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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 285, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/285\hfil Impulsive fractional functional differential equations]
{Impulsive fractional functional differential equations
 with a weakly continuous nonlinearity}

\author[Y. Wang, F. Gao, P. Kloeden \hfil EJDE-2017/285\hfilneg]
{Yejuan Wang, Fengshuang Gao, Peter Kloeden}

\address{Yejuan Wang (corresponding author) \newline
School of Mathematics and Statistics,
Gansu Key Laboratory of Applied
Mathematics and Complex Systems,
 Lanzhou University,
Lanzhou 730000, China}
\email{wangyj@lzu.edu.cn}

\address{Fengshuang Gao \newline
School of Mathematics and Statistics,
Gansu Key Laboratory of Applied
Mathematics and Complex Systems,
Lanzhou University,
Lanzhou 730000, China.\newline
 Department of Mathematics, Tsinghua University,
Beijing 10084,  China}
\email{gfs16@mails.tsinghua.edu.cn}

\address{Peter Kloeden \newline
School of Mathematics and Statistics,
Huazhong University of Science $\&$ Technology,
Wuhan 430074,  China}
\email{kloeden@math.uni-frankfurt.de}

\dedicatory{Communicated by Zhaosheng Feng}

\thanks{Submitted October 5, 2016. Published November 14, 2017.}
\subjclass[2010]{34K45, 34G20}
\keywords{Impulsive fractional delay differential equation; global solution;
\hfill\break\indent Caputo fractional time derivative}

\begin{abstract}
 A general theorem on the local and global existence of solutions is
 established for an  impulsive fractional delay differential equation
 with  Caputo fractional substantial derivative  in a separable Hilbert
 space under the assumption that the nonlinear term is weakly continuous.
 The uniqueness of solutions is also considered under an  additional
 Lipschitz assumption.
\end{abstract}

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

\section{Introduction}

Fractional differential equations have been used to establish a more accurate
 model in diverse  fields  such as engineering, physics, chemistry,
signal analysis and economics. It  is applied widely in nonlinear
oscillations of earthquakes,  physical phenomena like seepage flow in
porous media and in fluid dynamic traffic models. We refer the reader
to \cite{Kilbas,Michalski,Miller,Podlubny} for more details on fractional calculus.
 In 2006, the concept of fractional substantial derivative was presented
by Friedrich et al. in \cite{Friedrich} when they considered  retardation
effects in Kramers-Fokker-Planck type equations.
Carmi et al.\ \cite{Carmi2} used them to study the distribution of
 functionals of  anomalous diffusion trajectories.
The fractional substantial integral is defined by \cite{Chen, Deng}
\[
  I_s^\nu f(x)=\frac{1}{\Gamma(\nu)}\int_a^x(x-\tau)^{\nu-1}
e^{-\beta (x-\tau)}f(\tau)d\tau,\quad \nu>0,
 \]
  and in the similar way \cite{Chauhan}, the Caputo fractional substantial
derivative is defined as
\[
  D_s^\mu f(x)=I_s^\nu[D_s^m f(x)],\quad \nu=m-\mu,
\]
where $\beta$ is a constant or a function independent of $x$,
say $\beta(y)$, $m$ is the smallest integer that exceeds $\mu$, and
\[
D_s^m=\Big(\frac{\partial }{\partial x}+\beta\Big)^m
=(D+\beta)^m.
\]

In the previous decades, the theory of impulsive differential  equations
has been studied with great interests mainly due to the important
role such equations play in studying evolution processes that are
subject to abrupt changes in their states, such as changes of populations,
transmission of diseases, and so on. The reader is referred to
\cite{Bainov,Benchohra1,Lakshmikantham1} for the basic theory of impulsive
differential equations.

In  this paper we establish some global existence theorems
for impulsive fractional delay differential equations on Hilbert spaces.
These results will be used by us in \cite{kloeden} to investigate the
asymptotic behavior of lattice models involving such  equations.
 The global existence of mild solutions to impulsive fractional functional
differential equations was discussed in \cite{Chauhan,Gautam}, while
in \cite{Guo} the existence and uniqueness of solutions for impulsive
fractional functional differential equations were  considered.
In addition, the Cauchy problem for fractional impulsive differential
equations with delay was  addressed in \cite{Zhang}. Moreover,
Benchohra and Berhoun  \cite{Benchohra}   investigated the existence of
solutions for impulsive fractional differential equations with state-dependent delay.

We consider the global existence of solutions of impulsive functional
differential equations with Caputo fractional substantial time derivative
\begin{equation}\label{ivp}
\begin{gathered}
D_s^{\alpha} u(t)=f(t,u_t),\quad t\ge 0,\;t\neq t_k,\\
u(s)=\phi(s),\quad \forall s\in[-h,0],\\
u(t_k^+)-u(t_k^-)=I_k(u(t_k^-)),\quad k=1,2,\dots
\end{gathered}
\end{equation}
in the separable Hilbert space $X$, where $D_s^\alpha$ is the Caputo
fractional substantial derivative with $0<\alpha<1$ and $\beta >0$.
We assume that the nonlinear term $f$ is weakly continuous in bounded sets.
This concept was given in \cite{Tomas} and delay differential equations
in Banach spaces with a classical derivative were treated.
In addition, we prove the uniqueness of solutions of \eqref{ivp} under Lipschitz
 conditions.


This article is structured as follows. Notation, some basic definitions
and preliminary  results are given in the next section, and then,
in Section 3  we present theorems of the local and  global existence
and also uniqueness  of solutions for \eqref{ivp} in a  separable Hilbert
space. Proofs  of these theorems are then given in Sections 4, 5  and 6.

\section{Preliminaries}

Let $X$ be a separable  Hilbert space with norm $\|\cdot\|$ and inner product
$(\cdot,\cdot)$. Let $PC_t$ $:=$ $PC([-h,t]; X)$, $h>0$, $t\geq 0$, be a
 Banach space of all such functions $u$ $:$ $[-h,t]\to X$,
which are continuous everywhere except for a finite number of points
$t_k$, $k=1,2,\dots, m$, at which $u(t_k^+)$ and $u(t_k^-)$ exist and
$u(t_k)=u(t_k^-)$, endowed with the norm
\[
\|u\|_{PC_t}=\sup_{-h\leq s\leq t}\|u(s)\|.
\]
For any $u\in PC_T=PC([-h,T]; X)$, we denote by $u_t$ the
element of $PC_0=PC([-h,0]; X)$ defined by
$u_t(\theta)=u(t+\theta)$, $\theta\in[-h,0]$. Here $I_k\in C(X,X)$
for each $k$, $u(t_k^+)=\lim_{h\to 0}u(t_k+h)$ and
$u(t_k^-)=\lim_{h\to 0}u(t_k-h)$ represent the right and left-hand limits
of $u(t)$ at $t=t_k$, respectively.

Let $X^\ast$  be the dual space of $X$ with the pairing between $X$ and $X^\ast$
denoted by $\langle\cdot,\cdot\rangle$,   and let $X_w$ be the space $X$
endowed with the weak topology. We consider the space
$PC_{0,w}=PC([-h,0]; X_w)$. Let $t \ge 0$ and  $\{u_{t}^n\}_{n=1}^\infty$
be a given  sequence.
We say that $u_{t}^n\to u_t\in PC_{0,w}$ in $PC_{0,w}$ if it satisfies
\begin{itemize}
\item[(1)] for any $s\in[-h,0]$ with  $t+s\neq t_k$, for  $k=1, 2,\dots$,
\[
u^n(t+s_n)\to u(t+s)\quad \text{in  $X_w$  as }  n\to \infty
\]
for any sequence $\{s_n\}_{n=1}^\infty$ with $s_n \to s$;

\item[(2)] for any $s\in[-h,0]$ with $t+s=t_k$ for some $k=1, 2,\dots$,
\[
u^n(t+s_n)\to u(t+s)\hspace{2mm} \quad \text{in  $X_w$ as }  n\to \infty
\]
 for any sequence $\{s_n\}_{n=1}^\infty$ with $s_n\leq s$ and $s_n\to s$.
We  say that the function $f :[0,\infty)\times PC_0 \to X$
is weakly continuous in bounded sets for each $t\in [0, \infty)$
if $u^n\to u$ in $PC_{0, w}$, and $\|u^n\|_{PC_0}\le M$ for all
$n\in \mathbb{N}$, imply that $f(t, u^n)\to f(t, u)$ in
$X_w$, and we say that the function $g:X\to X$ is weakly continuous
in bounded sets if $v^n \to v$ in $X_w$ and
$\|v^n\|\le M$ for all $n \in \mathbb{N}$, imply that
$g(v^n) \to g(v)$ in $X_w$.
\end{itemize}

\begin{definition} \label{def2.1}\rm
A function $u\in PC_T$ is called a solution of  initial value problem
\eqref{ivp} if  $u(t)=\phi(t)$ for  $t\in[-h,0]$ with
$\phi \in PC_0$, and, for $t\in[0,T]$, $u(t)$ satisfies the integral equation
\[
u(t)=\begin{cases}
 \phi(0)e^{-\beta t}+\frac{1}{\Gamma(\alpha)}
\int_0^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
&t\in[0,t_1],\\[4pt]
\big(u(t_1^-)+I_1(u(t_1^-))\big) e^{-\beta (t-t_1)}\\
+\frac{1}{\Gamma(\alpha)}
\int_{t_1}^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau
,&t\in(t_1,t_2],\\
\dots\\
\big(u(t_m^-)+I_m(u(t_m^-))\big) e^{-\beta (t-t_m)}\\
+\frac{1}{\Gamma(\alpha)}
\int_{t_m}^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
&t\in(t_m, T],
\end{cases}
\]
where $t_m=\max\{t_k : t_k <T, k=0, 1, 2, \dots\}$ and
$t_0=0$. Here and elsewhere $\Gamma$ denotes the Gamma function.
\end{definition}

\begin{lemma}\label{lemma2.2}
A function $u\in PC_T$ is a solution of initial value problem \eqref{ivp}
 if and only if
\begin{equation}\label{eq2.1}
u(t)=\begin{cases}
\phi(t), &t\in[-h,0],\\
  \phi(0)e^{-\beta t}+\frac{1}{\Gamma(\alpha)}
\int_0^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
 &t\in[0,t_1],\\[4pt]
\phi(0)e^{-\beta t}+I_1(u(t_1^-))e^{-\beta(t-t_1)}\\
 +\frac{1}{\Gamma(\alpha)}\int_0^{t_1} (t_1-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau\\
 +\frac{1}{\Gamma(\alpha)}\int_{t_1}^t (t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
&t\in(t_1,t_2],\\
\dots\\
 \phi(0)e^{-\beta t}+\mathop\sum _{k=1}^m I_k(u(t_k^-))e^{-\beta(t-t_k)}\\
 +\frac{1}{\Gamma(\alpha)}\mathop\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
  (t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau\\
 +\frac{1}{\Gamma(\alpha)}\int_{t_m}^t (t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
&t\in(t_m, T].
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
Assume that $u$ is a solution of the initial value problem \eqref{ivp}.
 Then by Definition \ref{def2.1}, we obtain
\[
u(t)=\phi(0)e^{-\beta t}+\frac{1}{\Gamma(\alpha)}
\int_0^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau, \quad
 \text{if } t\in[0,t_1],
\]

\begin{align*}
u(t) =&\Big(\phi(0)e^{-\beta t_1}+\frac{1}{\Gamma(\alpha)}
\int_0^{t_1} (t_1-\tau)^{\alpha-1}e^{-\beta(t_1-\tau)}
 f(\tau,u_\tau)d\tau\Big)e^{-\beta(t- t_1)}\\
&+I_1(u(t_1^-))e^{-\beta(t-t_1)}
+\frac{1}{\Gamma(\alpha)}\int_{t_1}^t (t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau\\
=&\phi(0)e^{-\beta t}+I_1(u(t_1^-))e^{-\beta(t-t_1)}
+\frac{1}{\Gamma(\alpha)}\int_0^{t_1} (t_1-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau\\
&+\frac{1}{\Gamma(\alpha)}\int_{t_1}^t (t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau, \quad\text{if   } t\in(t_1,t_2],\\
&\dots
\end{align*}
\begin{equation}\label{eq2.2}
\begin{split}
u(t)=&\phi(0)e^{-\beta t}+\mathop\sum _{k=1}^{m} I_k(u(t_k^-))e^{-\beta(t-t_k)}\\
&+\frac{1}{\Gamma(\alpha)}\mathop\sum_{k=1}^{m}\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau\\
& +\frac{1}{\Gamma(\alpha)}\int_{t_m}^t (t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau, \quad  \text{if   } t\in(t_m,T].
\end{split}
\end{equation}
In a similar way, if \eqref{eq2.1} holds, then  we can prove that $u$
is the solution of \eqref{ivp}, and thus the proof of this lemma is complete.
\end{proof}

\begin{definition} \label{def2.3}\rm
A set $\Lambda$ is said to be quasi-equicontinuous in $[0, T]$ if for any
$\varepsilon>0$, there exists $\delta'>0$ such that if $u \in \Lambda$,
$k \in \mathbb{N}$, $s_1, s_2 \in (t_{k-1}, t_k] \cap [0, T]$ and
$|s_1-s_2|<\delta'$, then
$\|u(s_1)-u(s_2)\|<\varepsilon$.
\end{definition}

\begin{theorem}[Leray-Schauder fixed point theorem]\label{theorem2.3}
 Let $F$ be a continuous and compact mapping of a
Banach space $X$ into itself, such that the set
$$
\{x \in X : x=\lambda Fx \text{ for some } 0 \leq\lambda \leq 1\}
$$
is bounded. Then $F$  has a fixed point.
 \end{theorem}

\section{Existence theorems}

In this section, we consider the existence and uniqueness of
 global  solutions of the initial value problem \eqref{ivp}.
 First we state some assumptions for the functions $f$
and $I_k$ in  \eqref{ivp}.
\begin{enumerate}
    \item[(H1)] The function $f:[0,\infty)\times PC_0\to X$ is weakly continuous
in bounded sets for each $t \in [0, \infty)$, and there exist $K_2>0$ and  a
function $K_1\in L^{1/\gamma}([0,\infty),\mathbb{R}_+)$ with
$\gamma<\alpha$  such that
    \[
    \|f(t,\psi)\|\leq K_1(t)+K_2\|\psi\|_{PC_0} \quad \text{for  all }
\psi \in PC_0 \text{ and } t \in [0, \infty) .
    \]
    \item[(H2)] The functions $I_k: X\to X$ are weakly continuous  in bounded
sets and there exist $J_1$, $J_2>0$ such that
        \[
         \|I_k(x)\|\leq J_1\|x\|+J_2 \quad \text{for  all $x\in X$ and }
k\in \mathbb{N}.
         \]

\item[(H3)] $\delta=\sup_{k\in\mathbb{N}}\{t_k-t_{k-1}\}<\infty$,
$\eta=\inf_{k\in\mathbb{N}}\{t_k-t_{k-1}\}>0$.

\item[(H4)] There  exists  $M_1>0$   such that
  \[
  \|f(t,\varphi)-f(t, \psi)\|\leq M_1\|\varphi-\psi\|_{PC_0} \quad
\text{for all }\varphi,\psi\in PC_0 \text{ and } t\in[0,\infty).
  \]

\item[(H5)] There exists  $N>0$   such that
  \[
  \|I_k(x)-I_k(y)\|\leq N\|x-y\| \quad \text{ for all } x,y\in X
\text{ and } k\in\mathbb{N}.
  \]
\end{enumerate}

In the sequel $C$ denotes an arbitrary positive constant, which
may be different from line to line and even in the same line.
We now state a theorem regarding the local existence of solutions
for problem \eqref{ivp}.

\begin{theorem}\label{theorem3.1}
Assume that $X$ is a separable Hilbert space, and  conditions {\rm (H1)--(H3)}
are satisfied. Then for every $\phi\in PC_0$,  initial value  problem \eqref{ivp}
 has at least one solution defined on $[0, b]$ with $b >t_1$, where $t_1$ is
 given by  \eqref{ivp}.
\end{theorem}

\begin{theorem}\label{thm3.3}
Assume  the conditions of Theorem \ref{theorem3.1}.
Then for every $\phi\in PC_0$,  initial value  problem \eqref{ivp} has
 at least one solution defined on $[0, \infty)$ in the sense of
Definition \ref{def2.1}.
\end{theorem}

We will also prove the uniqueness of solutions.

\begin{theorem} \label{theorem3.4}
Assume  the hypotheses of Theorem \ref{theorem3.1}.
Also, suppose that the conditions {\rm (H4), (H5)} are satisfied.
 Then for every $\phi\in PC_0$,  problem \eqref{ivp} possesses a unique
solution $u(\cdot)$ defined on $[0, \infty)$ in the sense of Definition \ref{def2.1}.
\end{theorem}

\section{Proof of Theorem \ref{theorem3.1}}

Since $X$ is separable, there exists a family of elements
$\{e_j\}_{j=1}^\infty$ of $X$ which are orthonormal in $X$.
 Let $X_{(n)}= \operatorname{span} \{e_1,\dots,e_n\}$ in $X$ and
$P_n:X\to X_{(n)}$ is an orthonormal projector.
Fix some $\phi\in PC_0$, and let $u_n=P_nu$, $\phi_n=P_n\phi$.
By Lemma \ref{lemma2.2}, for every $n$ we introduce the mapping
$T_n: PC_b\to PC_b$ defined by
\begin{equation}\label{eq3.1}
(T_nu_n)(t)=\begin{cases}
\phi_n(t), & t\in[-h,0],\\[4pt]
\phi_n(0)e^{-\beta t}+\sum_{0<t_k<t} P_nI_k(u_n(t_k^-))e^{-\beta(t-t_k)}\\
+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau\\
+\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau, & t\in[0,b],
\end{cases}
\end{equation}
where $t_m=\max\{t_k:t_k<t, k=0, 1, 2, \dots\}$ and $t_0=0$.
Now we show that the operator $T_n$ is continuous and completely continuous.
Since the proof of the case $m=0$ is similar, we assume $m\geq 1$.

\subsection*{Step 1:  $T_n$ maps bounded sets into bounded sets in $PC_b$}

Indeed, it is enough to show that for any $\rho>0$, there  exists a positive
 constant $\rho'$ such that for each $u_n \in B(\rho)$ one has
$\sup_{t \in [0, b]}\|T_n u_n(t)\| \le \rho'$, where
\[
B(\rho)=\{u_n\in PC_b ~:~ u_n(t)=\phi_n(t) \text{ on } [-h,0] \text{ and }
\sup_{t\in[0,b]}\|u_n(t)\|\leq \rho\}.
\]
Let $u_n\in B(\rho)$,   by (H1) and (H2) we obtain for each  $t\in [0,b]$,
\begin{align}
& \|(T_nu_n)(t)\| \nonumber \\
&\leq\|\phi_n(0)\|e^{-\beta t}+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
\|P_nf(\tau,u_{n\tau})\|d\tau \nonumber \\
&\quad+\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}\|P_nf(\tau,u_{n\tau})\|d\tau \nonumber \\
&\quad +\sum_{0<t_k<t}\|P_nI_k(u_n(t_k^-))\|e^{-\beta(t-t_k)} \nonumber \\
&\leq\|\phi(0)\|+\frac{1}{\Gamma(\alpha)}
 \sum _{k=1}^m\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}\Big(K_1(\tau)+K_2\rho\Big)d\tau \nonumber \\
&\quad +\sum_{k=1}^m\Big(J_1\rho+J_2\Big) e^{-\beta(t-t_k)} \nonumber \\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
\Big(K_1(\tau)+K_2\rho\Big)d\tau. \label{eq3.2}
\end{align}
We proceed to estimate the three last terms in \eqref{eq3.2}.
First, by (H3) we have
\begin{equation}
\begin{aligned}
&\frac{K_2\rho}{\Gamma(\alpha)}\Big(\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}d\tau
+\int_{t_m}^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}d\tau\Big)
\\
&=-\frac{K_2\rho}{\Gamma(\alpha+1)}\Big(\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
 e^{-\beta(t-\tau)}d(t_k-\tau)^\alpha
 +\int_{t_m}^t e^{-\beta(t-\tau)}d(t-\tau)^\alpha\Big)
\\
&\le \sum_{k=1}^m\frac{K_2\rho}{\Gamma({\alpha}+1)}
\Big((t_k-t_{k-1})^\alpha e^{-\beta(t-t_{k-1})}
+\beta\int_{t_{k-1}}^{t_k}(t_k-\tau)^\alpha e^{-\beta(t-\tau)}d\tau\Big)
\\
&\quad  +\frac{K_2\rho}{\Gamma({\alpha}+1)}\Big((t-t_m)^\alpha
e^{-\beta(t-t_m)}+\beta\int_{t_m}^t(t-\tau)^\alpha e^{-\beta(t-\tau)}d\tau\Big)
\\
&\leq\frac{K_2\rho}{\Gamma({\alpha}+1)}\Big(\sum_{k=1}^m(t_k-t_{k-1})^\alpha
e^{-\beta(m-k+1)\eta}+(t-t_m)^\alpha \\
&\quad +\beta\sum_{k=1}^m\int_{t-t_k}^{t-t_{k-1}}z^\alpha e^{-\beta z}dz
 +\beta\int_0^{t-t_m}z^\alpha e^{-\beta z}dz\Big) \\
&\leq\frac{K_2\rho \delta^\alpha e^{\beta \eta}}{\Gamma(\alpha+1)(e^{\beta \eta}-1)}+\frac{K_2\rho}{\beta^\alpha},\label{eq3.3}
\end{aligned}
\end{equation}
and in the similar way,
\begin{equation} \label{eq3.4}
\sum_{k=1}^m (J_1\rho+J_2)e^{-\beta(t-t_k)} \le (J_1 \rho+J_2)
\sum_{k=1}^m e^{-\beta(m-k)\eta} \le (J_1\rho+J_2)
\frac{e^{\beta \eta}}{e^{\beta \eta}-1}.
\end{equation}
Define  $K_1^\ast=\Big(\int_0^\infty (K_1(t))^{1/\gamma}dt\Big)^\gamma$.
Then, using  H\"{o}lder's inequality, (H1) and (H3), we obtain
\begin{align}
&\frac{1}{\Gamma(\alpha)}\sum _{k=1}^m\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}K_1(\tau)d\tau+\frac{1}{\Gamma(\alpha)}
\int_{t_m}^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}K_1(\tau)d\tau
\nonumber \\
&\leq\frac{1}{\Gamma(\alpha)}\sum_{k=1}^m\Big(\int_{t_{k-1}}^{t_k}
(t_k-\tau)^{\frac{\alpha-1}{1-\gamma}}
e^{\frac{-\beta(t-\tau)}{1-\gamma}}d\tau\Big)^{1-\gamma}
\Big(\int_{t_{k-1}}^{t_k}(K _1(\tau))^{1/\gamma}d\tau\Big)^\gamma
\nonumber\\
&\quad +\frac{1}{\Gamma(\alpha)}
\Big(\int_{t_m}^t(t-\tau)^{\frac{\alpha-1}{1-\gamma}}
e^{\frac{-\beta(t-\tau)}{1-\gamma}}
d\tau\Big)^{1-\gamma}\Big(\int_{t_m}^t(K_1(\tau))^{1/\gamma}d\tau
\Big)^\gamma
\nonumber\\
&\leq\frac{1}{\Gamma(\alpha)}\Big(\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\frac{\alpha-1}{1-\gamma}}
e^{\frac{-\beta(t-\tau)}{1-\gamma}}d\tau+\int_{t_m}^t
 (t-\tau)^{\frac{\alpha-1}{1-\gamma}}e^{\frac{-\beta(t-\tau)}{1-\gamma}}
d\tau\Big)^{1-\gamma}
\nonumber\\
&\quad\times\Big(\sum_{k=1}^m\int_{t_{k-1}}^{t_k}(K_1(\tau))^{1/\gamma}d\tau
+\int_{t_m}^t(K_1(\tau))^\frac{1}{\gamma}d\tau\Big)^{\gamma}
\nonumber\\
&\leq\frac{K_1^\ast}{\Gamma(\alpha)}\Big(-\sum_{k=1}^m
 \frac{1-\gamma}{\alpha-\gamma}\int_{t_{k-1}}^{t_k}
e^{\frac{-\beta(t-\tau)}{1-\gamma}}d(t_k-\tau)^{\frac{\alpha-\gamma}{1-\gamma}}
\nonumber \\
&\quad -\frac{1-\gamma}{\alpha-\gamma}\int_{t_m}^t
e^{\frac{-\beta(t-\tau)}{1-\gamma}}d(t-\tau)^{\frac{\alpha-\gamma}{1-\gamma}}
 \Big)^{1-\gamma}
\nonumber\\
&\leq\frac{K_1^\ast(\frac{1-\gamma}{\alpha-\gamma})^{1-\gamma}}{\Gamma(\alpha)}
\Big(\sum_{k=1}^m(t_k-t_{k-1})^{\frac{\alpha-\gamma}{1-\gamma}}
e^{\frac{-\beta(m+1-k)\eta}{1-\gamma}}+(t-t_m)^{\frac{\alpha-\gamma}{1-\gamma}}
\nonumber\\
&\quad +\frac{\beta}{1-\gamma}\Big(\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\frac{\alpha-\gamma}{1-\gamma}}
e^{\frac{-\beta(t-\tau)}{1-\gamma}}d\tau+\int_{t_m}^t
 (t-\tau)^{\frac{\alpha-\gamma}{1-\gamma}}e^{\frac{-\beta(t-\tau)}{1-\gamma}}
 d\tau\Big)\Big)^{1-\gamma}
\nonumber\\
&\leq\frac{K_1^\ast(\frac{1-\gamma}{\alpha-\gamma})^{1-\gamma}}{\Gamma(\alpha)}
\Big(\sum_{k=1}^m \delta^{\frac{\alpha-\gamma}{1-\gamma}}
 e^{\frac{-\beta(m+1-k)\eta}{1-\gamma}}+\delta^{\frac{\alpha-\gamma}{1-\gamma}}
\nonumber \\
&\quad  +\frac{\beta}{1-\gamma}\sum_{k=1}^m\int_{t-t_k}^{t-t_{k-1}}
 z^{\frac{\alpha-\gamma}{1-\gamma}}
e^{\frac{-\beta z}{1-\gamma}}dz
  +\frac{\beta}{1-\gamma}\int_0^{t-t_m}z^{\frac{\alpha-\gamma}{1-\gamma}}
e^{\frac{-\beta z}{1-\gamma}}dz\Big)^{1-\gamma}
\nonumber\\
&\leq\frac{K_1^{\ast}}{\Gamma(\alpha)}\Big(\frac{1-\gamma}{\alpha-\gamma}
\delta^{\frac{\alpha-\gamma}{1-\gamma}}
\frac{e^{\frac{\beta \eta}{1-\gamma}}}{e^{\frac{\beta\eta}{1-\gamma}}-1}
+\frac{\beta}{\alpha-\gamma}\frac{\Gamma(\frac{\alpha+1-2\gamma}{1-\gamma})}
{(\frac{\beta}{1-\gamma})
^{\frac{\alpha+1-2\gamma}{1-\gamma}}}\Big)^{1-\gamma}. \label{eq3.5}
\end{align}
It then follows from \eqref{eq3.2}--\eqref{eq3.5} that for any $n\in \mathbb{N}$,
 $u_n\in B(\rho)$ and $t\in [0,b]$,
\begin{align*}
\|(T_nu_n)(t)\|\leq
&\|\phi(0)\|+\Big(\frac{1-\gamma}{\alpha-\gamma}
 \delta^{\frac{\alpha-\gamma}{1-\gamma}}
 \frac{e^{\frac{\beta \eta}{1-\gamma}}}{e^{\frac{\beta\eta}{1-\gamma}}-1}
+\frac{\beta}{\alpha-\gamma}\frac{\Gamma(\frac{\alpha+1-2\gamma}{1-\gamma})}
 {(\frac{\beta}{1-\gamma})^{\frac{\alpha+1-2\gamma}{1-\gamma}}}\Big)^{1-\gamma}
\frac{K_1^{\ast}}{\Gamma(\alpha)}
\\
&\quad +\frac{K_2\rho\delta^\alpha e^{\beta \eta}}{\Gamma(\alpha+1)(e^{\beta \eta}-1)}
+\frac{K_2\rho}{\beta^\alpha}+\frac{J_1\rho e^{\beta \eta}}{e^{\beta \eta}-1}
+\frac{J_2e^{\beta \eta}}{e^{\beta \eta}-1}=\rho'.
\end{align*}
Therefore, $T_nu_n\in B(\rho')$.

\subsection*{Step 2: $T_n$ maps bounded sets into quasi-equicontinuous sets
of $PC_b$}

 Let $B(\rho)$ be a bounded set of $PC_b$ as in Step 1. We show that
$T_n(B(\rho))=\{T_nu_n: u_n\in B(\rho)\}$ is a quasi-equicontinuous
family of functions, that is, for any $\varepsilon>0$ there exists
$\delta'>0$ such that if $n\in\mathbb{N}$, $s_1,s_2\in(t_{k-1}$,
 $t_k]\cap[-h,b]$ and $|s_1-s_2|$ $<$ $\delta'$, then
$\|(T_nu_n)(s_2)-(T_nu_n)(s_1)\|<\varepsilon$.

 Since the proof of the case $m=0$ is similar, we assume that
$t_m<s_1<s_2\le t_{m+1}$ for some $m\in\{1, 2,\dots\}$. Then  from (H1)-(H3)
and \eqref{eq3.1},  for all $n\in\mathbb{N}$ and $u_n\in B(\rho)$  we obtain  that
 \begin{align}
 &\|(T_nu_n)(s_2)-(T_nu_n)(s_1)\| \nonumber \\
&\leq\|\phi_n(0)\||e^{-\beta s_2}-e^{-\beta s_1}|
 +\sum_{k=1}^m\|P_nI_k(u_n(t_k^-))\|e^{-\beta(s_2-t_k)}|1-e^{-\beta(s_1-s_2)}|
\nonumber \\
 &\quad+\frac{1}{\Gamma(\alpha)}\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\alpha-1}e^{-\beta(s_2-\tau)}|1-e^{-\beta(s_1-s_2)}|
 \|P_nf(\tau,u_{n\tau})\|d\tau \nonumber\\
 & \quad +\frac{1}{\Gamma(\alpha)}\int_{t_m}^{s_1}
\Big((s_1-\tau)^{\alpha-1}e^{-\beta(s_1-\tau)}-(s_2-\tau)^{\alpha-1}
e^{-\beta(s_2-\tau)}\Big)\|P_nf(\tau,u_{n\tau})\|d\tau
\nonumber \\
 & \quad +\frac{1}{\Gamma(\alpha)}\int_{s_1}^{s_2}(s_2-\tau)^{\alpha-1}
e^{-\beta(s_2-\tau)}\|P_nf(\tau,u_{n\tau})\|d\tau
\nonumber \\
&\leq \rho|e^{-\beta s_2}-e^{-\beta s_1}|
 +(J_1\rho+J_2) |1-e^{-\beta(s_1-s_2)}|\sum_{k=1}^me^{-\beta(m-k)\eta}
\nonumber \\
 & \quad +\frac{1}{\Gamma(\alpha)}\sum_{k=1}^m\int_{t_{k-1}}^{t_k}
(t_k-\tau)^{\alpha-1}e^{-\beta(s_2-\tau)}(e^{-\beta(s_1-s_2)}-1)
\Big(K_1(\tau)+K_2\rho\Big)d\tau
\nonumber \\
 &\quad +\frac{1}{\Gamma(\alpha)}\int_{t_m}^{s_1}\Big((s_1-\tau)^{\alpha-1}
e^{-\beta(s_1-\tau)}-(s_2-\tau)^{\alpha-1}e^{-\beta(s_2-\tau)}\Big)
\Big(K_1(\tau)+K_2\rho\Big)d\tau
\nonumber \\
 &\quad +\frac{1}{\Gamma(\alpha)}\int_{s_1}^{s_2}(s_2-\tau)^{\alpha-1}
e^{-\beta(s_2-\tau)}\Big(K_1(\tau)+K_2\rho\Big)d\tau
\nonumber \\
&\leq\rho|e^{-\beta s_2}-e^{-\beta s_1}|+(J_1\rho+J_2)
|1-e^{-\beta(s_1-s_2)}|\frac{e^{\beta \eta}}{e^{\beta\eta}-1}
+E_1+E_2+E_3. \label{eq3.6}
\end{align}
 For $E_1$, by the similar argument as in \eqref{eq3.3} and \eqref{eq3.5},
we have
 \begin{equation}\label{eq3.7}
 E_1\leq C\Big(e^{-\beta(s_1-s_2)}-1\Big)\to 0 \quad  \text{as } s_2\to s_1.
 \end{equation}
For $E_2$, by  H\"{o}lder's inequality  and (H1), we obtain
\begin{align}
E_2&\leq\frac{1}{\Gamma(\alpha)}\int_{t_m}^{s_1}\Big((s_1-\tau)^{\alpha-1}
 -(s_2-\tau)^{\alpha-1}\Big)\Big(K_1(\tau)+ K_2\rho\Big)d\tau
\nonumber \\
& \quad +\frac{1}{\Gamma(\alpha)}\int_{t_m}^{s_1}(s_2-\tau)^{\alpha-1}
\Big(e^{-\beta(s_1-\tau)}-e^{-\beta(s_2-\tau)}\Big)
\Big(K_1(\tau)+K_2\rho\Big)d\tau
\nonumber\\
&\leq\frac{1}{\Gamma(\alpha)}\Big(\int_{t_m}^{s_1}
 \Big((s_1-\tau)^{\alpha-1}-(s_2-\tau)^{\alpha-1}\Big)
^{\frac{1}{1-\gamma}}d\tau\Big)^{1-\gamma}
\Big(\int_{t_m}^{s_1}(K_1(\tau))^{1/\gamma}d\tau\Big)^\gamma
\nonumber \\
& \quad +\frac{1}{\Gamma(\alpha)}\Big(\int_{t_m}^{s_1}(s_2-\tau)^{\frac{\alpha-1}
{1-\gamma}}
\Big(e^{-\beta(s_1-\tau)}-e^{-\beta(s_2-\tau)}
\Big)^{\frac{1}{1-\gamma}}d\tau\Big)^{1-\gamma}  \nonumber \\
&\quad\times \Big(\int_{t_m}^{s_1}(K_1(\tau))^{1/\gamma}d\tau\Big)^\gamma
  +\frac{K_2\rho}{\Gamma(\alpha)}\int_{t_m}^{s_1}(s_1-\tau)^{\alpha-1}
-(s_2-\tau)^{\alpha-1}d\tau
\nonumber\\
&\quad +\frac{K_2\rho}{\Gamma(\alpha)}
 \int_{t_m}^{s_1}(s_2-\tau)^{\alpha-1}\Big(e^{-\beta(s_1-\tau)}
-e^{-\beta(s_2-\tau)}\Big)d\tau
\nonumber \\
&\leq\frac{K_1^\ast}{\Gamma(\alpha)}\Big(\int_{t_m}^{s_1}
\Big((s_1-\tau)^{\frac{\alpha-1}{1-\gamma}}-(s_2-\tau)
^{\frac{\alpha-1}{1-\gamma}}\Big)d\tau\Big)^{1-\gamma}
+\frac{K_2\rho}{\Gamma(\alpha+1)}(s_2-s_1)^{\alpha}
\nonumber\\
&\quad  +\frac{K_1^\ast(s_2-s_1)^{\alpha-1}}{\Gamma(\alpha)}
\Big(\int_{t_m}^{s_1}\Big(e^{\frac{-\beta(s_1-\tau)}{1-\gamma}}
-e^{\frac{-\beta(s_2-\tau)}{1-\gamma}}\Big)d\tau\Big)^{1-\gamma}
\label{eq3.8} \\
&\quad +\frac{K_2\rho}{\beta\Gamma(\alpha)}(s_2-s_1)^{\alpha-1}
(1-e^{-\beta(s_2-s_1)})
\nonumber\\
&\leq\frac{K_1^\ast}{\Gamma(\alpha)}
 \Big(\frac{1-\gamma}{\alpha-\gamma}(s_1-t_m)^{\frac{\alpha-\gamma}{1-\gamma}}
-\frac{1-\gamma}{\alpha-\gamma}(s_2-t_m)^{\frac{\alpha-\gamma}{1-\gamma}}
 +\frac{1-\gamma}{\alpha-\gamma}(s_2-s_1)
^{\frac{\alpha-\gamma}{1-\gamma}}\Big)^{1-\gamma}
\nonumber\\
&\quad +\frac{K_1^\ast}{\Gamma(\alpha)}
 \Big(\frac{1-\gamma}{\beta}\Big)^{1-\gamma}(s_2-s_1)^{\alpha-1}
\left(1-e^{\frac{-\beta(s_2-s_1)}{1-\gamma}}\right)^{1-\gamma}
+\frac{K_2\rho}{\Gamma(\alpha+1)}(s_2-s_1)^{\alpha}
\nonumber\\
&\quad +\frac{K_2\rho}{\beta\Gamma(\alpha)}(s_2-s_1)^{\alpha-1}
 (1-e^{-\beta(s_2-s_1)})
\nonumber\\
&\leq\frac{K_1^\ast}{\Gamma(\alpha)}
 \Big(\frac{1-\gamma}{\alpha-\gamma}(s_2-s_1)^{\frac{\alpha-\gamma}{1-\gamma}}
 \Big)^{1-\gamma}
 +\frac{K_2\rho}{\Gamma(\alpha)}(s_2-s_1)^{\alpha}
\nonumber\\
&\quad  +\frac{K_2\rho}{\beta\Gamma(\alpha)}(s_2-s_1)^{\alpha-1}(\beta(s_2-s_1)
+o(s_2-s_1))
\nonumber\\
 & \quad +\frac{K_1^\ast}{\Gamma(\alpha)}
 \left(\frac{1-\gamma}{\beta}\right)^{1-\gamma}(s_2-s_1)^{\alpha-1}
 \Big(\frac{\beta(s_2-s_1)}
 {1-\gamma}+o(s_2-s_1)\Big)^{1-\gamma}
\nonumber\\
&\to 0  \quad \text{as } s_2 \to s_1, \nonumber
\end{align}
where $\lim_{s_2-s_1\to 0}\Big(\frac{o(s_2-s_1)}{s_2-s_1}\Big)= 0$.

For $E_3$, by  H\"{o}lder's inequality and (H1), we find that
\begin{equation}
\begin{aligned}
E_3
&\leq\frac{1}{\Gamma(\alpha)}\Big(\int_{s_1}^{s_2}(s_2-\tau
 )^{\frac{\alpha-1}{1-\gamma}}
e^{\frac{-\beta(s_2-\tau)}{1-\gamma}}d\tau\Big)^{1-\gamma}
 \Big(\int_{s_1}^{s_2}(K_1(\tau))^{1/\gamma}d\tau\Big)^\gamma
\\
&\quad+\frac{K_2\rho}{\Gamma(\alpha)}\int_{s_1}^{s_2}(s_2-\tau)^{\alpha-1}
e^{-\beta(s_2-\tau)}d\tau
\\
&\leq\frac{K_1^\ast}{\Gamma(\alpha)}
 \Big(\frac{1-\gamma}{\alpha-\gamma}\Big)^{1-\gamma}(s_2-s_1)^{\alpha-\gamma}
+\frac{K_2\rho}{\Gamma(\alpha+1)}(s_2-s_1)^\alpha \\
&\to 0 \quad \text{ as } s_2 \to s_1.
\end{aligned}\label{eq3.9}
\end{equation}
Therefore, \eqref{eq3.6}-\eqref{eq3.9} imply that $T_n(B(\rho))$
is quasi-equicontinuous.

\subsection*{Step 3:  $T_n$ is continuous}

Let $\{u_n^j\}_{j=1}^\infty$ be a sequence such that $u_n^j\to v$ in $PC_b$
as $j\to\infty$. Since for any $\tau\in [0,b]$,
\begin{equation}
\begin{aligned}
\|u^j_{n\tau}-v_\tau\|_{PC_0}
&=\sup_{\theta\in[-h,0]}\|u_n^j(\tau+\theta)-v(\tau+\theta)\| \\
&\leq\sup_{t\in[0,b]}\|u_n^j(t)-v(t)\|\to 0\quad \text{as } j\to\infty,
\end{aligned}\label{eq3.10}
\end{equation}
by the weak continuity of the nonlinear terms $f$ and $I_k$, we obtain that
 for any $\tau\in[0,b]$,
\begin{equation}\label{eq3.11}
\lim_{j\to\infty}P_nf(\tau,u^j_{n\tau})=P_nf(\tau,v_{\tau}),
\end{equation}
and for each $k \in \mathbb{N}$,
\begin{equation}\label{eq3.12}
\lim_{j\to \infty}P_nI_k(u^j_n(t_k^-))=P_nI_k(v(t_k^-)).
\end{equation}
On the other hand, by (H1) and \eqref{eq3.10} we conclude that  for all
$\tau\in[0,b]$ and $j$ sufficiently large,
\begin{equation} \label{eq3.13}
\begin{aligned}
\|P_nf(\tau,u_{n\tau}^j)-P_nf(\tau,v_\tau)\|
& \leq  2K_1(\tau)+K_2\|u^j_{n\tau}\|_{PC_0}
+K_2\|v_\tau\|_{PC_0}  \\
& \le  2K_1(\tau)+C+2K_2\|v\|_{PC_b}.
\end{aligned}
\end{equation}
Then, by \eqref{eq3.3}-\eqref{eq3.5}, \eqref{eq3.11} and \eqref{eq3.13},
we deduce from Lebesgue's theorem that
\begin{equation}\label{eq3.14}
\lim_{j\to\infty}\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}\|P_nf(\tau,u^j_{n\tau})-P_nf(\tau,v_\tau)\|=0,
\end{equation}
and
\begin{equation}\label{eq3.15}
\lim_{j\to\infty}\int_{t_m}^{t}(t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}\|P_nf(\tau,u^j_{n\tau})-P_nf(\tau,v_\tau)\|=0.
\end{equation}
Then by \eqref{eq3.1}, \eqref{eq3.12} and \eqref{eq3.14}-\eqref{eq3.15},
we find that for any $t\in[0,b]$,
\begin{equation}
\begin{aligned}
&\|(T_nu_n^j)(t)-(T_nv)(t)\| \\
&\leq\sum_{0<t_k<t}\|P_nI_k(u_n^j(t_k^-))-P_nI_k(v(t_k^-))\|
\\
&\quad+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}
 (t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
\|P_nf(\tau,u^j_{n\tau})-P_nf(\tau,v_\tau)\|d\tau
\\
&\quad+\frac{1}{\Gamma(\alpha)}\int_{t_m}^{t}(t-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}\|P_nf(\tau,u^j_{n\tau})-P_nf(\tau,v_\tau)\|
d\tau\to 0
\end{aligned}\label{eq3.16}
\end{equation}
 as $j\to\infty$.
By the proof of Step 2, we see that $\{T_nu_n^j\}_{j=1}^\infty$ is a
quasi-equicontinuous family of functions. Hence, the Arzel\`a-Ascoli
theorem yields  $T_nu_n^j\to T_nv$ in $PC_b$.

As a consequence of Steps 1-3, and the Arzel\`a-Ascoli theorem, we can conclude that
 $T_n:PC([-h,b]; X_{(n)})\to PC([-h,b]; X_{(n)})$ is continuous and
completely continuous.

\subsection*{Step 4: A priori bounds}

 We show there exists an open set $U\subseteq PC([-h,b];X_{(n)})$
 with $u_n\neq\lambda T_nu_n$ for $\lambda\in(0,1)$ and $u_n\in\partial U$.
Let $u_n\in PC([-h,b];X_{(n)})$ and $u_n=\lambda T_nu_n$ for some
$0<\lambda<1$. Then for each $t\in[0,b]$ we have
\begin{equation} \label{eq3.17}
\begin{aligned}
u_n(t)
&=\lambda\Big\{\phi_n(0)e^{-\beta t}+\sum_{0<t_k<t} P_nI_k(u_n(t_k^-))
e^{-\beta(t-t_k)}\\
&\quad  +\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
 e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau \\
&\quad  +\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau\Big\},
\end{aligned}
\end{equation}
where  $t_m=\max\{t_k:t_k<t, k=0, 1, 2, \dots\}$. For $t \in [0, t_1]$,
replacing $t$ by $t+\theta$ (where $\theta \in [-h, 0]$) in \eqref{eq3.17},
and arguing as in the proof of \eqref{eq3.5}, in view of
(H1), (H3) and  H\"{o}lder's inequality, we obtain
\begin{align}
&\|u_n(t+\theta)\|  \nonumber \\
&\leq \|\phi(0)\|e^{-\beta(t+\theta)}
 +\frac{1}{\Gamma(\alpha)}\int_0^{t+\theta}(t+\theta-\tau)^{\alpha-1}
e^{-\beta(t+\theta-\tau)}\Big(K_1(\tau)+K_2\|u_{n\tau}\|_{PC_0}\Big)d\tau
\nonumber\\
&\leq \|\phi(0)\|e^{-\beta(t+\theta)}+\frac{1}{\Gamma(\alpha)}
\Big(\int_0^{t+\theta}(t+\theta-\tau)^{\frac{\alpha-1}{1-\gamma}}
 e^{\frac{-\beta(t+\theta-\tau)}{1-\gamma}}d\tau\Big)^{1-\gamma}
\nonumber\\
&\quad\times \Big(\int_0^{t+\theta}(K_1(\tau))^{1/\gamma}d\tau\Big)^\gamma
 \label{eq3.18}\\
& \qquad +\frac{K_2}{\Gamma(\alpha)}\Big(\int_0^{t+\theta}(t+\theta-\tau
)^{\frac{\alpha-1}{1-\gamma}}d\tau\Big)^{1-\gamma}
\Big(\int_0^{t+\theta} e^{\frac{-\beta(t+\theta-\tau)}{\gamma}}
\|u_{n\tau}\|_{PC_0}^{1/\gamma}d\tau\Big)^\gamma
\nonumber\\
& \leq \|\phi(0)\|e^{-\beta(t+\theta)}+C_1^\ast+C_2^\ast
\Big(\int_0^{t+\theta}e^{\frac{-\beta(t+\theta-\tau)}{\gamma}}\|u_{n\tau}\|_{PC_0}
^{1/\gamma}d\tau\Big)^\gamma, \nonumber
\end{align}
where we have used the notation
\begin{gather*}
C_1^\ast := \frac{K_1^\ast}{\Gamma(\alpha)}
\Big(\frac{1-\gamma}{\alpha-\gamma}\delta^{\frac{\alpha-\gamma}{1-\gamma}}
+\frac{\beta}{\alpha-\gamma}
\frac{\Gamma(\frac{\alpha+1-2\gamma}{1-\gamma})}{(\frac{\beta}{1-\gamma}
)^{\frac{\alpha+1-2\gamma}{1-\gamma}}}\Big)^{1-\gamma}, \\
 C_2^\ast := \frac{K_2}{\Gamma(\alpha)}
\Big(\frac{1-\gamma}{\alpha-\gamma}\Big)^{1-\gamma} \delta^{\alpha-\gamma}.
\end{gather*}
Note that if $t+\theta<0$, then
\[
\|u_n(t+\theta)\|=\|\phi_n(t+\theta)\|
\leq\|\phi\|_{PC_0}
\leq \|\phi\|_{PC_0}e^{-\beta(t+\theta)}.
\]
Therefore,
\[
\|u_{nt}\|_{PC_0}\leq e^{\beta h}\|\phi\|_{PC_0}
e^{-\beta t}+C_1^\ast+C_2^\ast e^{\beta h}
\Big(\int_0^te^{\frac{-\beta(t-\tau)}{\gamma}}
\|u_{n\tau}\|_{PC_0}^{1/\gamma}d\tau\Big)^\gamma,
\]
and
\[
e^{\frac{\beta t}{\gamma}}\|u_{nt}\|_{PC_0}^{1/\gamma}
\leq 3^{\frac{1-\gamma}{\gamma}}e^{\frac{\beta h}{\gamma}}
\|\phi\|^{1/\gamma}_{PC_0}+3^{\frac{1-\gamma}{\gamma}}
(C_1^\ast)^{1/\gamma}e^{\frac{\beta t}{\gamma}}
+C_3^\ast\int_0^te^{\frac{\beta\tau}{\gamma}}\|u_{n\tau}
\|_{PC_0}^{1/\gamma}d\tau,
\]
where we have used the notation
$$
C_3^\ast := 3^{\frac{1-\gamma}{\gamma}}(C_2^\ast)^{1/\gamma}
e^{\frac{\beta h}{\gamma}}.
$$
Applying  Gronwall's inequality, we have for $t \in [0, t_1]$,
\begin{equation}\label{eq3.19}
\begin{aligned}
\|u_{n t}\|_{PC_0}^{1/\gamma}
&\leq 2\times3^{\frac{1-\gamma}{\gamma}}e^{\frac{\beta h}{\gamma}}
\|\phi\|_{PC_0}^{1/\gamma}e^{(C_3^\ast-\frac{\beta }{\gamma})t}\\
&\quad +3^{\frac{1-\gamma}{\gamma}}(C_1^\ast)^{1/\gamma}
+3^{\frac{1-\gamma}{\gamma}}(C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|t}}{|\frac{\beta}{\gamma}
-C_3^\ast|},
\end{aligned}
\end{equation}
and thus
\begin{equation}\label{eq3.20}
\begin{aligned}
\|u_{n}(t_1)\|^{1/\gamma}
&\leq 2\times 3^{\frac{1-\gamma}{\gamma}}e^{\frac{\beta h}{\gamma}}
 \|\phi\|_{PC_0}^{1/\gamma}e^{(C_3^\ast-\frac{\beta }{\gamma})t_1} \\
&\quad +3^{\frac{1-\gamma}{\gamma}}(C_1^\ast)^{1/\gamma}
 +3^{\frac{1-\gamma}{\gamma}}(C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|t_1}}{|\frac{\beta}{\gamma}
-C_3^\ast|}=D_1^\ast.
\end{aligned}
\end{equation}
For $t\in(t_1,t_2]$, similar to \eqref{eq3.18} and \eqref{eq3.19},
in view of (H1)-(H3), we find for $t+\theta$ $>$ $t_1$
(where $\theta\in[-h, 0]$) that
\begin{align}
\|u_n(t+\theta)\|
&\le \|u_n(t_1^-)+I_1(u_n(t_1^-))\|e^{-\beta(t+\theta-t_1)} \nonumber\\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_1}^{t+\theta}(t+\theta-\tau)^{\alpha-1}
e^{-\beta(t+\theta-\tau)}\|f(\tau, u_{n\tau})\|d\tau \nonumber\\
&\leq(1+J_1)\|u_n(t_1^-)\|e^{-\beta(t+\theta-t_1)}+J_2e^{-\beta(t+\theta-t_1)}
 \label{eq3.21}\\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_1}^{t+\theta}(t+\theta-\tau)^{\alpha-1}
e^{-\beta(t+\theta-\tau)} \Big(K_1(\tau)+K_2\|u_{n\tau}\|_{PC_0}\Big)d\tau
 \nonumber\\
&\leq(1+J_1)\|u_n(t_1^-)\|e^{-\beta(t+\theta-t_1)}+J_2e^{-\beta(t+\theta-t_1)}
+C_1^\ast \nonumber\\
&\quad +C_2^\ast\Big(\int_{t_1}^{t+\theta}e^{\frac{-\beta(t+\theta-\tau)}{\gamma}}
\|u_{n\tau}\|_{PC_0}^{1/\gamma}d\tau\Big)^\gamma. \nonumber
\end{align}
Hence,
\begin{equation} \label{eq3.22}
\begin{aligned}
\|u_{n}(t+\theta)\|^{1/\gamma}
&\leq 3^{\frac{1-\gamma}{\gamma}}(1+J_1)^{1/\gamma}
e^{\frac{\beta h}{\gamma}}\|u_n(t_1^-)\|^{1/\gamma}
e^{-\frac{\beta(t-t_1)}{\gamma}} \\
&\quad +3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
 +C_3^\ast\int_{t_1}^te^{-\frac{\beta(t-\tau)}{\gamma}}
 \|u_{n\tau}\|_{PC_0}^{1/\gamma}d\tau.
\end{aligned}
\end{equation}
It follows from \eqref{eq3.19} and \eqref{eq3.20} that if $t \in (t_1, t_2]$
and $t+\theta \le t_1$, then we have
\begin{equation} \label{eq3.23}
\begin{aligned}
 \|u_{n}(t+\theta)\|^{1/\gamma}
& \leq 2\times 3^{\frac{1-\gamma}{\gamma}}
 e^{\frac{\beta h}{\gamma}}\|\phi\|_{PC_0}^{1/\gamma}
e^{(C_3^\ast-\frac{\beta }{\gamma})(t+\theta)}
 +3^{\frac{1-\gamma}{\gamma}}(C_1^\ast)^{1/\gamma}
\\
& \quad +3^{\frac{1-\gamma}
{\gamma}}(C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|(t+\theta)}}
{|\frac{\beta}{\gamma}-C_3^\ast|}
\\
& \leq  D_1^\ast e^{-\frac{\beta}{\gamma}(t+\theta-t_1)}.
\end{aligned}
\end{equation}
Using $\theta \in [-h, 0]$ we get from \eqref{eq3.22} and \eqref{eq3.23} that
\begin{equation} \label{eq3.24}
\begin{aligned}
e^{\frac{\beta}{\gamma}t}\|u_{nt}\|_{PC_0}^{1/\gamma}
&\leq 3^{\frac{1-\gamma}{\gamma}}(1+J_1)^{1/\gamma}D_1^\ast
e^{\frac{\beta h}{\gamma}}e^{\frac{\beta t_1}{\gamma}}
+3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
e^{\frac{\beta}{\gamma}t}
 \\
&\quad +C_3^\ast\int_{t_1}^te^{\frac{\beta \tau}{\gamma}}
\|u_{n\tau}\|_{PC_0}^{1/\gamma}d\tau.
\end{aligned}
\end{equation}
By using Gronwall's inequality, we have that for $t \in (t_1, t_2]$,
\begin{equation} \label{eq3.25}
\begin{aligned}
\|u_{n t}\|_{PC_0}^{1/\gamma}
&\leq 2 \times 3^{\frac{1-\gamma}{\gamma}}(1+J_1)^{1/\gamma}D_1^\ast
e^{\frac{\beta h}{\gamma}}e^{(C_3^\ast-\frac{\beta }{\gamma})(t-t_1)}
+3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
\\
&\quad +3^{\frac{1-\gamma}{\gamma}}
(J_2+C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|(t-t_1)}}
{|\frac{\beta}{\gamma}-C_3^\ast|},
\end{aligned}
\end{equation}
and consequently,
\begin{equation} \label{eq3.26}
\begin{aligned}
&\|u_{n}(t_2)\|^{1/\gamma}\\
&\leq 2 \times 3^{\frac{1-\gamma}{\gamma}}
(1+J_1)^{1/\gamma}D_1^\ast e^{\frac{\beta h}{\gamma}}
e^{(C_3^\ast-\frac{\beta }{\gamma})(t_2-t_1)}\\
& \quad +3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
 +3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast
-\frac{\beta}{\gamma}|(t_2-t_1)}}{|\frac{\beta}{\gamma}-C_3^\ast|}
=D_2^\ast.
\end{aligned}
\end{equation}
In  a similar way as above, we obtain that for $t \in (t_m, t_{m+1}]$ with
$m \ge 2$,
\begin{equation} \label{eq3.27}
\begin{aligned}
\|u_{n t}\|_{PC_0}^{1/\gamma}
&\leq 2 \times 3^{\frac{1-\gamma}{\gamma}}
(1+J_1)^{1/\gamma}D_m^\ast e^{\frac{\beta h}{\gamma}}
e^{(C_3^\ast-\frac{\beta }{\gamma})(t-t_m)}
+3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}\\
& \quad +3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|(t-t_m)}}
{|\frac{\beta}{\gamma}-C_3^\ast|},
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq3.28}
\begin{aligned}
&\|u_{n}(t_{m+1})\|^{1/\gamma} \\
& \leq 2 \times 3^{\frac{1-\gamma}{\gamma}}(1+J_1)^{1/\gamma}D_m^\ast
e^{\frac{\beta h}{\gamma}}e^{(C_3^\ast-\frac{\beta }{\gamma})(t_{m+1}-t_m)}
+3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}\\
&\quad + 3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|(t_{m+1}-t_m)}}
{|\frac{\beta}{\gamma}-C_3^\ast|}=D_{m+1}^\ast.
\end{aligned}
\end{equation}
For convenience, let
\begin{gather*}
B_1^\ast = 2 \times 3^{\frac{1-\gamma}{\gamma}}(1+J_1)^{1/\gamma}
 e^{\frac{\beta h}{\gamma}},  \\
B_2^\ast= 3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
 + 3^{\frac{1-\gamma}{\gamma}}(J_2+C_1^\ast)^{1/\gamma}
C_3^\ast \frac{1+e^{|C_3^\ast-\frac{\beta}{\gamma}|\delta}}
{|\frac{\beta}{\gamma}-C_3^\ast|}.
\end{gather*}
Then by using  mathematical induction, we find for $m \ge 2$ that
\begin{equation} \label{eq3.29}
\begin{aligned}
D_m^\ast & \leq  B_1^\ast D_{m-1}^\ast e^{(C_3^\ast
-\frac{\beta }{\gamma})(t_{m}-t_{m-1})}+B_2^\ast\\
&  \leq (B_1^\ast)^{m-1}  e^{(C_3^\ast-\frac{\beta }{\gamma})
(t_{m}-t_{1})}D_1^\ast +B_2^\ast\sum_{k=2}^m
e^{(C_3^\ast-\frac{\beta }{\gamma})(t_{m}-t_{k})}(B_1^\ast)^{m-k}.
\end{aligned}
\end{equation}
Note that (H3) implies  $(m-1)\eta$ $\le$ $t_m-t_1 \le (m-1)\delta$ and
$(m-k)\eta$ $\le$ $t_m-t_k \le (m-k)\delta$. It follows from \eqref{eq3.29} that
\begin{equation} \label{eq3.30}
\begin{aligned}
D_m^\ast
& \leq (B_1^\ast)^{\frac{t_m-t_1}{\eta}}  e^{(C_3^\ast
 -\frac{\beta }{\gamma})(t_{m}-t_{1})}D_1^\ast
+B_2^\ast\sum_{k=2}^m   e^{(C_3^\ast
 -\frac{\beta }{\gamma})(t_{m}-t_{k})}(B_1^\ast)^{\frac{t_m-t_k}{\eta}}
\\
&= e^{(C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta})(t_{m}-t_{1})}D_1^\ast
+B_2^\ast\sum_{k=2}^m   e^{|C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta}|(t_{m}-t_{k})}
\\
&\le  e^{(C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta})(t_{m}-t_{1})}D_1^\ast
+B_2^\ast\sum_{k=2}^m   e^{|C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta}|(m-k)\delta}
\\
&\le  e^{\left(C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta}\right)(t_{m}-t_{1})}D_1^\ast
+B_2^\ast\frac{e^{|C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta}|\frac{\delta}{\eta}(t_m-t_2)}}
{1-e^{-|C_3^\ast-\frac{\beta }{\gamma}+\frac{\ln B_1^\ast}{\eta}|\delta}}.
\end{aligned}
\end{equation}
Therefore, by \eqref{eq3.27} and \eqref{eq3.30} we deduce that for
$t \in (t_m, t_{m+1}]$ with $m \ge 2$,
\begin{align*}
\|u_{nt}\|_{PC_0}^{1/\gamma}
&  \leq B_1^\ast D_m^\ast e^{(C_3^\ast-\frac{\beta }{\gamma})(t-t_{m})}
+B_2^\ast \\
&\le C e^{(C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta})(t-t_{1})}D_1^\ast
 +Ce^{|C_3^\ast-\frac{\beta }{\gamma}
+\frac{\ln B_1^\ast}{\eta}| \frac{\delta}{\eta}(t-t_{2})}+C.
\end{align*}
Combining this with \eqref{eq3.19}, \eqref{eq3.20} and \eqref{eq3.25},
we obtain that for all $t \ge 0$,
\begin{equation} \label{eq3.31}
\|u_{nt}\|_{PC_0}^{1/\gamma}  \leq C\|\phi\|_{PC_0}^{1/\gamma}
e^{(C_3^\ast-\frac{\beta }{\gamma}+\frac{\ln B_1^\ast}{\eta})t}+Ce^{C t}+C.
\end{equation}
Hence, we can find a $W^\ast>0$ such that for all $t \in [0, b]$,
$$
\|u_{nt}\|_{PC_0}\leq W^\ast.
$$
Set
\[
U=\{u_n\in PC([-h,b]; X_{(n)}): \|u_n\|_{PC_b}<W^\ast+1\}.
\]
Note that $T_n:\overline{U}\to PC([-h,b]; X_{(n)})$ is continuous and
completely continuous. From the choice of $U$, there is no
$u_n\in \partial U$ such that $u_n=\lambda T_n(u_n)$ for
$\lambda\in(0,1)$. As a consequence of Theorem \ref{theorem2.3},
we deduce that $T_n$ has a fixed point $u_n$ in $U$, which is a local
solution of \eqref{ivp} in $X_{(n)}$.

\subsection*{Step 5: Existence of local solutions for \eqref{ivp} in $X$}

We pass now to the case of a general separable Hilbert space $X$.
 We form the approximating equations
\begin{equation}\label{eq3.32}
\begin{gathered}
 D_s^\alpha u_n(t)=P_n(f(t,u_{nt})), \quad t\geq 0,\; t\neq t_k,\\
u_n(s)=\phi_n(s), \quad \forall s\in[-h, 0],\\
u_n(t^+_k)-u_n(t_k^-)=P_n(I_k(u_n(t^-_k))), \quad k=1,2,\dots.\\
 \end{gathered}
\end{equation}
It follows from the preceding discussion that we may find a solution
$u_n$ of the approximating equation on $0\leq t\leq b$ such that for
all $n \in \mathbb{N}$,
  \begin{equation}\label{eq3.33}
  \sup_{t\in[0,b]}\|u_n(t)\|\leq C,
\end{equation}
and for $t,s\in (t_m,t_{m+1}]$ for each $m\in \{0,1,2,\dots\}$, we have
\begin{equation}\label{eq3.34}
\|u_n(t)-u_n(s)\|\leq C|e^{-\beta t}-e^{-\beta s}|+C|t-s|^{\alpha-\gamma}.
\end{equation}
Since $X$ is a Hilbert space, from \eqref{eq3.33} we deduce that for any
$t\in[0,b]$, $\{u_n(t)\}_{n=1}^\infty$ is relatively compact in $X_w$.
Using the diagonal method one can choose a subsequence of
$\{u_n(\cdot)\}_{n=1}^\infty$ and a function
$u:\big(\mathbb{Q}\cup \{t_k\}_{k=1}^\infty\big)\cap [0,b] \to X$
such that $u_n(t)\to u(t)$ in $X_w$ for any
$t \in \big(\mathbb{Q}\cup \{t_k\}_{k=1}^\infty\big)\cap [0,b]$. Since
$$
\|u(t)-u(s)\|\leq\liminf\|u^n(t)-u^n(s)\|
\leq C|e^{-\beta t}-e^{-\beta s}|+C|t-s|^{\alpha-\gamma}
$$
for all $t, s \in \mathbb{Q}\cap(t_m,t_{m+1}]$ and for each
 $m \in \{0, 1, 2,\dots\}$, the function $u$ can be extended to a
piecewise continuous function (denote again $u$ $:$ $[0,b]\to X$) such that
\begin{equation}\label{eq3.35}
\|u(t)-u(s)\|\leq C|e^{-\beta t}-e^{-\beta s}|+C|t-s|^{\alpha-\gamma}
\end{equation}
for all $t,s\in (t_m,t_{m+1}]$ and for each $m\in\{0,1,2,\dots\}$.

We shall prove that $u_n(s_0)\to u(s_0)$ in $X_w$. Indeed, for any
$s_0\in \big((t_m,t_{m+1})\setminus\mathbb{Q}\big)\cap[0,b]$ for some
$m\in \{0,1,2,\dots\}$ and $v\in  X^\ast$, we have
\[
\langle u_n(s_0)-u(s_0),v\rangle
=\langle u_n(s_0)-u_n(s_m),v\rangle+\langle u_n(s_m)-u(s_m),v\rangle
+\langle u(s_m)-u(s_0),v\rangle,
\]
where $s_m\in \mathbb{Q}$ are such that $s_m\to s_0$. For any
$\varepsilon>0$ there exist $m(\varepsilon)$ and
 $N(m(\varepsilon),\varepsilon)$ such that for all $n\geq N$,
\begin{gather*}
\big|\langle u_n(s_0)-u_n(s_m),v\rangle \big|
\leq\|u_n(s_0)-u_n(s_m)\|\|v\|<\frac{\varepsilon}{3}, \\
\big|\langle u(s_m)-u(s_0),v\rangle\big|
\leq\|u(s_m)-u(s_0)\|\|v\|<\frac{\varepsilon}{3}, \\
\big|\langle u_n(s_m)-u(s_m),v\rangle\big|<\frac{\varepsilon}{3}.
\end{gather*}
Thus, $|\langle u_n(s_0)-u(s_0),v\rangle|<\varepsilon$,
and consequently $u_n(s_0)\to u(s_0)$ in $X_w$. In fact,
we have that for any $s_0\in [0,b]\setminus\{t_k\}_{k=1}^\infty$,
\begin{equation}\label{eq3.36}
u_n(s_n)\to u(s_0)\quad \text{in } X_w \text{ if }s_n\to s_0,
\end{equation}
and for  $s_0=t_k\cap [0,b]$ for some $k=1,2,\dots,$
\begin{equation}\label{eq3.37}
u_n(s_n)\to u(s_0)\quad \text{ in } X_w \text{ if } s_n\leq s_0 \text{  and }
s_n\to s_0.
\end{equation}
By a similar argument, \eqref{eq3.36} and \eqref{eq3.37} can be obtained
from the equality
\[
\langle u_n(s_n)-u(s_0),v\rangle
=\langle u_n(s_n)-u_n(s_0),v\rangle+\langle u_n(s_0)-u(s_0),v\rangle.
\]
Then \eqref{eq3.36} and \eqref{eq3.37} imply that for any $\tau\in [0,b]$
and $s\in [-h,0]$, $u_{n\tau}(s_n)=u_n(\tau+s_n)\to u(\tau+s)=u_{\tau }(s)$ in
$X_w$ for $\tau+s_n\to \tau+s\in [0,b]\setminus\{t_k\}_{k=1}^\infty$ or
$\tau+s_n\to \tau+s$ with $\tau + s\in \{t_k\}_{k=1}^\infty$ and
$s_n$ $\leq$ $s$, so that $u_{n\tau}\to u_\tau$ in $PC_{0,w}$.

Finally, we show that $u(\cdot)$ is a solution of \eqref{ivp}.
 For this aim we will pass to the limit in the integral
\[
u_n(t)=\begin{cases}
\phi_n(t), &  t\in[-h,0],\\[4pt]
 \phi_n(0)e^{-\beta t}+\sum_{0<t_k<t} P_nI_k(u_n(t_k^-))e^{-\beta(t-t_k)} \\
+\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
e^{-\beta(t-\tau)} P_nf(\tau,u_{n\tau})d\tau \\
+\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau, & t\in[0,b],
\end{cases}
\]
where $t_m=\max\{t_k:k=0,1,2,\dots, t_k<t\}$. Since $f$ and $I_k$ are
 weakly continuous in bounded sets, for any $\tau\in[0,t]$ we have
\begin{equation}\label{eq3.38}
f(\tau,u_{n\tau})\to f(\tau,u_\tau) \quad \text{in $X_w$ as }n\to\infty,
\end{equation}
and for each $k$,
\begin{equation}\label{eq3.39}
I_k(u_n(t_k^-))\to I_k(u(t_k^-))\quad \text{in $X_w$ as } n\to\infty.
\end{equation}
Using the Riesz representation  theorem, we obtain that for any
$v\in X^\ast$, there exists an element $w \in X$ corresponding to
 $v$ such that
\[
\langle u,v\rangle=(u,w)\quad \text{for all } u\in X,
\]
in view  of $\|f(\tau,u_{n\tau})\|\leq K_1(\tau)+K_2C$ and
$\|I_k(u_n(t_k^-))\|\leq J_1C+J_2$, we have
\begin{equation}\label{eq3.40}
\begin{split}
&|\langle P_nf(\tau,u_{n\tau}),v\rangle-\langle f(\tau,u_{n\tau}),v\rangle|
=|(P_nf(\tau,u_{n\tau})- f(\tau,u_{n\tau}),w)|
 \\
& =|(f(\tau,u_{n\tau}), (I-P_n)w)|\leq(K_1(\tau)+K_2C)\|(I-P_n)w\|\to 0
\end{split}
\end{equation}
 as $n\to \infty$, and
\begin{equation}\label{eq3.41}
|\langle P_nI_k(u_n(t_k^-)),v\rangle-\langle I_k(u_n(t_k^-)),
v\rangle|\leq(J_1C+J_2)\|(I-P_n)w\|\to 0
\end{equation}
as $n \to \infty$.
Then by \eqref{eq3.38}, \eqref{eq3.40} and Lebesgue's theorem we obtain
for any $v\in X^\ast$ that
\begin{equation}\label{eq3.42}
\begin{split}
&\Big\langle \int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}P_nf(\tau,u_{n\tau})d\tau,v\Big\rangle
\\
&=\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}\langle
 P_nf(\tau,u_{n\tau}),v\rangle d\tau
\\
&\to\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}\langle
f(\tau,u_{\tau}),v\rangle d\tau
\\
&=\Big\langle\int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,v\Big\rangle,
\end{split}
\end{equation}
and in a similar way,
\begin{equation}\label{eq3.43}
\begin{aligned}
&\Big\langle \int_{t_m}^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
P_nf(\tau,u_{n\tau})d\tau,v\Big\rangle \\
&\to \Big\langle\int_{t_m}^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
f(\tau,u_\tau)d\tau,v\Big\rangle.
\end{aligned}
\end{equation}
Therefore, \eqref{eq3.39} and \eqref{eq3.41}-\eqref{eq3.43} imply that
for any $v\in X^\ast$,
 \begin{align*}
\langle u(t),v\rangle
&=\langle u(0)e^{-\beta t},v\rangle+\sum_{0<t_k<t}
 \langle I_k(u(t_k^-))e^{-\beta(t-t_k)},v\rangle
 \\
&\quad  +\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \Big\langle \int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}
  e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,v\Big\rangle
  \\
&\quad  +\frac{1}{\Gamma(\alpha)}
\Big\langle \int_{t_m}^t(t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau,
v\Big\rangle.
 \end{align*}
 As $v\in X^\ast$ is arbitrary, we get the equality
 \begin{align*}
 u(t)&=\phi(0)e^{-\beta t}+\sum_{0<t_k<t}I_k(u(t_k^-))
 e^{-\beta(t-t_k)}\\
&\quad +\frac{1}{\Gamma(\alpha)}\sum_{0<t_k<t}
 \int_{t_{k-1}}^{t_k}(t_k-\tau)^{\alpha-1}e^{-\beta(t-\tau)}
f(\tau,u_\tau)d\tau
\\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_m}^t(t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}f(\tau,u_\tau)d\tau  \quad \text{ for all } t\in[0,b].
\end{align*}
This completes the  proof of  Theorem \ref{theorem3.1}.

\section{Proof of Theorem \ref{thm3.3}}

\begin{proof}
Thanks to Theorem \ref{theorem3.1}, we  there exists at least
one solution $u^{(1)}(t)$ such that
\begin{equation}
u^{(1)}(t)=\begin{cases}
 \phi(t), & t\in[-h,0],\\
\phi(0)e^{-\beta t}+\frac{1}{\Gamma(\alpha)}
\int_0^t (t-\tau)^{\alpha-1}e^{-\beta(t-\tau)}f(\tau,u^{(1)}_\tau)d\tau,
& t\in[0,t_1].
\end{cases}
\end{equation}
Arguing as in the proof of Theorem \ref{theorem3.1}, we obtain the
 existence of $u^{(2)}(t)$ satisfying
\begin{equation}
u^{(2)}(t)=\begin{cases}
 u^{(1)}(t), & t\in[t_1-h,t_1],\\[4pt]
(u^{(1)}(t_1^-)+I_1(u^{(1)}(t_1^-)))
e^{-\beta (t-t_1)}\\
+\frac{1}{\Gamma(\alpha)}\int_{t_1}^t (t-\tau)^{\alpha-1}
e^{-\beta(t-\tau)}f(\tau,u^{(2)}_\tau)d\tau, & t\in(t_1,t_2].
\end{cases}
\end{equation}
Continuing in this way, we obtain  a global solution of \eqref{ivp}
 in the sense of Definition \ref{def2.1}.
\end{proof}

\section{Proof of Theorem \ref{theorem3.4}}

 By Theorem \ref{thm3.3} there exists at least one solution defined on
$[0, \infty)$. We will show that this solution is unique.

If  $u(\cdot)$, $v(\cdot)$  are two solutions of problem \eqref{ivp} with the
 initial value $\phi$,
 then for $t\in[0,t_1]$, by Definition \ref{def2.1} and (H4), we get
\begin{equation}\label{eq3.44}
\|u(t)-v(t)\|\leq \frac{M_1}{\Gamma(\alpha)}\int_0^t(t-\tau)^{\alpha-1}
e^{-\beta (t-\tau)}\|u_\tau-v_\tau\|_{PC_0}d\tau.
\end{equation}
Let $\theta\in [-h, 0]$. Replacing $t$ by $t+\theta$ in \eqref{eq3.44},
noticing that $\|u(t+\theta)-v(t+\theta)\|=0$ if $t+\theta$ $<$ $0$,
and for $t+\theta$ $\ge $ $0$,  using H\"{o}lder's inequality we have
\begin{align*}
&\|u(t+\theta)-v(t+\theta)\| \\
&\leq\frac{M_1}{\Gamma(\alpha)}
\Big(\int_0^{t+\theta}(t+\theta-\tau)^{(\alpha-1)p}d\tau\Big)^{1/p}
\Big(\int_0^{t+\theta}e^{-\beta(t+\theta-\tau)q}\|u_\tau-v_\tau\|^q_{PC_0}d\tau
\Big)^{1/q},
\end{align*}
where $p$, $q>1$, $(\alpha-1)p >-1$ and $\frac{1}{p}+\frac{1}{q}=1$. Thus,
\begin{align*}
\|u_t-v_t\|_{PC_0}^q
&=\sup_{\theta \in [-h, 0]}\|u(t+\theta)-v(t+\theta)\|^q \\
&\leq \frac{M_1^q\delta^{\alpha q-1}}{\Gamma^q(\alpha)(p\alpha-p+1)^{q/p}}
e^{\beta q h}\int_0^te^{-\beta q(t-\tau)}\|u_\tau-v_\tau\|_{PC_0}^qd\tau.
\end{align*}
By Gronwall's inequality we find that
\begin{equation}\label{eq3.45}
 \|u_t-v_t\|_{PC_0}=0\quad \text{for all }  t \in [0, t_1].
\end{equation}
By using  mathematical induction and arguing in a  similar way as above,
in view of (H5) and \eqref{eq3.45},  we obtain that for $t\in (t_m,t_{m+1}]$
with $m \ge 1$,
\[
\|u_t-v_t\|_{PC_0}^q\leq \frac{M_1^q\delta^{\alpha q-1}}
{\Gamma^q(\alpha)(p\alpha-p+1)^{q/p}}e^{\beta q h}\int_{t_m}^t
e^{-\beta q(t-\tau)}\|u_\tau-v_\tau\|_{PC_0}^qd\tau.
\]
Again by Gronwall's inequality, we have that
\begin{equation}\label{eq3.46}
 \|u_t-v_t\|_{PC_0}=0\quad \text{for all } t \in (t_m, t_{m+1}].
\end{equation}
Since $m$ is arbitrary, we deduce that $u \equiv v$.

\subsection*{Acknowledgments}
The authors express their sincere thanks to the anonymous reviewer
for his/her careful reading of the paper, giving valuable comments
and suggestions. It is their contributions that greatly improve the
paper. The authors also thank the editors for their kind help.
This work was supported by NSF of China (Grant Nos. 11571153 and 11571125),
the Fundamental Research Funds for the Central Universities under
Grant Nos. lzujbky-2017-ct01  and lzujbky-2016-ct12, the Scientific Research
Foundation for the Returned Overseas Chinese Scholars, State Education Ministry,
and Creative Experimental Project of National Undergraduate Students under
Grant No. 201510730062.

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