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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 241, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/241\hfil piecewise continuous mild solutions]
{Existence of piecewise continuous mild solutions for impulsive 
functional differential equations with iterated deviating arguments}

\author[P. Kumar, D. N. Pandey, D. Bahuguna \hfil EJDE-2013/241\hfilneg]
{Pradeep Kumar, Dwijendra N. Pandey, Dhirendra Bahuguna}  


\address{Pradeep Kumar \newline
 Department of Mathematics and Statistics, Indian Institute of
Technology Kanpur, Pin 208016, India}
\email{prdipk@gmail.com}

\address{Dwijendra N. Pandey \newline
Department of Mathematics, Indian Institute of Technology Roorkee,
Pin 247667, India}
\email{dwij.iitk@gmail.com}

\address{Dhirendra Bahuguna \newline
Department of Mathematics and Statistics, Indian Institute of
Technology Kanpur, Pin 208016, India}
\email{dhiren@iitk.ac.in,  Tel. +91-512-2597053, Fax +91-512-2597500}

\thanks{Submitted  May 20, 2013. Published October 31, 2013.}
\subjclass[2000]{34K45, 34A60, 35R12, 45J05}
\keywords{Impulsive functional differential equation; 
 deviating argument; \hfill\break\indent 
analytic semigroup; fixed point theorem}

\begin{abstract}
 The objective of this article is to prove the existence of piecewise
 continuous  mild solutions to impulsive functional differential equation
 with iterated deviating arguments in a Banach space.
 The results are obtained by using the theory of analytic semigroups  and
 fixed point theorems.
\end{abstract}

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


\section{Introduction}

In the theory of differential equations with deviating arguments, we
study differential equations involving variables (arguments) as
well as unknown functions and its derivative;  generally speaking,
under different values of the variables (arguments). It is very
important and significant branch of nonlinear analysis with numerous
applications to physics, mechanics, control theory, biology,
ecology, economics, theory of nuclear reactors, engineering, natural
sciences, and many other areas of science and technology. A
comprehensive coverage can be found in the book by El'sgol'ts and
Norkin \cite{norkin}.  For more details  on recent works in this direction,
we refer to  \cite{gal,rhd,haloi12,haloi12d,haloi12j} and
the references cited therein.

Impulsive effects are common phenomena due to short-term
perturbations whose duration is negligible in comparison with the
total duration of the original process. The governing equations of
such phenomena may be  modelled as impulsive differential equations.
In recent years, there has been a growing interest in the study of
impulsive differential equations as  these equations approach  the
simulation processes in the control theory, physics, chemistry,
population dynamics, biotechnology, economics and so on. The
investigation of existence and uniqueness of mild solutions for
impulsive differential equations have been discussed by many authors
(see \cite{cynj,dwwy,rmn,cya,bmh,mf,mgm,lzlj,wwc,ygs,jt,jt2,wgp,bdd,bdd2}
and references cited therein).

However, due to theoretical and practical difficulties, the study of
impulsive differential equations with deviating arguments has been
developed rather slowly. Recently, the study of impulsive
differential equations with deviating arguments has studied by
 some authors.  Guobing et al. \cite{ygs}
established the existence solution of periodic boundary value
problems for a class of impulsive neutral differential equations
with multi-deviation arguments. Jankowski \cite{jt2} discussed
the existence of solutions for second-order impulsive differential
equations with deviating arguments (see also \cite{lzlj, wwc, jt,
wgp, bdd, bdd2} and the references cited therein).


Liu \cite{ljh} studied the existence of mild solutions of the
following impulsive evolution equation by using semigroup theory
\begin{equation} \label{e1.1}
\begin{gathered}
u'(t) = Au(t) + f(t, u(t)), \quad 0 < t < T_0, \; t\neq t_i, \\
\Delta u(t_i) = I_i(u(t_i))=u(t_i^+)-u(t_i^-),\quad i=1,2,\dots ,\;
0 < t_1 < t_2< \dots  < T_0, \\
u(0)=u_0,
\end{gathered}
\end{equation}
where $A$ is the infinitesimal generator of a $C_0$-semigroup,
$u(t_i^+),u(t_i^-)$ represent the right and left hand limits
respectively at $t=t_i$, $I_i$'s are some operators and $f$ is a
suitable function.


Chang et al. \cite{{cynj}} investigated the existence of
mild solutions for the following impulsive problem
\begin{gather} \label{e1.2}
\begin{aligned}
\frac{d}{dt}(u(t)-F(t,u(h_1(t))))
&\in  A\Big[u(t)+\int_0^t f(t-s)u(s)ds\Big] \\
&\quad +G(t,u(h_2(t)))),\quad t\in[0,a],\; t\neq t_k,
\end{aligned}\\
\Delta u|_{t=t_k}= I_k(u(t_k^-)),\quad k=1,\dots ,m, \label{e1.3}\\
u(0)+g(u)= u_0, \label{e1.4}
\end{gather}
where $A$ is the infinitesimal generator of a
compact analytic resolvent operator $R(t)$, $t>0$ in a Banach space
$H$, $u_0\in H$, $0=t_0<t_1<\dots  t_m<t_{m+1}=a$,
$\Delta u|_{t=t_k}=u(t_k^+)-u(t_k^-)$, $u(t_k^+),u(t_k^-)$ represent the
right and left hand limits respectively at $t=t_k$.
$h_i:[0,a]\to[0,a]$, $i=1,2$, $f(t)$, $t\in[0,a]$ is a bounded linear
operator, and $F,G,g$ are suitable functions.

With strong motivation from the above work, in this article, we
consider the following impulsive functional differential equations
with iterated deviating arguments in a Banach space $(H,\|\cdot\|)$:
\begin{equation} \label{mainp}
\begin{gathered}
{\frac{d}{dt}}u(t)+Au(t) =f(t,u(t),u[w_1(t,u(t))]),\quad
  t\in I=[0,T_0],\; t \neq t_k,\\
\Delta u|_{t=t_k}= I_k(u(t_k^{-})),\; k=1,2,\dots ,m, \\
u(0)=u_0,
\end{gathered}
\end{equation}
where
$$
w_1(t,u(t))=h_1\big(t,u(h_2(t,\dots ,u(h_m(t,u(t)))\dots ))\big)
$$
 and $-A$ is the infinitesimal generator of an analytic semigroup of
bounded linear operators, $\{S(t),t\ge 0\}$ on $H$. Functions $f$
and $h_i$ are suitably defined and satisfying certain conditions to
be stated later. $0 = t_0 < t_1 < \dots  < t_m < t_{m+1} = T_0$,
$I_k \in C(H, H) (k = 1, 2, \dots  ,m)$, are bounded functions and
$\Delta u|_{t=t_k} = u(t^+_ k ) - u(t^-_ k )$, $u(t^-_ k )$ and
$u(t^+_ k )$ represent the left and right limits of $u(t)$ at
$t = t_k $, respectively.

The paper is organized as follows. In Section 2,
 we provide some basic definitions, notation,
lemmas and proposition which are used throughout the paper. In
Sections 3 and 4, we  prove the existence and uniqueness results
concerning the piecewise continuous mild solutions. In the
last section, we give an example to
demonstrate the application of the main results.


\section{Preliminaries and assumptions}

In this section, we will
introduce some basic definitions, notation, lemmas and proposition
which are used throughout this paper.

It is assumed that $-A$ generates an analytic semigroup of bounded
operators, denoted by $S(t)$, $t \ge 0$.
It is  known that there
exist constants $\tilde{M}\geq 1$ and $\omega \ge 0$ such that
$$
\|S(t)\| \leq\tilde{M}e^{\omega t}, \quad t \geq 0.
$$

If necessary,  we may assume without loss of generality that
$\|S(t)\|$ is uniformly bounded by $M$; i.e., $\|S(t)\|\leq M$ for
$t\ge 0$, and that $0 \in \rho(-A)$; i.e., $-A$ is invertible. In
this case, it is possible to define the fractional power
$A^{\alpha}$ for $0 \le \alpha \le 1$ as closed linear operator with
domain $D(A^{\alpha})\subseteq H$. Furthermore, $D(A^{\alpha})$ is
dense in $H$ and the expression
$$
\|x\|_{\alpha}=\|A^{\alpha}x\|,
$$
defines a norm on $D(A^{\alpha})$. Henceforth, we denote the
space $D(A^{\alpha})$  by $H_{\alpha}$ endowed with the norm
$\|\cdot\|_{\alpha}$. Also, for each $\alpha>0$, we define
$H_{-\alpha}=(H_\alpha)^{*}$, the dual space of $H_\alpha$, is a
$\|x\|_{-\alpha}=\|A^{-\alpha}x\|$. For more details, we refer to
the book by Pazy \cite{pazy}.


\begin{lemma}[{\cite[pp. 72,74,195-196]{pazy}}] \label{bslem1}
Suppose that $-A$ is the infinitesimal generator of an analytic
semigroup $S(t)$, $t \ge 0$ with $\|S(t)\| \le M$ for $t \ge 0$ and
$0 \in \rho(-A)$.
Then we have the following:
\begin{itemize}
\item[(i)] $H_{\alpha}$ is a Banach space for $0 \le \alpha \le 1$;

\item[(ii)]  For any $0<\delta\le\alpha$ implies $D(A^\alpha)\subset
D(A^\delta)$, the embedding $H_{\alpha} \hookrightarrow H_{\delta}$
is continuous;
\item[(iii)] The operator $A^\alpha S(t)$ is bounded for every $t>0$ and
$$
\|A^{\alpha}S(t)\|\leq C_{\alpha}t^{-\alpha}.
$$
\item[(iv)] For $\alpha\le 0$, $A^\alpha$ is bounded.
\end{itemize}
\end{lemma}


We define the space
\begin{align*}
X=\mathcal{PC}(H_\alpha)
=\Big\{&u:[0,T_0]\to H_\alpha:u\in
C((t_k,t_{k+1}],H_\alpha),k=0,1,\dots ,m,  \\
& u(t^-_k),u(t^+_k) \text{ exist and }
u(t^-_k)=u(t_k)\Big\},
\end{align*}
is a Banach space endowed with the
supremum norm
$$
\|u\|_{\mathcal{PC}} :=\sup_{t\in I}\|u(t)\|_\alpha.
$$
 We shall use the
following conditions on $f$ and $h_i$ in its arguments:
\begin{itemize}
  \item [(H1)]  Let $W\subset \operatorname{Dom}(f)$ be an open subset of
$\mathbb{R_{+}}\times H_\alpha \times H_{\alpha-1}$, where
$0\le \alpha<1$. For each $(t,u,v)\in W$, there is a neighborhood
$V_1\subset W$ of $(t,u,v)$, such that the nonlinear map
$f:\mathbb{R_{+}}\times H_\alpha \times H_{\alpha-1} \to H$
satisfies the  condition
\[
\|f(t,u,v)-f(s,u_1,v_1)\| \le L_f
\{|t-s|^{\theta_1}+\|u-u_1\|_{\alpha}+ \|v-v_1\|_{\alpha-1} \}
\]
for all $(t,u,v),(s,u_1,v_1)\in V_1$, where $L_f=L_{f}(t,u,v,V_{1})> 0$
and $0<\theta_1\le 1$ are constants.

\item[(H2)]  Let $U_{h_i}\subset \operatorname{Dom}(h_i)$ be open
subsets of $\mathbb{R}_{+} \times H_{\alpha -1} $,
where  $0\le \alpha<1$.
 For each $(t,u)\in U_{h_i}$, there is a neighborhood $V_{h_i}\subset
U_{h_i}$ of $(t,u)$, such that $h_i(0,\cdot )=0$ for each
$i=1,2,\dots ,m$, $h_i: \mathbb{R_{+}} \times H_{\alpha-1}
\to \mathbb{R_{+}}$ satisfies the  condition
\begin{equation} \label{e2.1}
|h_i(t,u)-h_i(s,v)| \le
L_{h_i}\{\|u-v\|_{\alpha-1}+|t-s|^{\theta_2}\}
\end{equation}
for all  $(t,u),  (s,v)\in V_{h_i}$,
$L_{h_i}=L_{h_i}(t,u,V_{h_i})> 0$ and $0<\theta_2\le 1$ are constants.

\item[(H3)] The functions $I_k:H_\alpha\to H_\alpha$ are
continuous and there exists $\Upsilon_k$ such that
$\|I_k(u)\|_\alpha\le\Upsilon_k$, $k=1,2,\dots ,m$.

\item[(H4)] There exist continuous nondecreasing
$d_k:\mathbb{R}_+\to\mathbb{R}_+$ such that
$\|I_k(u)-I_k(v)\|_\alpha\le d_k\|u-v\|_\alpha,k=1,2,\dots ,m$.
\end{itemize}

\subsection*{New concept of solutions}
Here, we prove a new concept of solutions \cite{mf,ljh} for the  problem
\begin{equation}
\begin{gathered}
u'(t)+Au(t)=r(t),\quad t\in[0,T_0], \; t\neq t_k,\\
u(0)=u_0\\
u(t_k)=I_k(u(t_k^-)),\quad k=1,2\dots ,m,
\end{gathered}\label{mild0}
\end{equation}
where $r\in H$. Let
\begin{equation}
\begin{gathered}
v'(t)+A v(t)= r(t),\quad t\in[0,T_0],\\
v(0)= v_0,
\end{gathered}\label{mild}
\end{equation}
 and
\begin{equation}
 \begin{gathered}
w'(t)+A w(t)=0,\quad t\in[0,T_0],\; t\neq t_k,\\
w(0)=0,\\
w(t_k)=I_k(u(t_k^-)),\quad k=1,2,\dots ,m,
\end{gathered}\label{mild2}
\end{equation}
be the decomposition of $u(.)=v(.)+w(.)$, where $v$ is
the continuous mild solution of \eqref{mild} and $w$ is the
piecewise continuous mild solution of \eqref{mild2}.

By a mild solution for \eqref{mild}, we mean a continuous function
$v:[0,T_0]\to H$ satisfying the  integral equation
\begin{equation} \label{e2.5}
v(t)=S(t)v_0+\int_0^t S(t-s)r(s)ds,\quad t\in [0,T_0].
\end{equation}
and by a piecewise continuous mild solution for \eqref{mild2}, we mean
a function $w\in \mathcal{PC}([0,T_0],D(A))$ satisfying
the  integral equation equivalent to the system
\eqref{mild2} (\cite[see Eq. (3.4)]{mf})
\begin{equation}
 w(t)=\begin{cases}
-\int_0^t Aw(s)ds, & t\in[0,t_1],\\
I_1(u(t^-_1))-\int_0^t Aw(s)ds, & t\in(t_1,t_2],\\
\sum_{i=1}^k I_i(u(t^-_i))-\int_0^t
Aw(s)ds, &t\in(t_k,t_{k+1}],\\
k=1,2,\dots ,m.
\end{cases}\label{mild3}
\end{equation}
The above equation can be expressed as
\begin{equation} 
 w(t)=\sum_{i=1}^k\chi_i(t)I_i(w(t_i^-))-\int_0^t
Aw(s)ds,\label{mild4}
\end{equation}
for $t\in[0,T_0]$, where
\begin{equation} \label{e2.8}
 \chi_i(t)=\begin{cases}
0,&\text{for } t\in[0,t_1],\\
1,&\text{for } t\in[t_k,t_{k+1}].
\end{cases}
\end{equation}

Taking Laplace transform of \eqref{mild4}, we obtain
\[
w(p)=\sum_{i=1}^k \frac{e^{-t_i p}}{p}I_i-\frac{Aw(p)}{p},
\]
this gives
\begin{equation} \label{e2.9}
w(p)=\sum_{i=1}^k {e^{-t_i p}}(pI+A)^{-1}I_i.
\end{equation}
We also note that $(pI+A)^{-1}=\int_0^\infty e^{-pt}S(t)dt$.
Thus we can obtain the mild solution for \eqref{mild2},
 \begin{equation} \label{e2.10}
w(t)=\sum_{i=1}^k\chi_i(t)S(t-t_i)I_i(w(t_i^-)).
\end{equation}
Hence, the mild solution for the problem \eqref{mild0} is
\begin{equation}
u(t)=S(t)u_0+\sum_{i=1}^k
\chi_i(t)S(t-t_i)I_i(u(t_i^-))+\int_0^t S(t-s)r(s)ds.\label{p1}
\end{equation}
We can rewrite equation \eqref{p1} as
\begin{equation}  \label{e2.12}
u(t)=\begin{cases}
S(t)u_0+\int_0^t S(t-s)r(s)ds,& t\in[0,t_1],\\
S(t)u_0+S(t-t_1)I_1(u(t^-_1))+\int_0^t S(t-s)r(s)ds,
 & t\in(t_1,t_2],\\
 S(t)u_0+\sum_{i=1}^k S(t-t_i)I_i(u(t^-_i))+\int_0^t
S(t-s)r(s)ds, &t\in(t_k,t_{k+1}],\\
k=1,2,\dots ,m.
\end{cases} %\label{2}
\end{equation}

\section{Local existence of mild solutions}

In this section, we  prove the  existence and uniqueness results concerning
piecewise continuous-mild solutions for system \eqref{mainp}.
For  $0\le \alpha<1$, we define
\[
 X_1=\{u\in X: \|u(t)-u(s)\|_{\alpha-1}\le L|t-s|,\forall
 t,s\in (t_k,t_{k+1}],k=0,1,\dots ,m\},
\]
which is needful for proving contraction principle (See \cite{gal,rhd}).
Where $L$ is a suitable positive constant to be specified later.

\begin{definition} \label{a}\rm
 A function $u:[0,T_0]\to H$ solution of the
problem \eqref{mainp},
\begin{equation} \label{e3.1}
u(t)=\begin{cases}
S(t)u_0+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds,& t\in[0,t_1],\\[4pt]
S(t)u_0+S(t-t_1)I_1(u(t^-_1))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, &t\in(t_1,t_2],\\
\dots\\
S(t)u_0+\sum_{i=1}^k S(t-t_i)I_i(u(t^-_i))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, &
t\in(t_k,t_{k+1}],\\
k=1,2,\dots ,m.
\end{cases} %\label{2}
\end{equation}
is said to be a piecewise continuous-mild solution.
\end{definition}

 For a fixed $R>0$, we define
$$
\mathcal{W}=\{u\in X\cap X_1:u(0)=u_0,
\|u-u_0\|_{\mathcal{PC}} \le R\}.
$$
Clearly, $ \mathcal{W}$ is a closed and bounded subset of $X_1$
and  is a Banach space.
We define a map $\mathcal{G}:\mathcal{W}\to\mathcal{W}$ by
\begin{equation} \label{e3.2}
(\mathcal{G}u)(t)= \begin{cases}
S(t)u_0+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds,&
t\in[0,t_1],\\[4pt]
 S(t)u_0+S(t-t_1)I_1(u(t^-_1))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, & t\in(t_1,t_2],\\
\dots\\
 S(t)u_0+\sum_{i=1}^k S(t-t_i)I_i(u(t^-_i))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, &t\in(t_k,t_{k+1}],\\
k=1,2,\dots ,m.
\end{cases} %\label{3}
\end{equation}

\begin{theorem} \label{b}
Let $0\le\alpha<1$, $u_0\in H_\alpha$ and the
assumptions {(H1)--(H4)} hold. Then  problem \eqref{mainp} has a
mild solution provided that
\begin{gather}
C_{\alpha}N{T_0^{1-\alpha}\over 1-\alpha}+M~\sum_{i=1}^k
\Upsilon_i\le \frac{R}{2},\label{assum2}
\\ \label{e3.4}
C_{\alpha }L_f(2+L~L_h ){T_0^{1-\alpha}\over 1-\alpha}+M\sum_{i=1}^m
d_i< 1,
\end{gather}
where $N$ and $L_h$ are  positive constants to be specified
later.
\end{theorem}

\begin{proof}
 Clearly, $\mathcal{G}:X\to X$. We begin with showing that
 $\mathcal{G}u\in X_1$ for each $u\in X_1$.
Let $u\in X_1$, then for each
$\tau_1,\tau_2\in[0,t_1]$, $\tau_1<\tau_2$ and $0\le \alpha<1$, we
have
\begin{equation}
\begin{aligned}
& \|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1} \\
&\leq \|[S(\tau_2)-S(\tau_1)]u_0\|_{\alpha-1} \\
&\quad +\int_0^{\tau_1}\|A^{\alpha-1}[S(\tau_2-s)-S(\tau_1-s)]\|\,
\|f(s,u(s),u(w_1(s,u(s))))\|ds\\
&\quad+\int_{\tau_1}^{\tau_2}\|A^{\alpha-1}S(\tau_2-s)\|\,
\|f(s,u(s),u(w_1(s,u(s))))\|ds.
\end{aligned}\label{c}
\end{equation}

Since $f(t,u(t),u(w_1(u(t),t)))$ is continuous, by
assumptions (H1), (H2), there exists a constant $N$, such that
\begin{equation}
\|f(t,u(t),u(w_1(t,u(t))))\|\le N,\quad u\in X,\; t\in[0,T_0].\label{b1}
\end{equation}
For the first term on the right-hand side of \eqref{c}, we have
\begin{equation}
\begin{aligned}
\|A^{\alpha-1}[S(\tau_2)-S(\tau_1)]u_0\|
&\le \int_{\tau_1}^{\tau_2}\|A^{\alpha-1}S'(s)u_0\|ds \\
&= \int_{\tau_1}^{\tau_2}\|A^\alpha
S(s)u_0\|ds \\
&\le M\|u_0\|_\alpha(\tau_2-\tau_1).
\end{aligned}\label{1}
\end{equation}
For the second term on the right-hand side of \eqref{c}, we have
\begin{equation}
\begin{aligned}
\|(S(\tau_2-s)-S(\tau_1-s))\|_{\alpha-1}
&\le
\int_0^{\tau_2-\tau_1}\|A^{\alpha-1}S^{\prime}(l)S(\tau_1-s)\|dl  \\
&= \int_0^{\tau_2-\tau_1}\|S(l)A^{\alpha}S(\tau_1-s)\|dl  \\
& \le  MC_{\alpha}(\tau_2-\tau_1)(\tau_1-s)^{-\alpha}.
\end{aligned}\label{new2}
\end{equation}
 Then using  \eqref{new2}, we obtain
the following bound for the second term on the right-hand side
of \eqref{c},
\begin{equation}
\begin{aligned}
&\int_0^{\tau_1}\|(S(\tau_2-s)-S(\tau_1-s))A^{\alpha-1}\|\,
\|f(s,u(s),u(w_1(s,u(s))))\|ds \\
&\le N M C_{\alpha}{T_0^{1-\alpha}\over {1-\alpha}}(\tau_2-\tau_1).
\end{aligned}\label{new3}
\end{equation}
The third term on the right-hand side of \eqref{c} is estimated as
\begin{equation}
\int_{\tau_1}^{\tau_2}\|S(\tau_2-s)A^{\alpha-1}\|
\|f(s,u(s),u(w_1(s,u(s))))\|ds \leq \|A^{\alpha-1}\|M N
(\tau_2-\tau_1).\label{new4}
\end{equation}
Thus from  inequalities \eqref{1}, \eqref{new3} and \eqref{new4},
we see that
\begin{equation}
\|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1}
\le M\{\|u_0\|_\alpha +N~C_\alpha\frac{T_0^{1-\alpha}}{1-\alpha}
+N \|A^{\alpha-1}\|\}(\tau_2-\tau_1).  \label{5}
\end{equation}
For $\tau_1,\tau_2\in(t_1,t_2]$, $\tau_1<\tau_2$ and $0\le \alpha<1$,
we have
\begin{equation}
\begin{aligned}
&\|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1} \\
&\leq \|[S(\tau_2-S(\tau_1)]u_0\|_{\alpha-1}+\|A^{\alpha-1}
 [S(\tau_2-t_1)-S(\tau_1-t_1)]I_1(u(t_1^-))\| \\
&\quad+\int_0^{\tau_1}\|A^{\alpha-1}[S(\tau_2-s)-S(\tau_1-s)]\|\,
 \|f(s,u(s),u(w_1(s,u(s))))\|ds \\
&\quad+\int_{\tau_1}^{\tau_2}\|A^{\alpha-1}S(\tau_2-s)\|\,
\|f(s,u(s),u(w_1(s,u(s))))\|ds.
\end{aligned}\label{d}
\end{equation}
The second term on the right-hand side of \eqref{d} is estimated as
\begin{equation}
\begin{aligned}
&\|A^{\alpha-1}[S(\tau_2-t_1)-S(\tau_1-t_1)]I_1(u(t_1^-))\|\\
&\le \int_{\tau_1}^{\tau_2}
\|A^{\alpha-1}S'(l-t_1)I_1(u(t_1^-))\|dl \\
&= \int_{\tau_1}^{\tau_2}\|A^{\alpha}S(l-t_1)I_1(u(t_1^-))\|dl \\
&\le M \|I_1(u(t_1^-))\|_\alpha(\tau_2-\tau_1).
\end{aligned}\label{4}
\end{equation}
Thus, from  inequalities \eqref{1}, \eqref{new3}, \eqref{new4}
and \eqref{4}, we obtain
\begin{equation}
\begin{aligned}
& \|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1} \\
& \le M\Big\{\|u_0\|_\alpha+\Upsilon_1
+N C_\alpha\frac{T_0^{1-\alpha}}{1-\alpha}
 +N \|A^{\alpha-1}\|\Big\}(\tau_2-\tau_1).
\end{aligned}\label{5b}
\end{equation}
Similarly, for $\tau_1,\tau_2\in(t_k,t_{k+1}]$, $\tau_1<\tau_2$,
$k=1,2,\dots ,m$ and $0\le \alpha<1$, we have
\begin{equation}
\begin{aligned}
& \|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1} \\
&\le M\Big\{\|u_0\|_\alpha+\sum_{i=1}^k \Upsilon_i
+NC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha}+N\|A^{\alpha-1}\|\Big\}
(\tau_2-\tau_1).
\end{aligned}\label{5c}
\end{equation}
Thus, for each $\tau_1,\tau_2\in[0,T_0]$, $\tau_1<\tau_2$ and
$0\le\alpha<1$, we have
\begin{equation} \label{e3.16}
\|(\mathcal{G}u)(\tau_2)-(\mathcal{G}u)(\tau_1)\|_{\alpha-1}
\le L(\tau_2-\tau_1),
\end{equation}
where
\[
L=\max\{M\|u_0\|_\alpha,M~\sum_{i=1}^m\Upsilon_i
,NMC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha},NM\|A^{1-\alpha}\|\}.
\]
 Therefore, $\mathcal{G}$ is piecewise
Lipschitz continuous on $[0,T_0]$ and so $\mathcal{G}:X_1\to X_1$.


Next we will show that $\mathcal{G}:\mathcal{W}\to \mathcal{W}$.
Let $u\in X\cap X_1$, then for each $t\in[0,t_1]$,
\begin{align*}
\|(\mathcal{G}u)(t)-u_0\|_{\alpha}
&\le \|(S(t)-I)A^{\alpha}u_0\| \\
&\quad + \int_0^t\|S(t-s)A^{\alpha}\|\|f(s,u(s),u(w_1(s,u(s))))\|ds \\
&\le  \|(S(t)-I)A^{\alpha}u_0\|+C_{\alpha}N{T_0^{1-\alpha}\over 1-\alpha}.
\end{align*}
Similarly, let $u\in X\cap X_1$, then for each $t\in (t_k,t_{k+1}],k=1,\dots ,m$,
\begin{align*}
&\|(\mathcal{G}u)(t)-u_0\|_{\alpha}\\
&\leq \|(S(t)-I)A^{\alpha}u_0\|
+ \int_0^t\|S(t-s)A^{\alpha}\|\|f(s,u(s),u(w_1(s,u(s))))\|ds \\
&\quad +\sum_{i=1}^{k}\|A^\alpha S(t-t_i)I_i(u(t^-_i))\| \\
&\leq \|(S(t)-I)A^{\alpha}u_0\|+C_{\alpha}N{T_0^{1-\alpha}\over
1-\alpha}+M \sum_{i=1}^k \Upsilon_i.
\end{align*}
Part (iii) of Lemma 2.1, implies that
\begin{equation}
\|(S(t)-I)A^{\alpha}(u_0)\|\leq {R\over 2}\label{assum1}.
\end{equation}
Thus, from \eqref{assum2} and \eqref{assum1}, it is clear that
$$
\|\mathcal{G}u- u_0\|_{\mathcal{PC}}\leq R.
$$
Therefore, $\mathcal{G}:\mathcal{W}\to \mathcal{W}$.

Finally, we will claim that $\mathcal{G}$ is a contraction map.
If $t\in [0,t_1]$ and $u, v\in \mathcal{W}$, then
\begin{equation}
\begin{aligned}
&\|(\mathcal{G}u)(t)-(\mathcal{G}v)(t)\|_{\alpha} \\
&\le \int_0^t\|S(t-s)A^{\alpha}\|~\|f(s,u(s),u(w_1(s,u(s))))
 -f(s,v(s),v(w(s,v(s))))\|ds.
\end{aligned} \label{bs11+}
\end{equation}
Also, we note that
\begin{equation} \label{e3.19}
\begin{aligned}
&\|f(t,u(t),u(w_1(t,u(t))))-f(t,v(t),v(w_1(t,v(t))))\| \\
& \leq L_f\{\|u(t)-v(t)\|_\alpha+\|u(w_1(t,u(t)))-v(w_1(t,v(t)))\|_{\alpha-1}\}  \\
&\leq L_f\Big[\|u(t)-v(t)\|_\alpha+\|A^{-1}\|\,
 \|u(w_1(t,v(t)))-v(w_1(t,v(t)))\|_{\alpha} \\
&\quad +\|u(w_1(t,u(t)))-u(w_1(t,v(t)))\|_{\alpha-1}\Big] \\
&\leq L_f\big\{2\|u-v\|_{\mathcal{PC}}+
L|w_1(t,u(t))-w_1(t,v(t))|\big\}.
\end{aligned}
\end{equation}
Now, let
$$
w_i(t,u(t))=h_i(t,u(h_{i+1}(t,\dots
u(t,h_m(t,u(t)))\dots ))),~~i=1,\dots ,m,
$$
with
$w_{m+1}(t,u(t))=t$. For more details, we refer to \cite[p. 2183]{ss}.

Hence, we have
\begin{align*}
|w_1(t,u(t))-w_1(t,v(t))|
&= |h_1(t,u(w_2(t,u(t))))-h_1(t,v(w_2(t,v(t)))) \\
&\le L_{h_1}\|u(w_2(t,u(t)))-u(w_2(t,u(t)))\|_{\alpha-1} \\
&\le L_{h_1}\Big\{\|u(w_2(t,u(t)))-u(w_2(t,v(t)))\|_{\alpha-1}\\
&\quad +\|u(w_2(t,v(t)))-v(w_2(t,v(t)))\|_{\alpha-1}\Big\} \\
&\le L_{h_1}\Big\{L|h_2(t,u(w_3(t,u(t)))-h_2(t,v(w_3(t,v(t)))|\\
&\quad +\|A\|^{-1}\|u-v\|_{\mathcal{PC}}\Big\} \\
&\dots  \\
&\le ( L^{m-1}L_{h_1}\dots  L_{h_m}+L^{m-2}L_{h_1}\dots
L_{h_{m-1}}+ \dots\\
&\quad  +LL_{h_1}L_{h_2}+ L_{h_1})\|A\|^{-1}\|u-v\|_{\mathcal {PC}}.
\end{align*}
Thus, we have
\begin{equation}
\begin{aligned}
&\|f(t,u(t),u(w_1(t,u(t))))-f(t,v(t),v(w_1(t,v(t))))\| \\
&\leq L_f\Big(2+LL_h\|A^{-1}\|\Big)\|u-v\|_{\mathcal{PC}} \\
&\leq L_f(2+LL_h)\|u-v\|_{\mathcal{PC}}.
\end{aligned}\label{f}
\end{equation}
where $L_h=( L^{m-1}L_{h_1}\dots  L_{h_m}+L^{m-1}L_{h_1}\dots  L_{h_{m-1}}+\dots
+ L_{h_1})>0$.
 We use \eqref{f} in \eqref{bs11+}, we obtain
\begin{equation}
\|(\mathcal{G}u)(t)-(\mathcal{G}v)(t)\|_{\alpha}
\le C_{\alpha }L_f(2+LL_h){T_0^{1-\alpha}\over
1-\alpha}\|u-v\|_{\mathcal{PC}}.\label{bs145}
\end{equation}
For $t\in(t_1,t_2]$, we have
\begin{equation}
\|(\mathcal{G}u)(t)-(\mathcal{G}v)(t)\|_{\alpha}
\le \Big[ C_{\alpha }L_f(2+LL_h){T_0^{1-\alpha}\over
1-\alpha}+Md_1\Big]\|u-v\|_{\mathcal{PC}}.  \label{bs146}
\end{equation}
For $t\in(t_k,t_{k+1}]$, $k=1,2,\dots ,m$, we have
\begin{equation}
\|(\mathcal{G}u)(t)-(\mathcal{G}v)(t)\|_{\alpha}
\le \Big[ C_{\alpha }L_f(2+LL_h){T_0^{1-\alpha}\over
1-\alpha}+M\sum_{i=1}^k d_i\Big]\|u-v\|_{\mathcal{PC}}. \label{bs146b}
\end{equation}
Thus, for each $t\in[0,T_0]$, we have
\begin{equation*}
\|(\mathcal{G}u)(t)-(\mathcal{G}v)(t)\|_{\alpha}
\le \Big[ C_{\alpha }L_f(2+LL_h){T_0^{1-\alpha}\over 1-\alpha}+M\sum_{i=1}^m
d_i \Big]\|u-v\|_{\mathcal{PC}}.
\end{equation*}
Therefore, the map $\mathcal{G}$ is a contraction map, hence
$\mathcal{G}$ has a unique fixed point $u\in \mathcal{W}$. That
is,  problem \eqref{mainp} has a unique mild solution.
\end{proof}

\section{Further existence results}

Theorem \ref{b} can be proved
if we omit the hypothesis (H1). In that case the  proof is based on
the  idea of Wang et al. \cite{mf}.

 \begin{theorem} \label{a1}
Assume the conditions (H3)-(H4) hold, the semigroup $\{S(t),t\ge 0\}$ is
compact, and $f:I\times H\times H\to H$ is continuous. For $u_0\in
H_\alpha$ there exists a constant $r>0$ such that
\begin{equation}
 M\{\|u_0\|_\alpha+\sum_{i=1}^m\Upsilon_i\}+C_\alpha
N_f \frac{T_0^{1-\alpha}}{1-\alpha} \le r,\label{e1}
\end{equation}
where
\[
N_f={\sup_{s\in[0,T_0],u\in\Omega}}\|f(s,u(s),u(w_1(s,u(s))))\|,
\]
and
$u\in\Omega=\{v\in \mathcal{PC}(H_\alpha):\|v\|_{\mathcal{PC}}\le r\}$.
Then there exists a mild solution $u\in\mathcal{PC}(H_\alpha)$
of problem \eqref{mainp}.
\end{theorem}

\begin{proof} Let us define a map
$ \Upsilon:\mathcal{PC}(H_\alpha)\to \mathcal{PC}(H_\alpha)$, by
\[
 (\Upsilon u)(t)=\begin{cases}
S(t)u_0+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, &
t\in[0,t_1],\\[4pt]
 S(t)u_0+S(t-t_1)I_1(u(t^-_1))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, & t\in(t_1,t_2],\\
\dots\\
 S(t)u_0+\sum_{i=1}^k S(t-t_i)I_i(u(t^-_i))\\
+\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds, &t\in(t_k,t_{k+1}],\\
 k=1,2,\dots ,m.
\end{cases}
\]

\noindent\textbf{Step 1.}
First we show that ${\Upsilon}$ is continuous. It follows from
the continuity of $f$ that
$$
\|f(s,u_n(s),u_n(w_1(s,u_n(s))))-f(s,u(s),u(w_1(s,u(s))))\|
\le \epsilon,\quad\text{as } n\to \infty,
$$
for $s\in[0,t]$, $t\in[0,T_0]$.

Now, for each $t\in[0,t_1]$, we have
\begin{equation} \label{e4.2}
\|(\Upsilon u_n)(t)-({\Upsilon}u)(t)\|_\alpha\le C_\alpha
\frac{T_0^{1-\alpha}}{1-\alpha}\epsilon\to 0,
\quad\text{as } n\to\infty.
\end{equation}
For, $t\in(t_1,t_2]$, we have
\begin{equation} \label{e4.3}
\begin{aligned}
&\|({\Upsilon}u_n)(t)-({\Upsilon}u)(t)\|_\alpha \\
&\le M\|I_1(u_n(t^-_1))-I_1(u(t^-_1))\|_\alpha+C_\alpha
\frac{T_0^{1-\alpha}}{1-\alpha}\epsilon\to 0,
\quad\text{as }n\to\infty.
\end{aligned}
\end{equation}
Similarly, for each $t\in(t_k,t_{k+1}]$, $k=1,2,\dots ,m$,
\begin{equation} \label{e4.4}
\begin{aligned}
&\|({\Upsilon}u_n)(t)-({\Upsilon}u)(t)\|_\alpha \\
&\le M\sum_{i=1}^k\|I_i(u_n(t^-_i))-I_i(u(t^-_i))\|_\alpha+C_\alpha
\frac{T_0^{1-\alpha}}{1-\alpha}\epsilon\to 0, \quad
\text{as }n\to\infty.
\end{aligned}
\end{equation}
We have that ${\Upsilon}$ is continuous.

\noindent\textbf{Step 2.}
Next we show that ${\Upsilon}$ maps bounded sets into
bounded sets in $\mathcal{PC}(H_\alpha)$.
Let $u\in \Omega$, then for $t\in[0,t_1]$, we have
\begin{equation} \label{e4.5}
\|({\Upsilon}u)(t)\|_\alpha
\le M\|u_0\|_\alpha+N_fC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha},
\end{equation}
For, $t\in(t_1,t_2]$, we have
\begin{equation} \label{e4.6}
\|({\Upsilon}u)(t)\|_\alpha\le
M\big\{\|u_0\|_\alpha+\Upsilon_1\big\}+N_fC_\alpha\frac{T_0^{1-\alpha}}
{1-\alpha}.
\end{equation}
Similarly, for each $t\in(t_k,t_{k+1}]$, $k=1,2,\dots ,m$, we have
\begin{equation} \label{e4.7}
\|({\Upsilon}u)(t)\|_\alpha\le
M\Big\{\|u_0\|_\alpha+\sum_{i=1}^k
\Upsilon_i\Big\}+N_fC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha} .
\end{equation}
Thus, from inequality \eqref{e1}, we see that ${\Upsilon}:\Omega\to
\Omega$.

\noindent\textbf{Step 3.} In this step, we show that ${\Upsilon}$
maps bounded sets into equicontinuous sets in $\mathcal{PC}(H_\alpha)$.
 Let $\tau_1,\tau_2\in[0,t_1],\tau_1<\tau_2$, we have
\begin{equation} \label{e4.8}
\begin{aligned}
&\|({\Upsilon}u)(\tau_2)-({\Upsilon}u)(\tau_1)\|_\alpha \\
&\le M\Big\{\|u_0\|_\alpha+N_fC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha}+
\|A^{\alpha-1}\|N_f\Big\}(\tau_2-\tau_1).
\end{aligned}
\end{equation}
Similarly, for each $\tau_1,\tau_2\in(t_k,t_{k+1}]$, $\tau_1<\tau_2$,
$k=1,2,\dots ,m$, we have
\begin{equation} \label{e4.9}
\begin{aligned}
&\|({\Upsilon}u)(\tau_2)-({\Upsilon}u)(\tau_1)\|_\alpha \\
&\le M\Big\{\|u_0\|_\alpha+\sum_{i=1}^k\Upsilon_i
+N_fC_\alpha\frac{T_0^{1-\alpha}}{1-\alpha}+
\|A^{\alpha-1}\|N_f\Big\}(\tau_2-\tau_1).
\end{aligned}
\end{equation}
As $\tau_2\to\tau_1$ the right-hand side of the above inequality
tends to zero. So, ${\Upsilon}(\Omega)$ is equicontinuous.


\noindent\textbf{Step 4.}
${\Upsilon}$ maps $\Omega$ into a compact set in $H_\alpha$.
For this purpose, we decompose ${\Upsilon}$ as
${\Upsilon}={\Upsilon_1}+{\Upsilon_2}$,
where
\[
 ({\Upsilon_1}u)(t)= S(t)u_0+(\mathcal{T}_1u)(t),\quad t\in I \backslash
\{t_1,\dots ,t_m\},
\]
where $(\mathcal{T}_1u)(t)=\int_0^t S(t-s)f(s,u(s),u(w_1(s,u(s))))ds$.
and
\[
 ({\Upsilon_2}u)(t)= \begin{cases}
0,& t\in[0,t_1],\\
 \sum_{i=1}^k S(t-t_i)I_i(u(t^-_i)),& t\in(t_k,t_{k+1}],\;
k=1,2,\dots ,m.
\end{cases}
\]
${\Upsilon_2}$ is a constant map and hence compact.

Finally, we need to prove that for $0\le t\le T_0$,
$({\Upsilon_1}u)(t)$ is relatively compact in $\Omega$. For each
$t\in [0,T_0]$, the set $\{S(t)u_0\}$ is precompact in $H_\alpha$
since $\{S(t),t\ge 0\}$ is compact.


For $t\in(0,T_0]$, and $\epsilon>0$ sufficiently small, we define
\[
({\mathcal T_1}^\epsilon u)(t)
=S(\epsilon)\int_0^{t-\epsilon}S(t-\epsilon-s)f(s,u(s),u(w_1(s,u(s))))
ds,\quad u\in\Omega.
\]
The set $\{({\mathcal{T}_1}^\epsilon u)(t):u\in\Omega\}$ is
precompact in $H_\alpha$ since $S(\epsilon)$ is compact. Moreover,
for any $u\in\Omega$, we have
\begin{align*}
\|(\mathcal{T}_1 u)(t)-({\mathcal{T}_1^\epsilon}u)(t)\|_\alpha
&\le  \int_{t-\epsilon}^t \|A^\alpha
S(t-s)\|\|f(s,u(s),u(w_1(s,u(s))))\| ds \\
&\le (C_\alpha N_f)\epsilon^{1-\alpha}.
\end{align*}
Therefore, $\{({\mathcal{T}_1^\epsilon}u)(t):u\in\Omega\}$ is
arbitrarily close to the set
$\{(\mathcal{T}_1 u)(t):u\in\Omega\},t>0$. Hence the set
$\{({\Upsilon_1}u)(t):u\in\Omega\}$ is precompact in $H_\alpha$.

Thus, ${\Upsilon_1}$ is a compact operator by Arzela-Ascoli theorem,
and hence ${\Upsilon}$ is a compact operator. Then Schauder fixed
point theorem ensures  that ${\Upsilon}$ has a fixed point, which
gives rise to a piecewise continuous-mild solution.
\end{proof}

\section{Applications}
We consider the following semi-linear heat equation  with a
deviating argument (See also \cite{gal}).
\begin{equation} 
\begin{gathered}
\frac{\partial u}{\partial t}= \frac{\partial^2 u}
 {\partial x^2}+\tilde{H}(x,u(x,t))+G(t,x,u(x,t)),\\
x\in (0,1),\quad t\in(0,\frac{1}{2})\cup (\frac{1}{2},1),\\
\Delta u|_{t=\frac{1}{2}} =  \frac{2u({\frac{1}{2}})^-}{2+u({\frac{1}{2}})^-} \\
u(0,t)=u(1,t)=0, \\
u(x,0)=u_0(x), \quad x\in (0,1),
\end{gathered} \label{exappgal}
\end{equation}
where
\begin{gather*}
\tilde{H}(x,u(x,t))=\int _0^{x}K(x,y)u(y,P(t)) dy,\\
P(t)=g_1(t)\Big|u\Big(x,g_2(t)|u(x,\dots  g_m(t)|u(x,t)|)|\Big)\Big|,
\end{gather*}
and the function $G: \mathbb{R_{+}} \times
[0,1]\times \mathbb{R}\to \mathbb{R}$ is measurable in $x$,
locally H\"{o}lder continuous in $t$, locally Lipschitz continuous
in $u$, uniformly in $x$. Assume that $g_i:\mathbb{R_{+}}\to
\mathbb{R_{+}}$ are locally H\"{o}lder continuous in $t$ with
$g_i(0)=0$ for each $i=1,2,\dots ,m$, and 
$K\in C^1([0,1]\times [0,1];\mathbb{R})$.
Let 
\begin{gather*}
X=L^2((0,1);\mathbb{R}), \quad
Au=\frac{d^2u}{dx^2}, \quad
D(A)=H^2(0,1)\cap H^1_0(0,1),\\
X_{1/2}=D((A)^{1/2})=H^1_0(0,1), \quad
X_{-1/2}=(H^1_0(0,1))^*=H^{-1}(0,1)\equiv H^1(0,1).
\end{gather*}
For $x\in (0,1)$, we define the
function $f:\mathbb{R_{+}}\times X_{1/2}\times X_{-1/2}\to X$ by
\begin{equation} 
f(t,u,\psi)(x)=  \tilde{H}(x,\psi)+G(t,x,u),\label{gs55}
\end{equation}
where $\tilde{H}:[0,1]\times X_{-1/2}\to X$ is given by
\begin{equation} 
\tilde{H}(x,\psi)=  \int_0^x K(x,y)\psi(y)dy,\label{gs55b}
\end{equation}
and $G:\mathbb{R_+}\times [0,1]\times X_{1/2}\to X$
satisfies 
\begin{equation} 
\|G(t,x,u)\|\le Q(x,t)(1+\|u\|_{1/2}),\label{gs55c}
\end{equation}
with $Q(.,t)\in X$ and $Q$ continuous in its second arguments.

For $u\in D(A) $ and $\lambda \in \mathbb{R}$ with 
$-Au=\lambda  u$, we have
\[
\langle -Au,u\rangle =\langle \lambda u,u\rangle,\quad 
\| u'\| _{L^2}=\lambda \| u\| _{L^2};
\]
so we have $\lambda >0$. The solution $u$ of $-Au=\lambda u$ is
\[
u( x) =D_1\cos ( \sqrt{\lambda }x) +D_2\sin ( \sqrt{\lambda }x).
\]

Using the boundary condition, we get $D_1=0$ and 
$\lambda= \lambda_n=n^2\pi^2$ for $n\in \mathbb{N}$.
Thus, for $n\in \mathbb{N}$, we have 
$$
u_n(x)=D_2 \sin(\sqrt{\lambda_n}x).
$$
Also $\langle u_n,u_{m} \rangle =0$, $m\neq n$ and
$ \langle u_n,u_n \rangle =1$. So, for $u\in D(A)$ there exists a sequence
$\alpha_n$ of real numbers such that  
$u(x)= \sum _{n\in \mathbb{N}} \alpha_nu_n(x)$ with   $\sum _{n\in
\mathbb{N}}(\alpha_n)^2<\infty$ and $\sum _{n\in
\mathbb{N}}(\alpha_n)^2(\lambda _n)^2<\infty$.

The semigroup is given by 
$$ 
S(t)u= \sum _{n\in \mathbb{N}}
\exp(-n^2t)\langle u,u_{m} \rangle u_{m}.
$$

Let $T>0$ be fixed. Now we will show that assumptions
(H1)  and (H2) are   satisfied. For (H1), we will show
that $\tilde{H}: [0,1]\times X_{-1/2}\to X$,
 where 
$$
\tilde{H}(x,\psi(x,t))=\int _0^{x}K(x,y)\psi(y,t)dy,
$$ 
with 
$\psi(x,t)= u(x,w_1(t,u(x,t)))$. For $x\in [0,1]$, we have
 \begin{align*}
  \|\tilde{H}(x,\psi_{1}(x,\cdot))-\tilde{H}(x,\psi_2(x,\cdot))\|
 & \leq  \int_0^{x}|K(x,y)|\|(\psi_{1}-\psi_2)(y,\cdot)\|dy \\
 & \leq \| K \| _{\infty} \int_0^{x}|(\psi_{1}-\psi_2)(y,\cdot)|dy.
 \end{align*}
Thus for $\psi_{1},\psi_2\in H^1(0,1)$ and by applying the
 Minkowski's integral inequality we obtain
\begin{align*}
\|\tilde{H}(x,\psi_{1}(x,\cdot))-\tilde{H}(x,\psi_2(x,\cdot))\|^2
 & \leq \| K \| _{\infty}^2\int_0 ^1
\int_0^{y}|(\psi_{1}-\psi_2)(y,\cdot)|^2dxdy \\
 & \leq \| K \| _{\infty} ^2\int_0^1 y |(\psi_{1}-\psi_2)(y,\cdot)|^2dy\\
 & \leq \| K \| _{\infty} ^2\| \psi_{1}-\psi_2 \| ^2_{{-1/2}}.
 \end{align*}
 Since, 
\[
\frac{\partial}{\partial x}\tilde{H}(x,\psi(x,\cdot))
=K(x,x)\psi (x,\cdot)+\int_0^{x}
 \frac{\partial K}{\partial x}(x,y)\psi(y,\cdot),
\]
we obtain, in similar way, that
$$
\| \frac{\partial}{\partial x}H(x,\psi_{1}(x,\cdot))-\frac{\partial}
  {\partial x}H(x,\psi_2(x,\cdot))\|\leq ( \| K \| _{\infty} +|
  |\frac{\partial K}{\partial x}\|_{\infty})\| \psi_{1}-\psi_2 \| _{{-1/2}}.
$$
 Thus
\begin{align*}
 \|\tilde{H}(x,\psi_{1}(x,\cdot))-\tilde{H}(x,\psi_2(x,\cdot))\|
 & \leq (2\|K\|_{\infty}+\|\frac{\partial K}{\partial x}\|_{\infty}) 
 \| \psi_{1}-\psi_2 \| _{{-1/2}} \\
 & \leq (2\|K\|_{\infty}+\|\frac{\partial K}{\partial x}\|_{\infty})
  \| \psi_{1}-\psi_2 \| _{{-1/2}}\\
 &= C_{1} \| \psi_{1}-\psi_2 \| _{{-1/2}},
\end{align*}
 where $C_{1}= (2\|K\|_{\infty}
 +\| \frac{\partial K}{\partial x}\|_{\infty})$.
Again the assumption on $G$ implies that there exist constants
$C_2>0$ and $0< \gamma \leq 1 $ such that
$$
\|G(t,x,u)-G(s,x,v)\|_{H^1_0(0,1)}\leq C_2
(|t-s|^{\gamma}+\|u-v\|_{{1/2}}),
$$  
for $t,s \in [0,T]$, $x\in (0,1)$ and $u,v \in H^1_0(0,1)$.
Thus $f: [0,T] \times H^1_0(0,1)\times H^1(0,1)\to L^2(0,1)$ defined by
$f=\tilde{H}+G$ satisfies the assumption (H1).

Our next aim is to prove that the functions
 $h_i:[0,T]\times X_{-1/2}\to [0,T]$ defined by
 $h_i(t,\phi)=g_i(t)|\phi (x,\cdot)|$ for each $i=1,2,\dots ,m$, 
satisfies (H2). For $t\in [0,T]$, we obtain
\[
|h_i(t,\phi)| 
=|g_i(t)|\,|\phi (x,\cdot)|
\leq \|g_i\|_{\infty} \|\phi\|_{L^\infty(0,1)}
\leq C \|\phi\|_{{-1/2}},
\]
where we used the embedding $H^1_0(0,1)\subset C[0,1]$ in
the last inequity, and $C$ is a constant depending on the bounds of
$g_i$'s. Since $g_i$'s are locally H\"{o}lder continuous, for some
constant $L_{g_i}>0$ and $0<\theta \leq 1$ we have
$|g_i(t)-g_i(s)|\leq L_{g_i}|t-s|^{\theta}$ for $t,s\in [0,T]$.
Moreover  $t,s\in [0,T]$ and $\phi_{1},\phi_2\in X_{-1/2} $, we
have
\begin{align*}
|h_i(t,\phi_{1})-h_i(t,\phi_2)| 
&=|g_i(t)(|\phi_{1} (x,\cdot)|-|\phi_2 (x,\cdot)|)
 +(g_i(t)-g_i(s))|\phi_2(x,\cdot)|\\
& \leq \|g\|_{\infty} \|\phi_{1}-\phi_2\|_{L^\infty(0,1)}
 +L_{g_i}|t-s|^{\theta}\|\phi_2\|_{L^\infty(0,1)}\\
& \leq C \|g\|_{\infty}\|\phi_{1}-\phi_2\|_{{-1/2}}
 +L_{g_i}|t-s|^{\theta}\|\phi_2\|_{{-1/2}}\\
& \leq  \max \{ C \|g\|_{\infty},L_{g_i}\|\phi_2\|_{\infty}
\}(\|\phi_{1}-\phi_2\|_{{-1/2}}+|t-s|^{\theta}).
\end{align*}
If $u, v\in D((-A)^{1/2})$, then 
$$
\|I_k(u)- I_k(v)\|_{1/ 2}\le {2\|u- v\|_{1/ 2}\over\|(2+ u)(2 + v)\|_{1/ 2}}
\le{ 1\over 2} \|u-v\|_{1/ 2}.
$$

Next for Theorem \eqref{a1} we define 
\[
f(t,u(t),\psi(t)))(x)={e^{-t}(\cos(u(x,t))+\sin(\psi(x,t)))
\over(2+t^2)(e^t+e^{-t})}+e^{-t},
\]
for $t\in[0,1/2)\cup(1/2,1]$, $u\in X$, and $x\in(0,1)$.  
 Clearly, 
\[
\|f(t,u,\psi)\|\le{e^{-t}\over(2+t^2)(e^t+e^{-t})}+e^{-t}=m(t), 
\]
with $m(t)\in L^\infty([0,1];\mathbb{R}_+)$.
Thus, we can apply all the results of this section to obtain the
main results.

\subsection*{Acknowledgements} We highly appreciate the valuable
comments and suggestions of the referees on our manuscript which
helped to considerably improve the quality of the manuscript. The
third author would like to acknowledge the financial aid from the
Department of Science and Technology, New Delhi, under its research
project SR/S4/MS:796.

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\end{document}