctan
(s^2+x(s)) ds,\quad t\in (0,1],\\
x(t)=\frac{1}{15+t^2}\ln(x(t)+1),\quad t\in (1,2].
\end{gather*}
For $t\in (0,1]$, we have
\begin{align*}
|y(t)-x(t)|^{1/2}
 &\leq [c_{\varphi}\exp\big(L_ft_1\big)]^\beta\varphi(t)^\beta\\
&\leq [c_{\varphi}\exp\big(L_ft_1\big)]^{1/2}e^{t/2}\\
&= e^{1/32} e^{t/2}.
\end{align*}
For $t\in (1,2]$, we have
\[
|y(t)-x(t)|^{1/2} \leq \frac{1}{4}|y(t)-x(t)|^{1/2}+1,
\]
which yields 
\[
|y(t)-x(t)|^{1/2}  \leq \frac{4}{3}.
\]
Summarizing, we have
\[
|y(t)-x(t)|^{1/2} \leq \frac{4}{3}(1+e^{t/2}),\quad t\in J.
\]
So the equation \eqref{E4.1} is generalized
$\frac{1}{2}$-Ulam-Hyers-Rassias stable with respect to
$(e^{t/2},1)$.


\subsection*{Acknowledgments}
This work is supported by National Natural Science Foundation of China
(11201091) and Doctoral Project of Guizhou Normal College (13BS010).
The authors want to thank the anonymous referees for their careful reading of the
manuscript and the insightful comments, which help us improve this article.
Also we would like to acknowledge the valuable
comments and suggestions from Professor Julio G. Dix.


\begin{thebibliography}{00}

\bibitem{Andras-NATMA}  Sz. Andr\'as, J. J. Kolumb\'an;
\emph{On the Ulam-Hyers stability of
first order differential systems with nonlocal initial conditions},
Nonlinear Anal. TMA, 82(2013), 1-11.

\bibitem{Andras-AMC} Sz. Andr\'as, A. R. M\'esz\'aros;
\emph{Ulam-Hyers stability of dynamic equations on time scales via
 Picard operators}, Appl. Math. Comput., 219(2013), 4853-4864.

\bibitem{Bain} D. D. Bainov, V. Lakshmikantham, P. S. Simeonov;
\emph{Theory of impulsive differential equations}, vol. 6 of Series in
Modern Applied Mathematics, World Scientific, Singapore, (1989).

\bibitem{Balachandran} V. K. Balachandran;
\emph{Topological algebras}, Narosa Publishing
House, New Delhi, 1999.

\bibitem{Bainov92} D. D. Bainov, P. S. Simeonov;
\emph{Integral inequalities and applications}, 
Kluwer Academic Publishers, Dordrecht, 1992.

\bibitem{Benchohra} M. Benchohra, J. Henderson,  S. Ntouyas;
\emph{Impulsive differential equations and inclusions}, 
vol. 2 of Contemporary Mathematics and Its
Applications, Hindawi, New York, NY, USA, 2006.

\bibitem{Cadariu1} L. C\u{a}dariu;
\emph{Stabilitatea Ulam-Hyers-Bourgin pentru ecuatii functionale}, 
Ed. Univ. Vest Timi\c{s}oara, Timi\c{s}ara, 2007.

\bibitem{Cieplinski} K. Ciepli\'{n}ski;
\emph{Stability of multi-additive mappings in
$\beta$-Banach spaces}, Nonlinear Anal.:TMA, 75(2012), 4205-4212.

\bibitem{Cimpean} D. S. Cimpean, D. Popa;
\emph{Hyers-Ulam stability of Euler's equation},
 Appl. Math. Lett., 24(2011), 1539-1543.

\bibitem{Fadd1}  P. Gavruta, S. M. Jung, Y. Li;
\emph{Hyers-Ulam stability for second-order linear
differential equations with boundary conditions}, Electron. J.
Differ. Eq., 2011(2011), No.80, 1-5.

\bibitem{Hegyi} B. Hegyi, S. M. Jung;
\emph{On the stability of Laplace's equation}, Appl.
Math. Lett., 26(2013), 549-552.

\bibitem{Hernandez} E. Hern\'andez, D. O'Regan;
\emph{On a new class of abstract impulsive
differential equations}, Proc. Amer. Math. Soc., 141(2013),
1641-1649.

\bibitem{Hyers41} D. H. Hyers;
\emph{On the stability of the linear functional equation},
Proc. Nat. Acad. Sci., 27(1941), 222-224.

\bibitem{Hyers}  D. H. Hyers, G. Isac, Th. M. Rassias;
\emph{Stability of functional equations in several variables}, 
Birkh\"{a}user, 1998.

\bibitem{Jun} K. W. Jun, Y. H. Lee;
\emph{A generalization of the Hyers-Ulam-Rassias
stability of the Pexiderized quadratic equation}, J. Math. Anal.
Appl.,  297(2004), 70-86.

\bibitem{Jung}  S. M. Jung, T. S. Kim, K. S. Lee;
\emph{A fixed point approach to the stability of quadratic functional equation},
 Bull. Korean Math. Soc., 43(2006), 531-541.

\bibitem{Jung2} S. M. Jung;
\emph{Hyers-Ulam-Rassias stability of functional equations in
mathematical analysis}, Hadronic Press, Palm Harbor, 2001.

\bibitem{Jung2-add} S. M. Jung;
\emph{Hyers-Ulam-Rassias stability of functional equations in
nonlinear analysis}, Springer, New York, 2011.

\bibitem{Fadd3} S. M. Jung;
\emph{Hyers-Ulam stability for Gegenbauer differential
equations}, Electron. J. Differ. Eq., 2013(2013), No.156, 1-8.

\bibitem{Fadd2} X. Li, J. Wang;
\emph{Ulam-Hyers-Rassias stability of semilinear differential equations 
with impulses}, Electron. J. Differ. Eq., 2013(2013), No.172, 1-8.

\bibitem{Fadd4} Y. Li, J. Huang;
\emph{Hyers-Ulam stability of linear second-order differential equations 
in complex Banach spaces}, Electron. J. Differ. Eq., 2013(2013), No.184, 1-7.

\bibitem{Lungu} N. Lungu, D. Popa;
\emph{Hyers-Ulam stability of a first order partial differential equation}, 
J. Math. Anal. Appl., 385(2012), 86-91.

\bibitem{Park} C. Park;
\emph{Hyers-Ulam-Rassias stability of homomorphisms in
quasi-Banach algebras}, Bull. Sci. Math., 132(2008), 87-96.

\bibitem{Pierri} M. Pierri, D. O'Regan, V. Rolnik;
\emph{Existence of solutions for semi-linear abstract differential equations 
with not instantaneous}, Appl. Math. Comput., 219(2013), 6743-6749.

\bibitem{Popa} D. Popa, I. Ra\c{s}a;
\emph{On the Hyers-Ulam stability of the linear differential equation},
J. Math. Anal. Appl., 381(2011), 530-537.

\bibitem{Rassias} Th. M. Rassias;
\emph{On the stability of linear mappings in Banach spaces}, 
Proc. Amer. Math. Soc., 72(1978), 297-300.

\bibitem{Rus2009}  I. A. Rus;
\emph{Ulam stability of ordinary differential equations}, 
Studia Univ. ``Babe\c{s} Bolyai"
Mathematica, 54(2009), 125-133.

\bibitem{Rus2010}  I. A. Rus;
\emph{Ulam stabilities of ordinary differential equations in a
Banach space}, Carpathian J. Math., 26(2010), 103-107.

\bibitem{Samo} A. M. Samoilenko, N. A. Perestyuk;
\emph{Impulsive differential equations}, vol. 14 of World Scientific 
Series on Nonlinear Science. Series A: Monographs and Treatises, 
World Scientific, Singapore, 1995.

\bibitem{Ulam} S. M. Ulam;
\emph{A collection of mathematical problems}, Interscience
Publishers, New York, 1968.

\bibitem{Wang-JMAA} J. Wang, M. Fe\v{c}kan, Y. Zhou;
\emph{Ulam's type stability of impulsive ordinary differential
equations}, J. Math. Anal. Appl., 395(2012), 258-264.

\bibitem{XuTZ}  T. Z. Xu;
\emph{On the stability of multi-Jensen mappings in
$\beta$-normed spaces},  Appl. Math. Lett., 25(2012), 1866-1870.

\end{thebibliography}

\end{document}