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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 56, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/56\hfil Existence and topological structure]
{Existence and topological structure of solution sets for $\phi$-Laplacian
 impulsive differential equations}

\author[J. Henderson,  A. Ouahab, S. Youcefi \hfil EJDE-2012/56\hfilneg]
{Johnny Henderson,  Abdelghani Ouahab, Samia Youcefi}  % in alphabetical order

\address{Johnny Henderson \newline
 Department of Mathematics,
Baylor University, Waco, TX 76798-7328, USA}
\email{Johnny\_Henderson@baylor.edu}

\address{Abdelghani Ouahab \newline
Laboratory of Mathematics, Sidi-Bel-Abb\`es University\\
PoBox 89, 22000 Sidi-Bel-Abb\`es, Algeria}
\email{agh\_ouahab@yahoo.fr}

\address{Samia Youcefi \newline
Laboratory of Mathematics, Sidi-Bel-Abb\`es University\\
PoBox 89, 22000 Sidi-Bel-Abb\`es, Algeria}
\email{youcefi.samia@yahoo.com}


\thanks{Submitted January 20, 2012. Published April 6, 2012.}
\subjclass[2000]{34A37, 34K45}
\keywords{ $\phi$-Laplacian; fixed point theorems; impulsive solution;
\hfill\break\indent compactness}

\begin{abstract}
 In this article, we present  results on the existence and the topological 
 structure  of the solution set for initial-value problems for the first-order
 impulsive differential equation
 \begin{gather*}
 (\phi(y'))' = f(t,y(t)),  \quad\text{a.e. }  t\in [0,b],\\
 y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})), \quad k=1,\dots,m,\\
 y'(t^+_{k})-y'(t^-_k)=\bar I_{k}(y'(t_{k}^{-})), \quad k=1,\dots,m,\\
 y(0)=A,\quad  y'(0)=B,
 \end{gather*}
 where  $0=t_0<t_1<\dots<t_m<t_{m+1}=b$,  $m\in\mathbb{N}$.
 The functions $I_k, \bar I_k$ characterize the jump in the
 solutions at impulse points $t_k$, $k=1,\dots,m$.
 For the final result of the paper, the hypotheses are modified so
 that the nonlinearity $f$ depends on $y'$, but the impulsive conditions
 and initial conditions remain the same.
\end{abstract}

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

\section{Introduction}

 The dynamics of many processes in physics, population
dynamics, biology and medicine may be subject to abrupt changes such
as shocks or perturbations (see for instance \cite{AgCoMaAnDa, Kr}
and the references therein). These perturbations may be seen as
impulses. For instance, in the periodic treatment of some diseases,
impulses correspond to the administration of a drug treatment or a
missing product. In environmental sciences, impulses correspond to
seasonal changes of the water level of artificial reservoirs. Their
models may be described by impulsive differential equations. The
mathematical study of boundary value problems for differential
equations with impulses was first considered in 1960 by Milman and
Myshkis \cite{MiMy} and then followed by a period of active research
which culminated in 1968 with the monograph by Halanay and Wexler
\cite{HaWe}.

Various mathematical results (existence, asymptotic behavior, and so on)
have been obtained so far
(see \cite{AgBeOrOu,AgOr,BaSi,DjGrOu-M,LaBaSi,SaPe,PePloSamSk}
and the references therein).

In this paper we shall establish an existence theory for  initial-value problems
with impulse effects. We will treat two cases.
For the first case, the problem has the form
\begin{gather}\label{1}
(\phi(y'(t)))'=f(t,y(t)), t\in J:=[0,b],\quad t\neq t_{k},\; k=1,\dots,m, \\
\label{2}
y(t_{k}^{+})-y(t_{k}^{-})=I_{k}(y(t_{k}^{-})),\quad k=1,\dots,m, \\
\label{3}
y'(t_{k}^{+})-y'(t_{k}^{-})=\bar I_{k}(y(t_{k}^{-})),\quad k=1,\dots,m, \\
\label{4}
y(0)=A,\quad y'(0)=B,
\end{gather}
where $f:[0,b]\times \mathbb{R}\to\mathbb{R}$ is a given function,
$I_{k},\bar I_{k}\in C(\mathbb{R},\mathbb{R})$, $\phi:\mathbb{R}\to\mathbb{R}$
is a suitable monotone homeomorphism, and $A, B\in\mathbb{R}$.  For this
setting, the proofs of the two results presented, while involving some cases,
are quite straight for word.
The second case is when the second member $f$ may depend on $y'$, and the problem
has the the form
\begin{gather}\label{5}
(\phi(y'(t)))'=f(t,y(t),y'(t)), t\in J:=[0,b],\quad t\neq t_{k},\; k=1,\dots,m, \\
\label{6}
y(t_{k}^{+})-y(t_{k}^{-})=I_{k}(y(t_{k}^{-})),\quad k=1,\dots,m, \\
\label{7}
y'(t_{k}^{+})-y'(t_{k}^{-})=\bar I_{k}(y(t_{k}^{-})),\quad k=1,\dots,m, \\
\label{8}
y(0)=A,\quad y'(0)=B,
\end{gather}
where $f:[0,b]\times \mathbb{R}\times \mathbb{R}\to\mathbb{R}$ is a given function,
$I_{k},\bar I_{k}, \phi$ and $A, B$ are as in problem \eqref{1}--\eqref{4}.
 Because of the dependency on $y'$, the proof of the result
presented is somewhat more involved.  Of course, the second case also covers the
first case when $f$ is independent of $y'$.

The goals of this article are to provide some existence results and to establish
the compactness of solution sets of the above problems.

\section{Preliminaries}

In this section, we recall from the literature some notation, definitions, and
auxiliary results which will be used throughout this paper.
Let $J=[0,b]$ be an interval of $\mathbb{R}$.
$C([0,b],\mathbb{R})$ is the Banach space of all continuous functions from $[0,b]$
into $\mathbb{R}$ with the norm
$$
\|y\|_{\infty}=\sup_{t\in[0,b]}|y(t)|.
$$
$L^{1}([0,b],\mathbb{R})$ denotes the Banach space of Lebesgue integrable functions,
 with the norm
$$
\|y\|_{L^{1}}=\int_0^{b}|y(s)|ds.
$$

\begin{definition} \label{def2.1} \rm
A map $f:[p,q]\times\mathbb{R}\to \mathbb{R}$ is said to be $L^{1}$-Carath\'eodory if
\begin{itemize}
  \item [(i)] $t\to f(t,y)$ is measurable for all $y\in \mathbb{R}$,
  \item [(ii)] $y\to f(t,y)$ is continuous for almost each $t\in[p,q]$,
  \item [(iii)] for each $r>0$, there exists $h_{r}\in L^{1}([p,q],\mathbb{R}_{+})$
 such that
$|f(t,y)|\leq h_{r}(t)$ for almost each $t\in[p,q]$ and for all $|y|\leq r$.
\end{itemize}
\end{definition}

\begin{lemma}[Gr\"{o}nwall-Bihari \cite{BaSm}] \label{l1}
 Let $I=[p,q]$ and let $u,g:I\to\mathbb{R}$ be positive continuous functions.
Assume there exist $c>0$ and a continuous nondecreasing function
$h: [0,\infty)\to(0,+\infty)$ such that
$$
u(t)\leq c+g(s)h(u(s))ds,\quad \forall t\in I.
$$
Then
$$
u(t)\leq H^{-1}\Big(\int_{p}^tg(s)ds\Big),\quad \forall t\in I,
$$
provided
$$
\int_{c}^{+\infty}\frac{dy}{h(y)}>\int_{p}^{q}g(s)ds,
$$
where $H^{-1}$ refers to inverse of the function
$H(u)=\int_{c}^{u}\frac{dy}{h(y)}$ for $u\geq c$.
\end{lemma}

\section{Main results}

Let $J_0=[0,t_1],\ J_{k}=(t_{k},t_{k+1}]$, $k=1,\dots,m$, and let $y_{k}$ be the
restriction of a function $y$ to $J_k$. To define  solutions for
 $\eqref{1}-\eqref{4}$, consider the space
\begin{align*}
PC&=\big\{y: [0,b]\to \mathbb{R},\; y_{k}\in C(J_{k},\mathbb{R}),\; k=0,\dots,m,
\text{ such that}\\
&\quad \text{$y(t^{-}_{k})$ and $y(t^{+}_{k})$ exist and satisfy
$ y(t^{-}_{k})=y(t_{k})$  for $ k=1,\dots,m$}\big\}.
\end{align*}
Endowed with the norm
$$
\|y\|_{PC}=\max\{\|y_{k}\|_{\infty},\, k=0,\dots,m \},\quad
\|y_k\|_{\infty}=\sup_{t\in J_k}|y(t)|,
$$
$PC$ is a Banach space.
\begin{align*}
PC^1&=\big\{y\in PC: y'_{k}\in C(J_{k},\mathbb{R}),\; k=0,\dots,m,
\text{ such that}\\
&\quad \text{$y'(t^{-}_{k})$  and $ y'(t^{+}_{k})$ exist and satisfy
$ y'(t^{-}_{k})=y'(t_{k})$  for $ k=1,\dots,m$}\big\}.
\end{align*}
is a Banach space with the norm
$$
\|y\|_{PC^1}=\max(\|y\|_{PC},\|y'\|_{PC}),\quad \text{or}\quad
 \|y\|_{PC^1}=\|y\|_{PC}+\|y'\|_{PC}.
$$

\begin{theorem}[Nonlinear Alternative \cite{DuGr}] \label{DuGr}
 Let $X$ be a Banach space with $C\subset X$ closed and convex.
Assume $U$ is a relatively open subset of $C$ with $0\in U$ and
$G:\overline{U}\to C$ is a compact map. Then either,
\begin{itemize}
\item[(i)] $G$ has a fixed point in $\overline{U}$; or
\item[(ii)] there is a point $u\in\partial U$ and $\lambda\in(0,1)$ with $u=\lambda G(u)$.
\end{itemize}
\end{theorem}


\begin{theorem}\label{t1}
Suppose that:
\begin{itemize}
  \item [(H1)] $f:[0,b]\times \mathbb{R}\to\mathbb{R}$ is an Carath\'eodory function
 and $I_{k},\bar I_{k}\in C(\mathbb{R},\mathbb{R})$.

  \item [(H2)] There exist $p\in L^{1}(J,\mathbb{R}_{+})$ such that
      $ |f(t,u)|\leq p(t)$  for  a.e.  $t\in J$
\end{itemize}
are satisfied. Then  \eqref{1}-\eqref{4} has at least one solution and the solutions set
$$
S=\{y\in PC([0,b],\mathbb{R}) : y \text{ is a solution of  \eqref{1}-\eqref{4}}\}
$$
is compact.
\end{theorem}

\begin{proof} The proof involves several steps.

\textbf{Step 1:} Consider the problem
\begin{equation}\label{9}
\begin{gathered}
    (\phi(y'))'=f(t,y)\quad t\in[0,t_1], \\
    y(0)=A,\quad y'(0)=B,
  \end{gathered}
\end{equation}
and the map
$N_1:C([0,t_1],\mathbb{R})\to C([0,t_1],\mathbb{R})$,
$$
y\mapsto (N_1y)(t)=A+\int_0^t\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]ds.
$$
Clearly the fixed points of $N_1$ are solutions of the problem \eqref{9}.

To apply the nonlinear alternative of Leray-Schauder type, we first show that
$N_1$ is completely continuous. The proof will be given in several steps.

\textbf{Claim 1:} $N$ sends bounded sets into bounded sets in $C([0,t_1], \mathbb{R})$.
Let
 $$
y\in D=\{y\in C([0,t_1], \mathbb{R}):\|y\|_{\infty}\leq q\}.
$$
Then for each $t\in[0,t_1]$, we have
$$
|(N_1y)(t)|\leq|A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)]|d\tau,
$$
since
\begin{align*}
|\phi(B)+\int_0^{s}f(\tau,y)d\tau|
&\leq |\phi(B)|+\int_0^{s}|f(\tau,y)|d\tau\\
&\leq |\phi(B)|+\int_0^{s}|p(\tau)|d\tau\\
&\leq |\phi(B)|+(\|p\|_{L^{1}})t_1,
\end{align*}
it follows that
 $$
[\phi(B)+\int_0^{s}f(\tau,y)d\tau]\in\overline{B}(0,l_1),
$$
where
$l_1=|\phi(B)|+(\|p\|_{L^{1}})t_1$.
Since $\phi^{-1}$ is continuous,
 $$
\sup_{x\in\overline{B}(0,l_1)}|\phi^{-1}(x)|<\infty.
$$
Thus
$$
\|N_1(y)\|_{\infty}\leq |A|+t_1\sup_{x\in\overline{B}(0,l_1)}|\phi^{-1}(x)|:=r
$$

\textbf{Claim 2:} $N_1$ maps bounded sets into equicontinuous sets.
Let $l_1,l_2\in[0,t_1]$, $l_1<l_2$
and $D$ be a bounded set of $C([0,t_1], \mathbb{R})$ as
in Claim 1.
Let $y\in D$.  Then
\begin{align*}
|(N_1y)'(t)|&= |\phi^{-1}[\phi(B)+\int_0^tf(s,y)ds]-\phi^{-1}(\phi(B))|\\
&\leq |\phi^{-1}[\phi(B)+\int_0^tf(s,y)ds]|+|B|\\
&\leq \sup_{x\in\overline{B}(0,l_1)}|\phi^{-1}(x)|+|B|:=r'.
\end{align*}
By the mean value theorem, we obtain
$$
|(N_1y)(l_2)-(N_1y)(l_1)|= |(N_1y)'(\xi)(l_2-l_1)| \leq r'|l_2-l_1|.
$$
As $l_2\to l_1$ the right-hand side of the above inequality tends to zero.

\textbf{Claim 3:} $N_1$ is continuous.
Let $(y_n)_{n\in\mathbb{N}}$ be a sequence such that $y_n\to y$ in $C([0,t_1],\mathbb{R})$.
 Then there is an integer $q$ such that $\|y_n\|_{\infty}\leq q$ for
all $n\in\mathbb{N}$ and $\|y\|_{\infty}\leq q$, $y_n\in D$ and $y\in D$. We have
\begin{align*}
&|(N_1y_n)(t)-(N_1y)(t)|\\
&\leq \int_0^t|\phi^{-1}[\phi(B) +\int_0^{s}f(\tau,y_n)d\tau]
 -\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]|ds.
\end{align*}
By the dominated convergence theorem, we have
$$
|\phi(B)+\int_0^{s}f(\tau,y_n)d\tau-\phi(B)
-\int_0^{s}f(\tau,y)d\tau|\to 0\quad \text{as}\ n\to\infty,
$$
and since $\phi^{-1}$ is continuous. Then
by the dominated convergence theorem, we have
\begin{align*}
&\|N_1(y_n)-N_1(y)\|_{\infty} \\
&\leq \int_0^{t_1}|\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y_n)d\tau]
 -\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]|ds\to0,
\end{align*}
as $n\to\infty$. Thus $N_1$ is continuous.

\textbf{Claim 4:} \emph{A priori} estimate.
Now we show that there exists a constant $M_0$ such that
$\|y\|_{\infty}\leq M_0$ where $y$ is a solution if the problem \eqref{9}.
 Let $y$ a solution of \eqref{9}:
$$
y(t)=A+\int_0^t\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y(\tau)d\tau)]ds.
$$
Then
\begin{align*}
|y(t)|
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y(\tau))d\tau]|ds\\
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}p(\tau)d\tau]|ds\\
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\|p\|_1t_1]|ds\\
&\leq |A|+\sup_{x\in\overline{B}(0,l_1)}\int_0^tds\\
&\leq |A|+t_1\sup_{x\in\overline{B}(0,l_1)}=:M_0.
\end{align*}
Thus,
$\|y\|_{\infty}=\sup_{t\in[0,t_1]}|y(t)|\leq M_0$.
Set
$$
U=\{y\in C([0,t_1],\mathbb{R}):\|y\|_{\infty}<M_0+1\}.
$$
As a consequence of Claims 1--4 and  the Ascoli-Arzela theorem, we can conclude
that the map $N_1:\overline{U}\to C([0,t_1],\mathbb{R})$ is compact.
From the choice of $U$ there is no $y\in\partial U$ such that
$y=\lambda N_1y$ for any $\lambda\in(0,1)$. As a consequence of
 the nonlinear alternative of Leray-Schauder we deduce that $N_1$ has a
fixed point denoted by $y_0\in{\overline{U}}$ which is solution of the problem \eqref{9}.

\textbf{Step 2:} Consider the problem
\begin{equation}\label{10}
\begin{gathered}
    (\phi(y'))'=f(t,y)\quad t\in(t_1,t_2], \\
    y(t_1^{+})=y_0(t_1^{-})+I_1(y_0(t_1^{-})),\\
    y'(t_1^{+})=y_0'(t_1^{-})+\bar I_1(y_0(t_1^{-})).
  \end{gathered}
\end{equation}
It is clear that all solutions of  \eqref{10}
are fixed points of the multi-valued operator
$N_2: C^{*}\to C^{*}$,
defined by
$$
(N_2y)(t)=A_1+\int_{t_1}^t\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]ds,
$$
where
$$
C^{*}=\{y\in C((t_1,t_2]) :  y(t_1^{+}),y'(t_1^{+}) \text{ exist}\}
$$
and
$$
A_1=y_1(t_1)+I_1(y_1(t_1)),\quad B_1=y'_1(t_1)+ \bar I(y_1(t_1)).
$$
As in Step 1, we can prove that $N_2$ at least one fixed point which is a
solution of \eqref{10}.

\textbf{Step 3:}
 We continue this process taking into account that
$y_{m}:=y\big|_{(t_{m},b]}$ is a solution of the problem
\begin{equation}\label{11}
\begin{gathered}
    (\phi(y'))'= f(t,y)\quad t\in(t_{m},b], \\
    y(t_{m}^{+})= y_{m-1}(t_{m}^{-})+I_{m}(y_{m-1}(t_{m}^{-})),\\
    y'(t_{m}^{+})= y_{m-1}'(t_{m}^{-})+\bar I_{m}(y_{m-1}(t_{m}^{-})).
  \end{gathered}
\end{equation}
A solution $y$ of problem \eqref{1}-\eqref{4} is ultimately defined by
$$
y(t)=\begin{cases}
y_0(t), & \text{if }t\in [0,t_1],\\
y_2(t), & \text{if }t\in (t_1,t_2], \\
\dots \\
 y_{m}(t), & \text{if }t\in (t_{m},t_{m+1}].
\end{cases}
$$

\textbf{Step 3:} Now we show that the set
$$
S=\{y\in PC([0,b],\mathbb{R}): y \text{ is a solution of \eqref{1}-\eqref{4}}\}
$$
is compact.
Let $(y_n)_{n\in\mathbb{N}}$ be a sequence in
$S$.
We put $B=\{y_n: n\in\mathbb{N}\}\subseteq PC([0,b],\mathbb{R})$.
Then from earlier parts of the proof of this theorem,
we conclude that $B$ is bounded and equicontinuous. Then from the Ascoli-Arzela theorem,
we can conclude that $B$ is compact.

Recall that $J_0=[0,t_1]$ and $J_{k}=(t_{k},t_{k+1}]$, $k=1,\dots,m$.  Hence:

$\bullet$ $y_n|_{J_0}$ has a subsequence
$$
(y_{n_{m}})_{{n_m}\in\mathbb{N}}\subset S_1=\{y\in C([0,t_1],\mathbb{R}): y
\textup{ is a solution of } \eqref{9}\}
$$
such that $y_{n_{m}}$ converges to $y$.
Let
$$
z_0(t)=A+\int_0^t\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]ds,
$$
and
\begin{align*}
&|y_{n_{m}}(t)-z_0(t)|\\
&\leq  \int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y_{n_m})d\tau]
-\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]|ds.
\end{align*}
As $n_{m}\to +\infty$, $y_{n_{m}}(t)\to z_0(t)$, and then
$$
y(t)=A+\int_0^t\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]ds.
$$
$\bullet$  $y_n|_{J_1}$ has a subsequence relabeled as  $(y_{n_{m}})\subset S_2$
converging to $y$ in $C^*$ where
$$
S_2=\{y\in C^{*}:  y \text{ is a solution of } \eqref{10}\}.
$$
Let
$$
z_1(t)=A_1+\int_{t_1}^t\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]ds,
$$
\begin{align*}
&|y_{n_{m}}(t)-z_1(t)|\\
&\leq  \int_{t_1}^t|\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y_{n_m})d\tau]
 -\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]|ds.
\end{align*}
As $n_{m}\to +\infty$, $y_{n_{m}}(t)\to z_1(t)$, and then
$$
y(t)=A_1+\int_{t_1}^t\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]ds.
$$
$\bullet$ We continue this process, and we conclude that $\{y_n\ |\ n\in\mathbb{N}\}$
has subsequence converging to
$$
z_m(t)=A_{m}+\int_{t_{m}}^t\phi^{-1}[\phi(B_{m})+\int_{t_{m}}^{s}f(\tau,y)d\tau]ds,
\quad t\in(t_m,b].
$$
Hence $S$ is compact.
\end{proof}

Next we replace (H2) in Theorem \ref{t1}  by
\begin{itemize}
\item[(H3)] There exists a continuous nondecreasing function
$\psi:[0,\infty)\to[0,\infty)$ and $p\in L^{1}(J,\mathbb{R}_{+})$ such that
 $$
 |f(t,u)|\leq p(t)\psi(|u|)\quad\text{a.e. t$\in J$  and $u\in\mathbb{R}$}.
$$
\end{itemize}

\begin{theorem} \label{thm3.3}
Under assumption {\rm (H3)}, problem \eqref{1}-\eqref{4} has at least one
solution and the solution set is compact.
\end{theorem}

\begin{proof}
As in the proof of Theorem \ref{t1} we can show that  \eqref{1}-\eqref{4} has at
least one solution by an application of the nonlinear alternative of Leray-Schauder.
We show only the estimation of a solution $y$ of \eqref{1}-\eqref{4}.

$\bullet$ For $t\in[0,t_1]$, we have
$$
y(t)=A+\int_0^t\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]ds.
$$
We put $m(r)=\max\{|y(r)|: r\in[0,t_1]\}$, and
\begin{align*}
|y(t)|&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}f(\tau,y)d\tau]|ds\\
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}p(\tau)\psi(|y(\tau)|)d\tau]|ds\\
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+\int_0^{s}p(\tau)\psi(m(r))d\tau]|ds\\
&\leq |A|+\int_0^t|\phi^{-1}[\phi(B)+t_1\|p\|_{L^{1}}\psi(m(r))]|ds.
\end{align*}
Then
$$
m(t)\leq|A|+\int_0^t\psi_1(m(s))ds,\quad t\in[0,t_1],
$$
where $\psi_1=(\phi^{-1}\circ \widetilde{\psi})$ and
$\widetilde{\psi}(u)=\phi(B)+t_1(\|p\|_{L^{1}})\psi(u)$.
By the nonlinear Gr\"{o}nwall-Bihari inequality (Lemma \ref{l1}), we infer the bound
$$
m(t)\leq H^{-1}(t)\leq M_0,
$$
where $H(t)=\int_{|A|}^t\frac{d\tau}{(\phi^{-1}\circ\widetilde{\psi})(\tau)}$.

$\bullet$ For $t\in(t_1,t_2]$, we have
$$
y(t)=A_1+\int_{t_1}^t\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]ds.
$$
We put $m(r)=\max\{|y(r)|: r\in(t_1,t_2]\}$, and
\begin{align*}
|y(t)|&\leq |A_1|+\int_{t_1}^t|\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}f(\tau,y)d\tau]|ds\\
&\leq |A_1|+\int_{t_1}^t|\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}p(\tau)\psi(|y(\tau)|)d\tau]|ds\\
&\leq |A_1|+\int_{t_1}^t|\phi^{-1}[\phi(B_1)+\int_{t_1}^{s}p(\tau)\psi(m(r))d\tau]|ds\\
&\leq |A_1|+\int_{t_1}^t|\phi^{-1}[\phi(B_1)+t_2(\|p\|_{L^{1}})\psi(m(r))]|ds.
\end{align*}
Then
$$
m(t)\leq|A_1|+\int_{t_1}^t\psi_1(m(s))ds,\quad t\in[t_1,t_2],
$$
where $\psi_1=(\phi^{-1}\circ \widetilde{\psi})$ and
 $\widetilde{\psi}(u)=\phi(B_1)+t_2(\|p\|_{L^{1}})\psi(u)$.

By the nonlinear Gr\"{o}nwall-Bihari inequality (Lemma \ref{l1}), we infer the bound
$$
m(t)\leq H^{-1}(t)\leq M_1,
$$
where $H(t)=\int_{|A_1|}^t\frac{d\tau}{(\phi^{-1}\circ\widetilde{\psi})(\tau)}$.

$\bullet$ For $t\in(t_{m},b]$, we have
$$
y(t)=A_{m}+\int_{t_m}^t\phi^{-1}[\phi(B_{m})+\int_{t_m}^{s}f(\tau,y)d\tau]ds.
$$
As in the pattern, there exists
$M_{m}>0$ such that
$$
m(t)\leq H^{-1}(t)\leq M_{m},
$$
where $H(t)=\int_{|A_{m}|}^t\frac{d\tau}{(\phi^{-1}\circ\widetilde{\psi})(\tau)}$.
Hence
$$
\begin{array}{llll}
\|y\|_{PC}&\leq \max(M_0,M_1,\dots,M_{m})=M.
\end{array}
$$
The proof is complete.
\end{proof}

For the next theorem we use the assumptions:
\begin{itemize}
\item[(H4)] $f: [0,b]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is a continuous  function.
\item[(H5)] There exist a continuous nondecreasing function
$\psi:\mathbb{R}_{+}\times\mathbb{R}_{+}\to(0,\infty)$ and $p\in L^{1}(J,\mathbb{R})$ such that
    $$
    |f(t,x,y)|\leq p(t)\psi(|x|,|y|)\quad for \quad all\quad x,y\in\mathbb{R},\quad t\in J
    $$
with
$$
\int_0^{b}p(s)ds<\int_{|A|+c|B|}^{\infty}\frac{du}{(\phi^{-1}\circ\psi)(u,u)}.
$$
\end{itemize}

\begin{theorem}\label{t2}
Under assumptions {\rm (H4), (H5)},  problem \eqref{5}-\eqref{8} has at least
one solution.
\end{theorem}

Prior to the proof of Theorem \ref{t2}, we present a useful lemma.

\begin{lemma}\label{l2}
The operator $L: D\to PC(J,\mathbb{R})$ defined by $L(y)=(\phi(y'))'$ where
$
D=\{y\in PC^1(J,\mathbb{R}): y(t_{k}^{+})
=y(t_{k})+I_{k}(y(t_{k})), y'(t_{k}^{+})
=y'(t_{k})+\bar I_{k}(y(t_{k})), y(t_{k})=y(t_{k}^{-}), y'(t_{k})
=y'(t_{k}^{-}), k=1,\dots,m, y(0)=A, y'(0)=B\}.
$
Assume that $L$ is well defined. Then $L$
is bijective and $L^{-1}$ is completely continuous.
\end{lemma}

\begin{proof}
\textbf{Step 1:} $L$ is bijective.

 $\bullet$ $L$ is injective. Let $y_1, y_2\in D$ be such that $L(y_1)=L(y_2)$.  Then
$$
(\phi(y'_1(t)))'=(\phi(y'_2(t)))', t\in[0,t_1],
$$
and thus
\begin{gather*}
\phi(y'_1(t))-\phi(y'_1(0)) = \phi(y'_2(t))-\phi(y'_2(0)),\quad t\in [0,t_1],\\
\phi(y'_1(t))-\phi(B) = \phi(y'_2(t))-\phi(B),\quad t\in[0,t_1].
\end{gather*}
Hence
$y'_1(t)=y'_2(t)$ for $t\in[0,t_1]$.
By integration of this equality, we obtain
\[
\int_0^ty'_1(s)ds = \int_0^ty'_2(s)ds, \quad t\in[0,t_1]
\]
which implies $y_1(t)-y_1(0)=y_2(t)-y_2(0)$, $t\in[0,t_1]$.
This implies that $y_1(t)=y_2(t)$, $t\in[0,t_1]$.

Next,
$$
\phi(y'_1(t))-\phi(y'_1(t_1)+\bar I_1(y_1(t_1)))=\phi(y'_2(t))-\phi(y'_2(t_1)
+\bar I_1(y_2(t_1))),\quad t\in (t_1,t_2]$$
implies
$y'_1(t)=y'_2(t)$, $t\in(t_1,t_2]$,
and so
$$
\int_{t_1}^ty'_1(s)ds=\int_{t_1}^ty'_2(s)ds,\quad t\in(t_1,t_2]
$$
implies
$ y_1(t)-(y_1(t_1)+I_1(y_1(t_1)))=y_2(t)-(y_2(t_1)+I_1(y_2(t_1)))$, $t\in (t_1,t_2]$,
and then
$$
y_1(t)=y_2(t),\quad t\in(t_1,t_2].
$$
Continuing this pattern,
$$
\phi(y'_1(t))-\phi(y'_1(t_{m})+\bar I_{m}(y_1(t_{m})))=\phi(y'_2(t))-\phi(y'_2(t_{m})
+\bar I_{m}(y_2(t_{m}))),\quad t\in(t_{m},b]
$$
implies
$y'_1(t)=y'_2(t)$, $t\in(t_{m},b]$,
and so
$$
\int_{t_{m}}^ty'_1(s)ds=\int_{t_{m}}^ty'_2(s)ds,\quad
t\in(t_{m},b]
$$
implies $y_1(t)-(y_1(t_{m})+I_{m}(y_1(t_{m})))=y_2(t)-(y_2(t_{m})+I_{m}(y_2(t_{m})))$,
$t\in (t_{m},b]$,
and hence
$y_1(t)=y_2(t)$, $t\in(t_{m},b]$.
This implies that $y_1=y_2$.

$\bullet$ $L$ is surjective. Let $h\in PC(J,\mathbb{R})$, then we define
\begin{equation} \label{12}
y(t)= \begin{cases}
    L_0(h)(t),&\text{if } t\in[0,t_1] ,\\
    L_1(h)(t),&\text{if } t\in(t_1,t_2] ,\\
    \dots\\
    L_{m-1}(h)(t),&\text{if } t\in(t_{m},b],
  \end{cases}
\end{equation}
where
\begin{gather*}
L_0(h)(t)= A+\int_0^t\phi^{-1}\Big[\phi(B)+\int_0^{s}h(\tau)d\tau\Big]ds,\quad t\in[0,t_1],
\\
\begin{aligned}
L_1(h)(t)&= L_0(h)(t_1)+I_1(L_0(h)(t_1))\\
&\quad+\int_{t_1}^t\phi^{-1}\Big[\phi(L'_0(h)(t_1)+\bar I_1(L_0(h)(t_1)))
+\int_{t_1}^{s}h(\tau)d\tau\Big]ds,\quad t\in(t_1,t_2],
\end{aligned}\\
\begin{aligned}
L_2(h)(t)&= L_1(h)(t_2)+I_2(L_1(h)(t_2))\\
&\quad+\int_{t_2}^t\phi^{-1}\Big[\phi(L'_1(h)(t_2)+\bar I_2(L_1(h)(t_2)))
 +\int_{t_2}^{s}h(\tau)d\tau\Big]ds,\quad t\in(t_2,t_{3}],
\end{aligned}\\
\dots\\
\begin{aligned}
L_{m}(h)(t)&= L_{m-1}(h)(t_{m})+I_{m}(L_{m-1}(h)(t_{m}))
 +\int_{t_{m}}^t\phi^{-1}\Big[\phi(L'_{m-1}(h)(t_{m})\\
&\quad +\bar I_{m}(L_{m-1}(h)(t_{m})))
 +\int_{t_{m}}^{s}h(\tau)d\tau\Big]ds, \quad t\in(t_{m},b].
\end{aligned}
\end{gather*}
From \eqref{12} we can easily check that
\begin{align*}
y(t)&= A+\sum_{0<t_{k}<t}I_{k}(L_{k-1}(h)(t_{k}))\\
&\quad +\int_0^t\phi^{-1}\Big[\phi(B+\sum_{0<t_{k}<t}
 \bar I_{k}(L_{k-1}(h)(t_{k})))+\int_0^{s}h(\tau)d\tau\Big]ds, \quad t\in J.
\end{align*}
Hence
$$
y'(t)=\phi^{-1}\Big[\phi(B+\sum_{0<t_2<t}\bar I_{k}(L_{k-1}(h)(t_{k})))
+\int_0^{s}h(s)ds\Big].
$$
From the definition of $y$ and $y'$ we can prove that
$y(0)=A$, $y'(0)=B$, $y(t_{k}^{+})=y(t_{k})+I_{k}(y(t_{k}))$,
$y'(t_{k}^{+})=y'(t_{k})+\bar I_{k}(y(t_{k}))$ and $y(t_{k})=y_{t_{k}^{-}}$,
$k=j,\dots,m$ and by using the fact that $I_{k}, \bar I_{k}$ are continuous
we can easily prove that $y, y'\in PC(J,\mathbb{R})$.

\textbf{Step 2:} $L^{-1}$ is completely continuous.

\textbf{Claim 1:} $L^{-1}$ is continuous. Let $h_n\in PC(J,\mathbb{R})$ be such that $h_n$
converges to $h$ in $PC(J,\mathbb{R})$ as $n\to\infty$. We show that $L^{-1}(h_n)$ converges
 to $L^{-1}(h)$. Let \{$y_n\}_{n\in\mathbb{N}}\subset D$
such that $\{L(y_n)\}_{n\in\mathbb{N}}=\{h_n\}_{n\in\mathbb{N}}$.
Then:

$\bullet$ For $t\in[0,t_1]$, we have
$$
y'_n(t)=\phi^{-1}\Big[\phi(B)+\int_0^th_n(s)ds\Big],
$$
and
$$
y_n(t)=
A+\int_0^t\phi^{-1}\Big[\phi(B)+\int_0^{s}h_n(\tau)d\tau\Big]ds
= A+\int_0^ty'_n(s)ds.
$$
Hence
$$
|y'_n(t)|\leq\Big|\phi^{-1}\Big[\phi(B)+\int_0^th_n(s)ds\Big]\Big|,
$$
since
\begin{align*}
|\phi(B)+\int_0^th_n(s)ds|
&\leq |\phi(B)|+\int_0^t|h_n(s)|ds\\
&\leq |\phi(B)|+t_1\|h_n\|_{PC}\\
&\leq |\phi(B)|+t_1M_*=K,
\end{align*}
where $\|h_n\|_{PC}\leq M_*$ for all $n\in\mathbb{N}$.
Then $[\phi(B)+\int_0^th_n(s)ds]\in \overline{B}(0,K)$. 
Since $\phi^{-1}$ is continuous
and $\overline{B}(0,K)$ is compact,
$$
\sup_{x\in\overline{B}(0,K)}|\phi^{-1}(x)|<\infty.
$$
Then
$$
\|y'_n\|_{\infty}\leq\sup_{x\in\overline{B}(0,K)}|\phi^{-1}(x)|:=M_0,
$$
and
$$
\|y_n\|_{\infty}\leq |A|+M_0t_1:=\overline{M_0}.
$$
So,  $\{y_n\}_{n\in\mathbb{N}}$ and $\{y'_n\}_{n\in\mathbb{N}}$ are bounded uniformly in $C([0,t_1],\mathbb{R})$.

We put $C=\{y_n:n\in\mathbb{N}\}\subseteq C([0,t_1],\mathbb{R})$.  We can easily show that $C$ is
bounded and equicontinuous,
and then from the Ascoli-Arzela theorem we conclude that $C$ is compact.
Then $y_n$ has a subsequence $(y_{n_{m}})$ converging to $y$.
Let
$$
z(t)=A+\int_0^t\phi^{-1}\Big[\phi(B)+\int_0^{s}h(\tau)d\tau\Big]ds
$$
so that
$$
|y_{n_{m}}(t)-z(t)|\leq\int_0^{t_1}|\phi^{-1}\Big[\phi(B)
+\int_0^{s}h_{n_{m}}(\tau)d\tau\Big] -\phi^{-1}\Big[\phi(B)
+\int_0^{s}h(\tau)d\tau\Big]|ds.
$$
Since $\phi^{-1}$ is continuous and as $n_{m}\to\infty$, $y_{n_{m}}\to z(t)$, then
$$
y(t)=A+\int_0^t\phi^{-1}\Big[\phi(B)+\int_0^{s}h(\tau)d\tau\Big]ds.
$$
By the same technique, we can prove that $\{y'_n\}$ converges to $y'(t)$
for $t\in[0,t_1]$.

$\bullet$ For $t\in(t_1,t_2]$, we have
$$
y'_n(t)=y'_n(t_1)+\phi^{-1}\Big[\phi(y'_n(t_1)+\bar I_1(y_n(t_1)))
+\int_{t_1}^th_n(s)ds\Big]
$$
and
\begin{align*}
y_n(t)&= y_n(t_1)+I_1(y_n(t_1))
 +\int_{t_1}^t\phi^{-1}\Big[\phi(y'_n(t_1)+\bar I_1(y_n(t_1)))
 +\int_{t_1}^{s}h_n(\tau)d\tau\Big]\\
&= y_n(t_1)+I_1(y_n(t_1))+\int_{t_1}^ty'_n(s)ds.
\end{align*}
Hence
\begin{align*}
|y'_n(t)|&\leq |y'_n(t_1)|+\Big|\phi^{-1}\Big[\phi(y'_n(t_1)
+\bar I_1(y_n(t_1)))+\int_{t_1}^th_n(s)ds\Big]\Big|\\
&\leq  M_0+\Big|\phi^{-1}\Big[\phi(y'_n(t_1)+\bar I_1(y_n(t_1)))
+\int_{t_1}^th_n(s)ds\Big]\Big|.
\end{align*}
Since
$$
|\phi(y'_n(t_1)+\bar I_1(y_n(t_1)))+\int_{t_1}^th_n(s)ds|
\leq|\phi(M_0+\sup_{x\in\overline{B}(0,M_0)}|\bar I_1(x)|)|+t_2K=K_*,
$$
then
$$
|\phi(y'_n(t_1)+\bar I_1(y_n(t_1)))+\int_{t_1}^th_n(s)ds|
\in\overline{B}(0,|\phi(M_0+\sup_{x\in\overline{B}(0,M_0)}|\bar I_1(x)|)|+t_2K).
$$
Since $\phi^{-1}$ is continuous and
$\overline{B}(0,|\phi(M_0+\sup_{x\in\overline{B}(0,M_0)}|\bar I_1(x)|)|+t_2K)$ 
is compact,
$$
\|y'_n\|_{\infty}\leq M_0+\sup_{x\in\overline{B}(0,|\phi(M_0
+\sup_{x\in\overline{B}(0,M_0)}|\bar I_1(x)|)|+t_2K)}|\phi^{-1}(x)|:=M_1,
$$
and
$$
\|y_n\|_{\infty}\leq\overline{M_0}+\sup_{x\in\overline{B}(0,M_0)}|I_1(z)|
+M_1t_2:=\overline{M_1}.
$$
Then $\{y_n\}_{n\in\mathbb{N}}$ and $\{y'_n\}_{n\in\mathbb{N}}$ are bounded uniformly
 in $C((t_1,t_2],\mathbb{R})$.
We put $C=\{y_n:n\in\mathbb{N}\}\subseteq C((t_1,t_2],\mathbb{R})$.  We can easily show again
that $C$ is bounded and equicontinuous, and then from the Ascoli-Arzela theorem
we conclude that $C$ is compact.
Then $y_n$ has a subsequence $(y_{n_{m}})$ converging to $y$.

Now, let
$$
z(t)=y(t_1)+I_1(y(t_1))+\int_{t_1}^t\phi^{-1}
\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))+\int_{t_1}^{s}h(\tau)d\tau\Big]ds.
$$
Then
\begin{align*}
|y_{n_{m}}(t)-z(t)|
&\leq |y_{n_{m}}(t_1)-y(t_1)|+|I_1(y_{n_m}(t_1))-I_1(y(t_1))|\\
&\quad \int_{t_1}^t|\phi^{-1}\Big[\phi(y'_{n_{m}}(t_1)
 +\bar I_1(y_{n_{m}}(t_1)))+\int_{t_1}^{s}h_{n_{m}}(\tau)d\tau\Big]\\
&\quad -\phi^{-1}\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))
 +\int_{t_1}^{s}h(\tau)d\tau\Big]|ds,
\end{align*}
Since $\phi^{-1}$ is continuous, $\{y_n\}_{n\in\mathbb{N}}$ and $\{y'_n\}_{n\in\mathbb{N}}$
converge to $y$ and $y'$, respectively, for $t\in[0,t_1]$, and as
 $n_{m}\to\infty$, $y_{n_{m}}\to z(t)$, then
$$
y(t)=y(t_1)+I_1(y(t_1))+\int_{t_1}^t\phi^{-1}
\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))+\int_{t_1}^{s}h(\tau)d\tau\Big]ds.
$$
By the same technique, we can prove that $\{y'_n\}$ converges to $y'(t)$
for $t\in(t_1,t_2]$.

$\bullet$ We continue this process until we get, for every $t\in(t_{m},b]$,
that $y_n(t)$ converges to $y(t)$ and
 $y'_n(t)$ converges to $y'(t)$. We conclude that $L(y)=h$, and this implies
that $L^{-1}$ is continuous.

\textbf{Claim 2:} $L^{-1}$ is compact. Let $\mathcal{D}$ be a bounded set
of $PC(J,\mathbb{R})$ and $\{y_n\}_{n\in\mathbb{N}}\subset L^{-1}(\mathcal{D})$.
Then there exists $\{h_n\}_{n\in\mathbb{N}}\subset\mathcal{D}$ such that $L(y_n)=h_n$,
for all $n\in\mathbb{N}$.

We show that
$|L_0^{-1}(h_n)(l_2)-L^{-1}(h_n)(l_1)|$ tends to zero as $l_2\to l_1$.
Since $L_0(y_n)(t)=h_n(t),\ t\in[0,t_1]$, it follows that
$$
y'_n(t)=\phi^{-1}\Big[\phi(B)+\int_0^th_n(s)ds\Big],\quad t\in[0,t_1].
$$
Using the fact that $h_n$ is bounded, thus there exist $M_0>0$  such that
$$
\|y_n\|_{\infty}, \|y'_n\|_{\infty}\leq M_0,\quad \text{for all } n\in\mathbb{N}.
$$

$\bullet$ Let $l_1,l_2\in [0,t_1]$, $l_1<l_2$.  Then, by the mean value theorem,
$$
|y_n(l_2)-y_n(l_1)|= |y'_n(\xi_n)(l_2-l_1)|\leq M_0|l_2-l_1|,
$$
and
$$
|\phi(y'_n)(l_2)-\phi(y'_n)(l_1)|
= \Big|\int_{l_1}^{l_2}h_n(s)ds\Big|
\leq \int_{l_1}^{l_2}|h_n(s)|ds
\leq |l_2-l_1|M_*,
$$
where
$\|h_n\|_{PC}\leq M_*$ for all $ n\in\mathbb{N}$.
As $l_2\to l_1$ the right hand side of the above inequality tends to zero.
 Then $\{y_n(\cdot)\}_{n\in\mathbb{N}}$ is equicontinuous and $\{y_n\}_{n\in\mathbb{N}}$  is bounded.
By the Ascoli-Arzela theorem there exist $y_0, z_0\in C([0,t_1],\mathbb{R})$ such that
$y_n$ and  $\phi(y_n')$ converge, respectively, to $y_0$ and $z_0$.
Since $\phi^{-1}$ is a continuous function,
$$
y'_n(t)\to \phi^{-1}(z_0)(t),\ n\to\infty.
$$
Set $$
y_*(t)=A+\int_0^t\phi^{-1}(z_0(s))ds,\ t\in[0,t_1],
$$
and from
$$
y_n(t)=A+\int_0^ty'_n(s)ds,\ t\in[0,t_1],
$$
we have
$$
\|y_n-y_*\|_{\infty}\leq \int_0^{t_1}|y_n'(s)-\phi^{-1}(z_0)(s)|ds.
$$
From the Lebesgue dominated convergence theorem, we deduce that
$\{y_n\}_{n\in\mathbb{N}}$ converges to $y_*$ in $C([0,t_1],\mathbb{R})$, and this implies that
$$
y_0(t)=A+\int_0^t\phi^{-1}(z_0(s))ds,\ t\in[0,t_1]\Rightarrow y'_0(t)
=\phi^{-1}(z_0(t)),\quad t\in[0,t_1].
$$
Then $y_n$ converges to $y_0$ in $C^1([0,t_1],\mathbb{R})$.

 For $t\in(t_1,t_2]$,
$$
y'_n(t)=\phi^{-1}\Big[\phi(y_n'(t_1)+\bar I_1(y_n(t_1)))+\int_{t_1}^th_n(s)ds\Big],
\quad t\in(t_1,t_2].
$$
Using the fact that $h_n$ is bounded, there exists $M_1>0$  such that
$$
 \|y_n\|_{\infty}, \|y'_n\|_{\infty}\leq M_1,\ \text{for all}\ n\in\mathbb{N}.
$$

$\bullet$ Let $l_1,l_2\in (t_1,t_2]$, $l_1<l_2$. Then
$$
|y_n(l_2)-y_n(l_1)|= |y'_n(\xi_n)(l_2-l_1)|\leq M_1|l_2-l_1|,
$$
and
$$
|\phi(y'_n)(l_2)-\phi(y'_n)(l_1)|\leq M_{*}|l_2-l_1|.
$$
As $l_2\to l_1$ the right hand side of the above inequality tends to zero,
then $\{y_n(\cdot)\}$ and $\{\phi(y'_n)(\cdot)\}$ are  equicontinuous.
By Ascoli-Arzela theorem there exist $y_1, z_1\in C([0,t_1],\mathbb{R})$ such that
$y_n$ and $\phi(y_n')$ converge, respectively, to $y_1, z_1$.
Since $\phi^{-1}$ and $I_1$ are a continuous functions, then
$y'_n(t)\to \phi^{-1}(z_1)(t)$ as $n\to\infty$.
Set
$$
y_{**}(t)=y_0(t_1)+I_1(y_0(t_1))+\int_{t_1}^t\phi^{-1}(z_1(s))ds,\quad t\in(t_1,t_2].
$$
From
$$
y_n(t)=y_0(t_1)+I_1(y_0(t_1))+\int_{t_1}^ty'_n(s)ds,\ t\in(t_1,t_2],
$$
we have
$$
\|y_n-y_{**}\|_{\infty}\leq \int_0^{t_1}|y_n'(s)-\phi^{-1}(z_1)(s)|ds.
$$
From the Lebesgue dominated convergence theorem, we deduce that
$\{y_n\}_{n\in\mathbb{N}}$ converges to $y_{**}$ in
$C_1=\{y\in C^1((t_1,t_2],\mathbb{R}):y(t_1^+), y'(t_1^+) \text{ exist}\}$.  This implies that
$$
y_1(t)=y_0(t_1)+I_1(y_0(t_1))+\int_{t_1}^t\phi^{-1}(z_0(s))ds,\quad t\in[0,t_1]
$$
implies $y'_1(t)=\phi^{-1}(z_1(t))$, \quad $t\in(t_1,t_2]$.
Then $y_n$ converges to $y_1$ in $C_1$.

$\bullet$ We continue this process until we have that there exists
$y_m\in C^{1}((t_m,b],\mathbb{R})$ such that
 $y_n$ converge to $y_m$ in
$C_m=\{y\in C^1((t_m,b],\mathbb{R})| y(t_m^+), y'(t_m^+) \text{ exist}\}$.
We define
\begin{equation} \label{102}
y(t)=  \begin{cases}
    y_0(t),&\text{if } t\in[0,t_1] ,\\
    y_1(t),&\text{if } t\in(t_1,t_2] ,\\
    \dots\\
    y_{m}(t),&\text{if } t\in(t_{m},b].
  \end{cases}
\end{equation}
It is clear that $\{y_n\}_{n\in\mathbb{N}}\subset PC$ and $y_n$ converges to $y$ in $PC$.
Using the fact that $L^{-1}$ is continuous, thus
$$
h_n=L^{-1}(y_n)\to L^{-1}(y)=h\quad \text{as } n\to \infty
$$
in $PC^1$. Then $L(y)=h$. Hence  $L^{-1}$ is compact.
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t2}]
 We consider the following fixed point problem that is equivalent
to  problem \eqref{5}--\eqref{8},
$$
y=(L^{-1}\circ F)(y),
$$
where $L^{-1}: PC(J,\mathbb{R})\to D$ and $F$ is the Nemystki operator given by
$F(t,y)=f(t,y,y')$.
From Lemma \ref{l2} we can prove that $(L^{-1}\circ F)$ is  compact.
 Now we show that $y\neq \lambda(L^{-1}\circ F)(y)$.

For $t\in[0,t_1]$, we have
\begin{gather*}
y'(t)=\phi^{-1}\Big[\phi(B)+\int_0^tf(s,y(s),y'(s))ds\Big], \\
y(t)=A+\int_0^ty'(s)ds.
\end{gather*}
Thus
$$
|y'(t)|\leq\Big|\phi^{-1}\Big[\phi(B)+\int_0^{s}p(s)\psi(|y(s)|,|y'(s)|)ds\Big]\Big|,
$$
\begin{align*}
|y(t)|&\leq |A|+\int_0^t|y'(s)|ds\\
&\leq |A|+\int_0^t\Big|\phi^{-1}
\Big[\phi(B)+\int_0^{s}p(\tau)\psi(|y(\tau)|,|y'(\tau)|)d\tau\Big]\Big|ds.
\end{align*}
Let $m(r)=\max(\sup_{t\in[0,t_1]}|y(t)|,\sup_{t\in[0,t_1]}|y'(t)|)$, then
\begin{align*}
|y(t)|&\leq |A|+\int_0^t\Big|\phi^{-1}
\Big[\phi(B)+\int_0^{s}p(\tau)\psi(m(r),m(r))d\tau\Big]\Big|ds\\
&\leq |A|+\int_0^t\Big|\phi^{-1}[\phi(B)+t_1\|p\|_{L^{1}}\psi(m(r),m(r))]\Big|ds.
\end{align*}
Then
$$
m(t)\leq|A|+\int_0^t\psi_1(m(s))ds,
$$
with $\psi_1=\phi^{-1}\circ\widetilde{\psi}$ and
$\widetilde{\psi}(u)=\phi(B)+t_1(\|p\|_{L^{1}})\psi(u)$.
By the nonlinear Gr\"{o}nwall-Bihari inequality (Lemma \ref{l1}), we infer the bound
$$
m(t)\leq H^{-1}(t)\leq M_0,
$$
where $$
H(t)=\int_{|A|}^t\frac{d\tau}{(\phi^{-1}\circ \widetilde{\psi})(\tau)}.
$$
For $t\in(t_1,t_2]$, we have
\begin{gather*}
y'(t)=y'(t_1)+\phi^{-1}\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))
+\int_{t_1}^tf(s,y(s),y'(s))ds\Big], \\
y(t)=y(t_1)+I_1(y(t_1))+\int_{t_1}^ty'(s)ds.
\end{gather*}
Thus
$$
|y'(t)|\leq|y'(t_1)|+\Big|\phi^{-1}\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))
+\int_{t_1}^tf(s,y(s),y'(s))ds\Big]\Big|.
$$
Let
$$
m(r)=\max(\sup_{t\in[0,t_1]}|y(t)|,\sup_{t\in[0,t_1]}|y'(t)|),
$$
and then
\begin{align*}
|y(t)|
&\leq |y(t_1)|+|I_1(y(t_1))|+t_2|y'(t_1)|\\
&\quad +\int_{t_1}^t\Big|\phi^{-1}\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))
 +\int_{t_1}^{s}p(\tau)\psi(|y(\tau)|,|y'(\tau)|)d\tau\Big]\Big|ds\\
&\leq M_0+|I_1(y(t_1))|+t_2|y'(t_1)|\\
&\quad \int_{t_1}^t\Big|\phi^{-1}\Big[\phi(y'(t_1)+\bar I_1(y(t_1)))
+\int_{t_1}^{s}p(\tau)\psi(m(r),m(r))d\tau\Big]\Big|ds.
\end{align*}
Then
$m(t)\leq M^{\ast}+\int_{t_1}^t\psi_1(m(s))ds$,
where
\begin{gather*}
M^{\ast}=(1+t_2)M_0+\sup_{z\in \overline{B}(0,M_0)}|I_1(z)|,\\
\psi_1=\phi^{-1}\circ\widetilde{\psi},\\
\widetilde{\psi}(u)=\phi(y'(t_1)+\bar I_1(y(t_1)))+t_2(\|p\|_{L^{1}})\psi(u,u).
\end{gather*}
 By the nonlinear Gr\"{o}nwall-Bihari inequality (Lemma \ref{l1}), we infer the bound
$$
m(t)\leq H^{-1}(t)\leq M_1,
$$
where
$$
H(t)=\int_{M^{\ast}}^t\frac{d\tau}{(\phi^{-1}\circ \widetilde{\psi})(\tau)}.
$$
For $t\in (t_{m},b]$, we have
\begin{gather*}
y'(t)=y'(t_{m})+\phi^{-1}\Big[\phi(y'(t_{m})+\bar I_{m}(y(t_{m})))
+\int_{t_{m}}^tf(s,y(s),y'(s))ds\Big],
\\
\begin{aligned}
y(t)&= y(t_{m})+I_{m}(y(t_{m}))\\
&\quad +\int_{t_{m}}^t\phi^{-1}\Big[\phi(y'(t_{m})+\bar I_{m}(y(t_{m})))
+\int_{t_{m}}^{s}f(\tau,y(\tau),y'(\tau))d\tau\Big]ds.
\end{aligned}
\end{gather*}
So, there exists
$ M_{m}>0$ such that $m(t)\leq H^{-1}(t)\leq M_{m}$,
where
$$
H(t)=\int_{M^{\ast\ast}}^t\frac{d\tau}{(\phi^{-1}\circ\widetilde{\psi})(\tau)}
$$
and
$$
M^{\ast\ast}=(1+b)M_{m-1}+\sup_{z\in \overline{B}(0,M_{m-1})}|I_{m}(z)|.
$$
Hence
$$
\|y\|_{\infty}\leq \max(M_0,M_1,\dots,M_{m}):=M.
$$
Let
$$
U=\{y\in PC^1(J,\mathbb{R}):\|y\|_{PC^1}<M+1\}.
$$
Then $L^{-1}\circ F:\overline{U}\to PC^1(J,\mathbb{R}$ is relatively compact.
Assume that there exists $\lambda\in(0,1)$ and $y\in\partial U$ such that
$y=\lambda(L^{-1}\circ F)(y)$.  Then $\|y\|_{PC^1}=M+1$, but $\|y\|_{PC^1}\leq M$.
Thus by the nonlinear alternative of Leray-Schauder, we conclude that
 $L^{-1}\circ F$ has a fixed point which is a solution of \eqref{5}-\eqref{8}.
\end{proof}

\begin{thebibliography}{00}
\bibitem{AgBeOrOu}{R. P. Agarwal, M. Benchohra, D. O'Regan, A. Ouahab};
 Second order impulsive dynamic equations
on time scales, \emph{Functional Differential Equations} \textbf{11} (2004), no. 3-4, 223-234.

\bibitem{AgOr} R. P. Agarwal, D. O'Regan;
 {Multiple nonnegative solutions for second order impulsive differential
equations}, \emph{Applied Mathematics and Computation}, \textbf{114} (2000), no. 1, 51-59.

\bibitem{AgCoMaAnDa} {Z. Agur, L. Cojocaru, G. Mazaur, R. M. Anderson, Y. L. Danon};
 {Pulse mass measles vaccination across age cohorts,} \emph{Proc. Nat. Acad. Sci. USA}
\textbf{90} (1993), 11698--11702.

\bibitem{BaSi} {D. D. Bainov, P. S. Simeonov};
\emph{Systems with Impulse Effect}, Ellis Horwood Ltd., Chichister,
1989.

\bibitem{BaSm} {D. D. Bainov, P. S. Simeonov};
\emph{Integral Inequalities and Applications, in: Mathematics and its Applications},
vol. 57, Kluwer Academic Publishers, Dordrecht, 1992

\bibitem{DuGr} {A. Granas, J. Dugundji};
 \emph{Fixed Point Theory}, Springer-Verlag, New York, 2003.

\bibitem{DjGrOu-M}{S. Djebali, L. Gorniewicz, A. Ouahab};
\emph{Existence and Structure of Solution Sets for Impulsive Differential Inclusions},
Lecture Notes,in Nonlinear Analysis, Nicolaus Copernicus University,
Toru\'{n}, No 13 , 2012.

\bibitem{HaWe} {A. Halanay, D. Wexler};
 \emph{Teoria Calitativa a systeme cu Impulduri}, { Editura Republicii Socialiste Romania,}
Bucharest, 1968.

\bibitem{Kr} {E.  Kruger-Thiemer};
 {Formal theory of drug dosage regiments,} \emph{J. Theo. Biol.} \textbf{13} (1966), 212--235.

\bibitem{LaBaSi} {V. Lakshmikantham, D. D. Bainov, P. S. Simeonov};
\emph{Theory of Impulsive Differential Equations},
World Scientific, Singapore, 1989.

\bibitem{MiMy} {V. D. Milman, A. A. Myshkis};
 {On the stability of motion in the presence of impulses (in Russian),} \emph{Sib. Math. J.}  \textbf{1} (1960), 233--237.

\bibitem{PePloSamSk} 
{N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, N. V. Skripnik};
\emph{Differential Equations with Impulse Effects Multivalued Right-hand Sides with Discontinuities,}
{ Walter de Gruyter \& Co.}, Berlin, 2011.

\bibitem{SaPe} {A. M. Samoilenko, N. A. Perestyuk};
\emph{Impulsive Differential Equations,} World Scientific,
Singapore, 1995.

\end{thebibliography}

\end{document}