\documentclass[reqno]{amsart}
\usepackage[notref,notcite]{showkeys}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 222, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/222\hfil Mild solutions]
{Mild solutions for non-autonomous impulsive semi-linear differential
equations with iterated deviating arguments}

\author[A. Chadha, D. N. Pandey \hfil EJDE-2015/222\hfilneg]
{Alka Chadha, Dwijendra N. Pandey}

\address{Alka Chadha \newline
Department of Mathematics,
Indian Institute of Technology Roorkee,
Roorkee, Uttarakhand 247667, India}
\email{alkachadda23@gmail.com}

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

\thanks{Submitted November 20, 2014. Published August 27, 2015.}
\subjclass[2010]{34K37, 34K30, 35R11, 47N20}
\keywords{Iterated deviated argument; analytic semigroup;
\hfill\break\indent  Banach fixed point theorem; impulsive differential equation}

\begin{abstract}
 In this work, we consider an impulsive non-autonomous semi-linear
 equation with iterated deviating arguments in a Banach space.
 We establish the existence and uniqueness of a mild  solution.
 Also we present an example that illustrates our main result.
\end{abstract}

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

\section{Introduction}

In the previous decades, impulsive differential equations have received
much attention of researchers mainly because its demonstrated applications
in widespread fields of science and engineering. Differential equation
systems which are characterized by the occurrence of an abrupt change in
the state of the system are known as impulsive differential equations.
These changes occur at certain time instants over a period of negligible duration.
Such process is investigated in various fields such as biology, physics,
control theory, population dynamics, economics, chemical technology,
medicine and so on. Impulsive differential equations are an appropriate model
to hereditary phenomena for which a delay argument arises in the modelling
equations. For more details for impulsive differential equation, we refer
to the monographs \cite{ben1,laks} and papers
\cite{ben2, ben3,  card,cue, kumar, liu, mach, sam} and references
given therein.

In this article, we investigate the existence and uniqueness of solution
for impulsive differential equation with iterated deviating arguments in
a complex Banach space $(E,\|\cdot\|)$.
We study the differential equation
\begin{gather}\label{geq1}
\begin{aligned}
\frac{d}{dt}[u(t)+G(t,u(a(t)))]
&= -A(t)[u(t)+G(t,u(a(t)))] \\
&\quad +F(t,u(t),u(h_1(t,u(t)))),\quad t>0
\end{aligned}\\
u(0)= u_0,\quad u_0\in E,  \\
\Delta u(t_i)= I_i(u(t_i)),\quad i=1,\dots,\delta\in \mathbb{N}, \label{geq2}
\end{gather}
where $h_1(t,u(t))=b_1(t, u(b_2(t,\dots,u(b_\delta(t,u(t)))\dots)))$ and
$-A(t):D(A(t))\subseteq E\to E$, $t\geq 0$ is a closed densely defined linear
operator. The functions $F$, $b_i$, $G$, $I_i:E\to E\;(i=1,\dots,\delta)$
 are appropriate functions to be mentioned later.
Here, $0=t_0<t_1<\dots<t_\delta<t_{\delta+1}=T$ are fixed numbers,
$0<T<\infty$, $\Delta u|_{t=t_i}=u(t_i^+)-u(t_i^-)$ and
$u(t_i^-)=\lim_{\varepsilon\to 0-}u(t_i+\varepsilon)$ and
$u(t_i^+)=\lim_{\varepsilon\to 0+}u(t_i+\varepsilon)$ denote the left and
right limits of $u(t)$ at $t=t_i$, respectively.
 In \eqref{geq1}, $-A(t)$ is assumed to be the infinitesimal generator of
an analytic semigroup of bounded linear operators on a Banach space $E$.

A Differential equation with a deviated argument is a valuable tool for
the modeling of many phenomena in several fields of science and engineering
such as theory of automatic control, biological systems, problems of long-term
planning in economics, theory of self-oscillating systems, study of problems
related to combustion in rocket motion and so on. The existence and uniqueness
of the solution to the differential equations with deviated argument have
been discussed by many authors (see \cite{gal}-\cite{haloi}).
For a detailed discussion of differential equations with iterated deviating
arguments, we refer to monograph \cite{elgol} and papers
\cite{gal, gal2, haloi, jan2, kumar, liu, muslim, pandey, stev, stev2}
and references given therein.

The existence and uniqueness of the solution for the following problem with
a deviated argument has been established by Gal \cite{gal},
\begin{gather}
u'(t)= Au(t)+F(t,u(t), u(h(u(t),t))),\quad t>0,\\
u(0)= u_0,\quad u_0\in E,
\end{gather}
in a Banach space $(E,\|\cdot\|)$. Where $-A$ generates an analytic semigroup
of bounded linear operators on $E$ and the function
$F:[0,\infty)\times E_\alpha\times E_{\alpha-1}\to E$,
$h:E_\alpha\times[0,\infty)\to[0,\infty)$ are H\"older continuous with exponent
 $\mu_1\in(0,1]$ and $\mu_2\in(0,1]$ respectively.
For $0<\alpha\leq 1$, $E_\alpha$ denotes the domain of $(-A)^\alpha$ which
is a Banach space with the norm $\|u\|_{\alpha}=\|(-A)^\alpha u\|$,
$u\in D((-A)^\alpha)$.

In \cite{haloi}, authors considered the following problem in a Banach
space $(E,\|\cdot\|)$,
\begin{gather}\label{req1}
u'(t)+A(t)u(t)= F(t,u(t),u(h(u(t),t))),\quad t>0,\\
u(0)= u_0,\quad u_0\in E,
\end{gather}
where $A$ is a closed, densely defined linear operator with domain
$D(A)\subset E$. In \eqref{req1}, $-A$ generates an analytic semigroup of
bounded linear operators on Banach space $E$.
The function $F:\mathbb{R}\times E_\alpha\times E_{\alpha-1}\to E$,
$h:E_\alpha\times\mathbb{R}_{+}\to[0,\infty)$ are appropriated functions.
The authors have established the existence of the solution for \eqref{req1}
by using Banach fixed point theorem.

The rest of this article is organized as follows:
Section $2$ provides some basic definitions, lemmas and theorems,
 assumptions as these are useful for proving our results.
Section $3$ focuses on the existence of a mild solution to
problem \eqref{geq1}-\eqref{geq2}. Section $4$ present an example to
illustrate the theory.

\section{Preliminaries}

In this section, we provide basic definitions, preliminaries, lemmas
and assumptions which are useful for proving main result in later section.

Throughout the work, we assume that $(E,\|\cdot\|)$ is a complex Banach space.
The notation $C([0,T],E)$ stands for the space of $E$-valued continuous functions
on $[0,T]$ with the norm $\|z\|=\sup\{\|z(\tau)\|, \tau\in[0,T]\}$ and
$L^1([0,T],E)$ denotes the space of $E$-valued Bochner integrable functions on
$[0,T]$ endowed with the norm
$\|\mathcal{F}\|_{L^1}=\int^T_0\|\mathcal{F}(t)\|dt,\;\mathcal{F}\in L^1([0,T],E)$.
We denote by $C^\beta([0,T];E)$ the space of all uniformly H\"older continuous
functions from $[0,T]$ into $E$ with exponent $\beta>0$.
It is easy to verify that $C^\beta([0,T];E)$ is a Banach space with the norm
\begin{equation}
\|y\|_{C^\beta([0,T];E)}=\sup_{0\leq t\leq T}\|y(t)\|
+\sup_{0\leq t,s\leq T,\;t\ne s}\frac{\|y(t)-y(s)\|}{|t-s|^{\beta}}.
\end{equation}
Let $\{A(t):0\leq t\leq T\}$, $T\in[0,\infty)$ be a family of closed linear
operators on the Banach space $E$. We impose following restrictions as \cite{fri}:
\begin{itemize}
\item[(P1)] The domain $D(A)$ of $\{A(t): t\in[0,T]\}$ is dense in $E$ and
$D(A)$ is independent of $t$.

\item[(P2)] For each $0\leq t\leq T $ and $\operatorname{Re}\lambda\leq 0$, the resolvent
$R(\lambda; A(t))$ exists and there exists a positive constant $K$
(independent of $t$ and $\lambda$) such that
$$
\| R(\lambda;A(t))\|\leq K /(|\lambda|+1),\quad
\operatorname{Re} \lambda\leq 0,\; t\in[0,T].
$$

\item[(P3)] For each fixed $\xi\in [0,T]$, there are constants $K>0$ and
$0<\mu\leq 1$ such that
 \begin{equation}
\|[A(\tau)-A(s)]A^{-1}(\xi)\|\leq K|\tau-s|^{\mu},\quad \text{for all }
\tau,s\in[0,T]
\end{equation}
where $\mu$ and $K$ are independent of $\tau,s$ and $\xi$.
\end{itemize}
The assumptions (P1)--(P3) allow the existence of a unique linear evolution
system (linear evolution operator) $U(t,s)$, $0\leq s\leq t\leq T$
which is generated by the family $\{A(t):t\in[0, T]\}$ and there exists
a family of bounded linear operators $\{\Phi(t,s):0\leq s\leq t\leq T\}$
such that $\|\Phi(t,s)\|\leq \frac{K}{|t-s|^{1-\mu}}$.
We also have that $U(t,s)$ can be written as
\begin{equation}
U(t,s)=e^{-(t-s)A(t)}+\int^t_s e^{-(t-\tau)A(\tau)}\Phi(\tau,s)d\tau.
\end{equation}
Assumption (P2) guarantees that $-A(s),\;s\in[0,T]$ is the infinitesimal generator
of a strongly continuous analytic semigroup $\{e^{-t A(s)}:t\geq 0\}$ in $B(E)$,
where the symbol $B(E)$ stands for the Banach algebra of all bounded linear
operators on $E$.

The assumptions (P1)--(P3) allow the existence of a unique fundamental solution
$\{U(t,s):0\leq s\leq t\leq T\}$ for the homogenous Cauchy problem such that
\begin{itemize}
\item[(i)] $U(t,s)\in B(E)$ and the mapping $(t,s)\to U(t,s)z$ is continuous for
 $z\in E$, i.e., $U(t,s)$ is strongly continuous in $t,s$ for all
 $0\leq s\leq t\leq T$.

\item[(ii)] For each $z\in E$, $U(t,s)z\in D(A)$, for all $0\leq s\leq t\leq T$.

\item[(iii)] $U(t,\tau)U(\tau,s)=U(t,s)$ for all $0\leq s\leq \tau\leq t\leq T$.

\item[(iv)] For each $0\leq s< t\leq T$, the derivative
 $\frac{\partial U(t,s)}{\partial t}$ exists in the strong operator topology
 and an element of $B(E)$, and strongly continuous in $t$, where $s<t\leq T$.

\item[(v)] $U(t,t)=I$.

\item[(vi)] $\frac{\partial U(t,s)}{\partial t}+A(t)U(t,s)=0$ for all
$0\leq s<t\leq T$.
\end{itemize}
We have also the following inequalities:
\begin{gather}
\|e^{-tA(\tau)}\|\leq Ke^{-d t},\;\;t\geq 0;\\
\|A(\tau)e^{-tA(\tau)}\| \leq \frac{Ke^{-d t}}{t},\;\;t>0,\\
\|A(t)U(t,\tau)\| \leq K|t-\tau|^{-1}, \;\;0\leq \tau\leq t\leq T.
\end{gather}
for all $\tau\in[0,T]$. Where $d$ is a positive constant.
For $\alpha>0$, we may define the negative fractional powers $A(t)^{-\alpha}$ as
\begin{equation}
A(t)^{-\alpha}=\frac{1}{\Gamma(\alpha)}\int^\infty_0 s^{\alpha-1}e^{-sA(t)}ds.
\end{equation}
Then, the operator $A(t)^{-\alpha}$ is a bounded linear and one to one operator
on $E$. Therefore, it implies that there exists an inverse of the operator
$A(t)^{-\alpha}$. We can define $A(t)^{\alpha}\equiv [A(t)^{-\alpha}]^{-1}$
which is the positive fractional powers of $A(t)$.
The operator $A(t)^{\alpha}\equiv [A(t)^{-\alpha}]^{-1}$ is a closed densely
defined linear operator with domain $D(A(t)^\alpha)\subset E$ and for
$\alpha<\beta$, we get $D(A(t)^\beta)\subset D(A(t)^\alpha)$.
Let $E_\alpha=D(A(0)^\alpha)$ be a Banach space with the norm
$\|y\|_\alpha=\|A(0)^\alpha y\|$. For $0<\omega_1\leq \omega_2$, we have
that the embedding $E_{\omega_2}\hookrightarrow E_{\omega_1}$ is continuous
and dense.
For each $\alpha>0$, we may define $E_{-\alpha}=(E_\alpha)^*$, which is the
 dual space of $E_\alpha$. The dual space is a Banach space with natural norm
 $\|y\|_{-\alpha}=\|A(0)^{-\alpha}y\|$.

In particular, by the assumption (P3), we conclude a constant $K>0$, such that
\begin{equation}
\|A(t)A(s)^{-1}\|\leq K, \quad \text{for all }0\leq s,t\leq T.
\end{equation}
For $0< \alpha\leq 1$, let $U_\alpha$ and $U_{\alpha-1}$ be open sets in
$E_\alpha$ and $E_{\alpha-1}$, respectively.
For every $v'\in U_\alpha$ and $v''\in U_{\alpha-1}$, there exist balls such that
$B_\alpha(v', r')\subset U_\alpha$ and $B_{\alpha-1}(v'', r'')\subset U_{\alpha-1}$,
for some positive numbers $r'$ and $r''$. Let $F$, $a$, $h$ and
$I_i\;(i=1,\dots,\delta)$ be the continuous functions satisfying following
conditions:
\begin{itemize}
\item[(P4)] The nonlinear map $F:[0,T]\times U_\alpha\times U_{\alpha-1}\to E$
is a H\"older continuous and there exist positive constants
$L_F\equiv L_F(t,v',v'',r',r'')$ and $0<\mu_1\leq 1$ such that
\begin{equation}\label{Feq1}
\begin{aligned}
&\|F(t,z_1,w_1)-F(s,z_2, w_2)\|\\
&\leq L_F(|t-s|^{\mu_1}+\|z_1-z_2\|_{\alpha}+\|w_1-w_2\|_{\alpha-1}),
\end{aligned}
\end{equation}
for all $(z_1,w_1), (z_2,w_2)\in B_\alpha\times B_{\alpha-1}$ and
$s,t\in[0,T]$.

\item[(P5)] The functions $b_i:[0,\infty)\times U_{\alpha-1}\to[0,\infty)$,
$(i=1,\dots, \delta)$ are continuous functions and there are positive constants
$L_{b_i}\equiv L_{b_i}(t,v', r')$ and $0<\mu_2\leq 1$ such that
 \begin{equation}\label{aeq1}
|b_i(t,z)-b_i(s,w)|\leq L_{b_i}(|t-s|^{\mu_2}+\|z-w\|_{\alpha-1}),
 \end{equation}
for all $(t,z),\;(s, w)\in [0,T]\times B_\alpha$.

\item[(P6)] For $0\leq\alpha<\beta<1$, $G:[0,T]\times U_{\alpha-1}\to E_\beta$
is a continuous map and there exists a positive constant
$L_G=L_G(t,v'',r'',\beta)$ such that
 \begin{gather}
\|A^\beta G(t_1, z_1)-A^\beta G(t_2, z_2)\|
\leq L_G[|t_1-t_2|+\|z_1-z_2\|_{\alpha-1}],\\
 4L_G\|A(0)^{\alpha-\beta-1}\|<1,
 \end{gather}
for each $(t_1,z_1),\;(t_2,z_2)\in [0,T]\times B_{\alpha-1}$.

 \item[(P7)] The function $a:[0,T]\to[0,T]$ is a continuous function and
satisfies the following conditions:
 \subitem(i) $a(t)\leq t$ for all $t\in[0,T]$.
 \subitem(ii) There exist a constant $L_a>0$ such that
 \begin{equation}
|a(t_1)-a(t_2)|\leq L_a|t_1-t_2|,
 \end{equation}
for all $t_1,t_2\in[0,T]$ and $L_a \|A^{-1}\|<1$.

 \item[(P8)] $I_i:U_\alpha\to U_\alpha\;(i=1,\dots,\delta)$ are continuous
functions and there exist positive constants $L_i\equiv L_i(t,v',r')$ such that
 \begin{gather}
\|I_i(z)-I_i(w)\|_\alpha\leq L_i\|z-w\|_\alpha, \quad i=1,\dots,\delta,\\
\|I_i(z)\|\leq C_i,\;\;i=1,\dots,\delta,
 \end{gather}
for all $z,w\in B_\alpha$, where  $C_i$ are positive constants.
\end{itemize}
Now, we turn to the Cauchy problem which is illustrated as
follows,
\begin{gather}\label{cheq1}
u'(t)= -A(t)u(t)+f(t),\\
\label{cheq2}
u(t_0)= u_0,\quad t\geq 0.
\end{gather}

\begin{theorem}[\cite{Pazy}]\label{thm1}
Assume that {\rm (P1)--(P3)} hold. If $f$ is a H\"older continuous function
from $[t_0,T]$ into $E$ with exponent $\beta$. Then, there exists a unique
solution of the problem \eqref{cheq1}-\eqref{cheq2} given by
\begin{equation}
u(t)=U(t,t_0)u_0+\int^t_{t_0}U(t,s)f(s)ds,\quad \forall t_0\leq t\leq T.
\end{equation}
Indeed, $u:[t_0,T]\to E$ is strongly continuously differentiable solution
on $(t_0, T]$.
\end{theorem}

We also have following results.

\begin{lemma}[\cite{fri}] \label{lem1}
Suppose that {\rm (P1)--(P3)} are satisfied. If $0\leq \gamma\leq 1$,
$0\leq\beta\leq\alpha<1+\mu$, $0<\alpha-\gamma\leq1$, then for any
$0\leq \tau<t<t+\Delta t\leq t_0$, $0\leq\zeta\leq T$,
\begin{equation}
\|A^{\gamma}(\zeta)[U(t+\Delta t,\tau)-U(t,\tau)]A^{-\beta}(\tau)\|
\leq K({\gamma,\beta,\alpha})(\Delta t)^{\alpha-\gamma}|t-\tau|^{\beta-\alpha}.
\end{equation}
\end{lemma}

\begin{lemma}[\cite{fri}] \label{lem2}
 Suppose that {\rm (P1)--(P3)} are satisfied and let $0\leq\gamma<1$.
Then for any $0\leq\tau\leq t\leq t+\Delta t\leq t_0$ and for any continuous
function $f(s)$,
\begin{equation}
\begin{aligned}
&\|A^\gamma (\zeta)[\int^{t+\Delta t}_t U(t+\Delta t, s)f(s)ds
 -\int^t_{\tau} U(t,s)f(s)ds]\| \\
&\leq K({\gamma})(\Delta t)^{1-\gamma}(|\log(\Delta t )|+1)
 \max_{\tau\leq s\leq t+\Delta t}\|f(s)\|.
\end{aligned}
\end{equation}
\end{lemma}

For more details, we refer to the monographs \cite{fri, Pazy}.

\section{Existence result}

In this section, the existence of mild solution for the problem
\eqref{geq1}--\eqref{geq2} is established by using fixed point theorem.
 Let $(E,\|\cdot\|)$ be a complex Banach space. The symbol $\mathcal{C}_\alpha^{T_0}$
denotes the Banach space of all $E_\alpha$-valued continuous functions on
$J=[0,T_0],\;0<T_0<T<\infty$ endowed with the sup-norm
$\sup_{t\in J}\|z(t)\|,\;z\in C(J;E_\alpha)$.

We choose $T_0$ sufficiently small, $0<T_0<T$ such that
\begin{gather}
\|(U(t,0)-I)(u_0+G(0,u_0))\|_\alpha+K(\alpha)\sum_{0<t_i<t}C_i
\leq \frac{r}{6},\quad \forall t\in[0,T_0], \\
\|G(t, u(a(t)))-G(0,u_0)\|_\alpha\leq \frac{r}{6},\quad t\in[0,T_0], \\
K(\alpha)N\frac{{T_0}^{1-\alpha}}{1-\alpha}\leq\frac{2r}{3},\quad \forall t\in[0,T_0].
\end{gather}
 We define
\begin{equation}
\begin{aligned}
Y&= PC([0,T_0];E_\alpha)=PC(E_\alpha) \\
&= \big\{u:J\to E_\alpha: u\in C((t_i,t_{i+1}],E_\alpha),\;
 i=1,\dots,\delta \\
&\quad \text{and } u(t_i^{+}), u(t_i^{-})=u(t_i) \text{ exist}\big\}.
\end{aligned}
\end{equation}
Clearly, the space $Y$ is a Banach space with the supremum norm
\begin{equation}
\|u\|_{PC,\alpha}=\max\{\sup_{t\in J}\|u(t+0)\|_{\alpha}, \; \sup_{t\in J}\|u(t-0)\|_{\alpha}\}.
\end{equation}
Consider
\begin{equation}
\begin{aligned}
Y_{\alpha-1}
&= {PC}_\mathcal{L}(J;E_{\alpha-1})
= \big\{u\in Y:\|u(t)-u(s)\|_{\alpha-1}\leq \mathcal{L}|t-s|, \\
&\quad \text{for all }t,s\in (t_i,t_{i+1}],\;i=0,1,\dots,\delta\big\},
\end{aligned}
\end{equation}
where $\mathcal{L}>0$ is a constant to be defined later. It is easy to see
that $Y_{\alpha-1}$ is a Banach space under the supremum norm of
$\mathcal{C}_\alpha^{T_0}=C(J,E_\alpha)$.

Before expressing and demonstrating the main result, we present the definition
of the mild solution to the problem \eqref{geq1}-\eqref{geq2}.

\begin{definition} \label{def3.1} \rm
A piecewise continuous function $u(\cdot):[0,T_0]\to E$ is called a mild
solution for the problem \eqref{geq1}-\eqref{geq2} if $u(0)=u_0$ and
$u(\cdot)$ satisfies the integral equation
\begin{equation}
\begin{aligned}
u(t)&= U(t,0)[u_0+G(0,u_0)]-G(t,u(a(t))) \\
&\quad +\int^t_0 U(t,s)F(s,u(s), u(h_1(s, u(s))))ds
 +\sum_{0<t_i<t}U(t,t_i)I_i(u(t_i^{-})).
\end{aligned}
\end{equation}
\end{definition}

Let $0<\eta<\beta-\alpha$ be the fixed constants. For $0<\alpha\leq 1$, let
\begin{equation}
\begin{aligned}
\mathcal{S}_\alpha
&= \big\{y\in Y\cap Y_{\alpha-1}:y(0)=u_0,\;
 \sup_{t\in J}\|y(t)-u_0\|_\alpha\leq r, \\
&\quad \|y(t_1)-y(t_2)\|_\alpha \leq P |t_1-t_2|^{\eta}
\text{ for all } t_1,t_2\in J \big\},
\end{aligned}
\end{equation}
where $ P$ and $r$ are positive constants to be defined later.
Thus, $\mathcal{S}_\alpha$ is a non-empty closed and bounded subset
of $Y_{\alpha-1}$. Next, we prove the following theorem for the existence
of a mild solution to the problem \eqref{geq1}. We adopt the ideas of
Friedman \cite{fri} and Gal \cite{gal} to prove the theorem.

\begin{theorem} \label{thm3.1}
Let $ u_0\in E_\beta $, where $0<\alpha<\beta \leq 1$.
Suppose that assumptions {\rm (P1)--(P8)} are satisfied and
\begin{equation}\label{thmeq1}
\|A(0)^{\alpha-\beta}\| L_G+K(\alpha)L_F(2+\mathcal{L} L_b)
\frac{T_0^{1-\alpha}}{(1-\alpha)}+K(\alpha)\sum_{i=1}^\delta L_i <1.
\end{equation}
 Then, there exists a unique solution $u(t)\in \mathcal{S}_\alpha$ for
the problem \eqref{geq1}-\eqref{geq2} on $[0,T_0]$.
\end{theorem}

\begin{proof}
Let us assume that $u_0\in E_\beta$. We define a map
$Q:\mathcal{S}_\alpha\to\mathcal{S}_\alpha$ by
\begin{equation}
\begin{aligned}
Q u(t)
&= U(t,0)(u_0+G(0,u_0))-G(t, u(a(t)))\\
&\quad +\int^t_0U(t,s)F(s,u(s), u(h_1(u(s),s)))ds \\
&\quad +\sum_{0<t_i<t}U(t,t_i)I_i(u(t_i^{-})),\quad u\in S_\alpha,\; t\in[0,T_0].
\end{aligned}
\end{equation}
We firstly claim that $Q(\mathcal{S}_\alpha)\subset \mathcal{S}_\alpha$.
Clearly, it can easily be shown that $Q u\in Y$. Now, we want to show that
$Q u\in Y_{\alpha-1}$. Indeed, if $\tau_1,\tau_2\in J$ with $\tau_2>\tau_1$,
then we have
\begin{equation}
\begin{aligned}
\|Q u(\tau_2)-Q u(\tau_1)\|_{\alpha-1}
&\leq \|[U(\tau_2,0)-U(\tau_1, 0)](u_0+G(0,u_0))\|_{\alpha-1} \\
&\quad +\|G(\tau_2, u(a(\tau_2)))-G(\tau_1, u(a(\tau_1)))\|_{\alpha-1} \\
&\quad +\Big\|\Big[\int^{\tau_2}_0U(\tau_2,s)F(s,u(s),u(h_1(u(s),s)))ds \\
&\quad -\int^{\tau_1}_0 U(\tau_1, s)F(s,u(s),u(h_1(u(s),s)))ds\Big]\Big\|_{\alpha-1} \\
&\quad +\sum_{0<t_i<t}\|[U(\tau_2,t_i)-U(\tau_1,t_i)]I_i(u(t_i^{-}))\|_{\alpha-1},
\end{aligned}\label{eqcts1}
\end{equation}
From  Lemma \ref{lem1} for the first term on the right hand side of
\eqref{eqcts1}, we obtain
\begin{equation}\label{fteq1}
\|[U(\tau_2,0)-U(\tau_1,0)](u_0+G(0,u_0))\|_{\alpha-1}
\leq K_1\|(u_0, G(0,u_0))\|_\alpha(\tau_2-\tau_1),
\end{equation}
where $K_1$ is some positive constant. By the assumptions $(P6)$ and $(P7)$,
it follows that
\begin{equation}
\|G(\tau_2, u(a(\tau_2)))-G(\tau_1, u(a(\tau_1)))\|_{\alpha-1}
\leq K_2 |\tau_2-\tau_1|,
\end{equation}
where $K_2=\|A(0)^{\alpha-\beta-1}\|L_G(1+\mathcal{ L} L_a)$ is a positive constant.
By using Lemma \ref{lem2} \cite[Lemma 14.4]{fri}, third term on the right hand side
of the inequality \eqref{eqcts1} can be calculated as
\begin{equation} \label{steq1}
\begin{aligned}
&\Big\|\Big[\int^{\tau_2}_0U(\tau_2,s)F(s,u(s),u(h_1(u(s),s)))ds\\
&-\int^{\tau_1}_0 U(\tau_1, s)F(s,u(s),u(h_1(u(s),s)))ds\Big]\Big\|_{\alpha-1} \\
&\leq K_3 N \big(|\log(\tau_2-\tau_1)|+1\big) (\tau_2-\tau_1),
\end{aligned}
\end{equation}
where $N=\displaystyle\sup_{0\leq s\leq T}\|F(s,u(s), u(h_1(u(s),s)))\|$ and
$K_3$ are positive constants depending on $\alpha$.
By Lemma \ref{lem1}, we conclude the last term of the right hand side of
\eqref{eqcts1},
\begin{equation}\label{tteq1}
\|[U(\tau_2,t_i)-U(\tau_1,t_i)]I_i(u(t_i^{-}))\|\leq K_4 C_i(\tau_2-\tau_1),
\end{equation}
where $K_4$ is some positive constant and
\[
\sum_{0<t_i<t}\|I_i(u(t_i^{-}))\|\leq \sum_{0<t_i<t} C_i, \quad i=1,\dots,\delta.
\]
From equations \eqref{eqcts1}-\eqref{steq1} and \eqref{tteq1}, we obtain
\begin{equation}
\|Q u(\tau_2)-Q u(\tau_1)\|_{\alpha-1}\leq \mathcal{L }|\tau_2-\tau_1|,
\end{equation}
where $\mathcal{L}$ is a constant such that
\begin{align*}
\mathcal{L}&=\max\Big\{K_1(u_0, G(0,u_0)),
 \frac{\|A(0)^{\alpha-\beta-1}\|L_G}{1-L_a L_G\|A(0)^{\alpha-\beta-1}\|}, \\
&\quad K_3 N(\log|(\tau_2-\tau_1)|+1), \sum_{0<t_i<t}K_4 C_i\},
\end{align*}
which depends on $K_1, K_2, K_3, N, T_0$. Thus, we get $Q u\in Y_\alpha$.

Next, we show that $\sup_{t\in J}\|(Q u)(t)-u_0\|_\alpha\leq r$ for $t\in[0,T_0]$.
Since $u_0\in E_\alpha$. For $u\in \mathcal{S}_\alpha$, we have
\begin{equation} \label{beq12}
\begin{aligned}
\|Q u(t)-u_0\|_\alpha
&\leq \|(U(t,0)-I)(u_0+G(0,u_0))\|_\alpha+\|G(t,u(a(t)))-G(0,u_0)\|_\alpha \\
&\quad +\int^t_0\|U(t,s)F(s,u(s),u(h_1(u(s),s)))\|_\alpha ds \\
&\quad +\sum_{0<t_i<t}\|U(t,t_i)I_i(u(t_i^{-}))\|_\alpha,
\end{aligned}
\end{equation}
Since $u_0\in E_\alpha$ and $u_0+G(0,u_0)\in E_\alpha$ and for $t\in[0,T_0]$,
we have following inequalities
\begin{gather}\label{beq2}
\|(U(t,0)-I)(u_0+G(0,u_0))\|_\alpha+K(\alpha)\sum_{0<t_i<t}C_i\leq \frac{r}{6},\quad
 \forall t\in[0,T_0], \\
\label{beq01} \|G(t, u(a(t)))-G(0,u_0)\|_\alpha\leq L_G[T_0+r]\leq \frac{r}{6},
\quad t\in[0,T_0], \\
\label{beq4} K(\alpha)N\frac{{T_0}^{1-\alpha}}{1-\alpha}\leq\frac{2r}{3},\quad
\forall t\in[0,T_0].
\end{gather}
We estimate the third term on the right hand side of equation \eqref{beq12}
as [see \cite[(14.13)page $160$ and Line 12  page 163]{fri}]
\begin{equation}\label{beq3}
\begin{aligned}
\|\int^t_0 U(t,s)F(s,u(s),u(h_1(u(s),s)))ds\|_{\alpha}
&\leq K(\alpha)N\int^t_0(t-s)^{-\alpha}ds \\
&\leq K(\alpha)N\frac{{T_0}^{1-\alpha}}{1-\alpha}.
\end{aligned}
\end{equation}
Thus, from \eqref{beq12}, \eqref{beq2},\eqref{beq4} and \eqref{beq3},
we conclude that
\begin{equation}
\sup_{t\in J}\|(Q u)(t)-u_0\|_{\alpha}\leq r,\quad t\in[0,T_0],
\end{equation}
Now, we show that $\|Q u(t+h)-Qu(t)\|_{\alpha}\leq Ph^\eta$ for $0<\eta<1$
and some positive constant $P$. If $0\leq t\leq t+h\leq T_0$, then
for $0\leq\alpha<\beta\leq 1$, we have
\begin{equation} \label{kp}
\begin{aligned}
&\|Q u(t+h)-Q u(t)\|_{\alpha}\\
&\leq \|[U(t+h,0)-U(t,0)](u_0+G(0,u_0)\|_{\alpha}
 +\|G(t+h, u(a(t)))-G(t, u(a(t)))\|_\alpha \\
&\quad +\Big\|\Big[\int^{t+h}_0U(t+h, s)F(s,u(s),u(h_1(u(s),s)))ds \\
&\quad -\int^t_0U(t,s)F(s,u(s),u(h_1(u(s),s)))ds\Big]\Big\|_{\alpha} \\
&\quad +\sum_{0<t_i<t}\|[U(t+h, t_i)-U(t,t_i)]I_i(u(t_i^{-}))\|_{\alpha},
\end{aligned}
\end{equation}
From  Lemmas \ref{lem1} and \ref{lem2}, \cite[Lemmas 14.1, 14.4]{fri},
 we obtain the following results
\begin{gather}\label{peq1}
\|[U(t+h,0)-U(t,0)]u_0\|_\alpha
\leq K(\alpha)\| u_0+ G(0,u_0))\|_\beta h^{\beta-\alpha}, \\
\label{peq2}
\|[U(t+h, t_i)-U(t,t_i)]I_i(u(t_i^{-}))\|_\alpha
\leq K(\alpha)h^{\beta-\alpha}\|I_i(u(t_i^{-}))\|_{\beta},\\
\label{peq21}
\|G(t+h, u(a(t+h)))-G(t, u(a(t)))\|_\alpha
 \leq \|A(0)^{\alpha-\beta}\| L_G(1+L_a \mathcal{L})h \\
\label{peq3}
\begin{aligned}
&\|\int^{t+h}_0U(t+h,s)F(s,u(s),u(h_1(u(s),s)))ds\\
&-\int^t_0U(t,s)F(s,u(s),u(h_1(u(s),s)))ds\|_\alpha \\
&\leq K(\alpha)N h^{1-\alpha}(1+|\log (h)|).
\end{aligned}
\end{gather}
Using  \eqref{peq1}-\eqref{peq3} in \eqref{kp}, we obtain
\begin{equation}
\begin{aligned}
&\|Q u(t+h)-Q u(t)\|_{\alpha}\\
&\leq h^{\eta}[K\|u_0+G(0, u_0)\|{T_0}^{\beta-\alpha-\eta}
 +\|A(0)^{\alpha-\beta}\| L_G(1+L_a \mathcal{L})h^{1-\gamma} \\
&\quad +K(\alpha) N{T_0}^\varsigma h^{1-\alpha-\eta-\varsigma}(1+|\log(h)|)
+K(\alpha)h^{\beta-\alpha-\eta}\|I_i(u(t_i^{-}))\|_\beta],
\end{aligned}
\end{equation}
where $\varsigma >0$ is a positive constant and $\varsigma<1-\alpha-\eta$.
Thus, for $t\in[0,T_0]$,
\begin{equation}
\|Q u(t+h)-Q u(t)\|\leq P h^\eta,
\end{equation}
for $P>0$ defined as
\begin{equation}
\begin{aligned}
P&= K\|u_0+G(0, u_0)\|{T_0}^{\beta-\alpha-\eta}
 +\|A(0)^{\alpha-\beta}\| L_G(1+L_a \mathcal{L})h^{1-\gamma} \\
&\quad +K(\alpha) N{T_0}^\varsigma h^{1-\alpha-\eta-\varsigma}
(1+|\log(h)|)+K(\alpha)h^{\beta-\alpha-\eta}\|I_i(u(t_i^{-}))\|_\beta.
\end{aligned}
\end{equation}
Hence $Q:\mathcal{S}_\alpha\to\mathcal{ S}_\alpha$. Now, it remains to
show that $Q$ is a contraction map.
 For $z_1, z_2\in \mathcal{S}_\alpha$ and $t\in[0, T_0]$, we have
\begin{equation} \label{conteq1}
\begin{aligned}
&\|(Q z_1)(t)-(Q z_2)(t)\|_\alpha \\
&\leq  \|G(t,z_1(a(t)))-G(t,z_2(a(t)))\|_\alpha \\
&\quad +K(\alpha)\int^t_0(t-s)^{-\alpha}[\|F(s,z_1(s), z_1(h_1(s,z_1(s))))\\
&\quad -F(s, z_2(s), z_2(h_1(s,z_2(s))))\|]ds \\
&\quad +\sum_{0<t_i<t}\|U(t,t_i)[I_i(z_1(t_i^{-}))-I_i(z_2(t_i^{-}))]\|_\alpha.
\end{aligned}
\end{equation}
Now, we estimate
\begin{equation}
\begin{aligned}
&\|F(t,z_1(t), z_1(h_1(t,z_1(t))))-F(t,z_2(t), z_2(h_1(t,z_2(t))))\| \\
&\leq L_F[\|z_1(t)-z_2(t)\|_\alpha
 +\|z_1(h_1(t,z_1(t)))-z_2(h_1(t,z_2(t)))\|_{\alpha-1}] \\
&\leq L_F[\|z_1(t)-z_2(t)\|_\alpha+\|A^{-1}\|\,
\|z_1(h_1(t,z_2(t)))-z_2(h_1(t,z_2(t)))\|_\alpha \\
&\quad +\|z_1(h_1(t,z_1(t)))-z_1(h_1(t,z_2(t)))\|_{\alpha-1}].
\end{aligned}
\end{equation}
Let
$$
h_j(t,u(t))=b_j(t, u(b_{j+1}(t,\dots u(t,b_\delta(t,u(t)))\dots))),\quad
j=1,2,\dots,\delta,\;u\in\mathcal{ S}_\alpha,
$$
with $h_{\delta+1}(t,u(t))=t$ \cite[p. 2183]{stev2}.

Using the bounded inclusion $E_\alpha\hookrightarrow E_{\alpha-1}$, we obtain
\begin{equation}
\begin{aligned}
&\|z_1(h_j(t, z_2(t)))-z_2(h_j(t, z_2(t)))\|_{\alpha-1}\\
&= \|A^{\alpha-1}z_1(h_j(t, z_2(t)))-z_2(h_j(t, z_2(t)))\|, \\
&\leq \|A^{-1}\|\times \|z_1(h_j(t, z_2(t)))-z_2(h_j(t, z_2(t)))\|_\alpha.
\end{aligned}
\end{equation}
Since $h_j\in \mathbb{R}^+$, we have
\[
\|z_1(t)-z_2(t)\|_\alpha
=\sup_{h_j(t,z_2(t))\in[0,t]}\|z_1(h_j(t, z_2(t)))-z_2(h_j(t, z_2(t)))\|_\alpha.
\]
Therefore,
\begin{align*}
\|z_1(h_j(t, z_2(t)))-z_2(h_j(t, z_2(t)))\|_{\alpha-1}
&\leq \|(A)^{-1}\|\sup_{t\in[0,T_0]}\|z_1(t)-z_2(t)\|_\alpha, \\
&\leq \|A^{-1}\|\times \|z_1-z_2\|_{\mathcal{PC},\alpha}.
\end{align*}
Thus, we can estimate
\begin{align*}
&|h_1(t,z_1(t))-h_1(t,z_2(t))|\\
&= |b_1(t, z_1(h_2(t,z_1(t))))-b_1(t, z_2(h_2(t,z_2(t))))|, \\
&\leq L_{b_1}\|z_1(h_2(t,z_1(t)))-z_2(h_2(t,z_2(t)))\|_{\alpha-1}, \\
&\leq L_{b_1}[\|z_1(h_2(t,z_1(t)))-z_1(h_2(t,z_2(t)))\|_{\alpha-1}, \\
&\quad  +\|z_1(h_2(t,z_2(t)))-z_2(h_2(t,z_2(t)))\|_{\alpha-1}], \\
&\leq L_{b_1}[\mathcal{L}|b_2(t,z_1(h_3(t,z_1(t)))-b_2(t,z_2(h_3(t,z_2(t)))|
 +\|A^{-1}\|\times \|z_1-z_2\|_{\mathcal{PC}, \alpha}], \\
&\dots, \\
&\leq [\mathcal{L}^{
\delta-1}L_{b_1}\dots L_{b_\delta}+\mathcal{L}^{\delta-2}L_{b_1}
 \dots L_{b_{\delta-1}}+\dots+\mathcal{L}L_{b_1} L_{b_2} \\
&\quad +L_{b_1}]\|A^{-1}\|\times\|z_1-z_2\|_{\mathcal{PC}, \alpha}.
\end{align*}
Therefore, we have
\begin{equation} \label{feq3}
\begin{aligned}
&\|F(t,z_1(t), z_1(h_1(t,z_1(t))))-F(t,z_2(t), z_2(h_1(t,z_2(t))))\| \\
&\leq L_F(2+\mathcal{L} L_b\|A^{-1}\|)\|z_1-z_2\|_{\mathcal{PC}, \alpha} \\
&\leq L_F(2+\mathcal{L} L_b)\|z_1-z_2\|_{\mathcal{PC},\alpha},
\end{aligned}
\end{equation}
where $L_b=[\mathcal{L}^{\delta-1}L_{b_1}\dots L_{b_\delta}
+\mathcal{L}^{\delta-2}L_{b_1}\dots L_{b_{\delta-1}}+\dots
+\mathcal{L}L_{b_1} L_{b_2}+L_{b_1}]>0$.
Similarly,
\begin{equation}\label{geq3}
\|G(t,z_1(a(t)))-G(t,z_2(a(t)))\|_\alpha
\leq \|A(0)^{\alpha-\beta}\| L_G\big[\|z_1(t)-z_2(t)\|_\alpha\big].
\end{equation}
Using  inequalities \eqref{feq3}, \eqref{geq3} in \eqref{conteq1}, we deduce that
\begin{equation}
\begin{aligned}
\|(Q z_1)(t)-(Q z_2)(t)\|_\alpha
&\leq \Big{[}\|A(0)^{\alpha-\beta}  \| L_G+K(\alpha)L_F(2+\mathcal{L} L_b)
 \frac{T_0^{1-\alpha}}{(1-\alpha)} \\
&\quad +K(\alpha)\sum_{i=1}^\delta L_i\Big{]}\sup_{t\in J}\|z_1(t)-z_2(t)\|_\alpha
\end{aligned}
\end{equation}
Thus, for $t\in[0,T_0]$,
\begin{align*}
&\|(Q z_1)-(Q z_2)\|_{\mathcal{PC},\alpha}\\
&\leq \Big{[}\|A(0)^{\alpha-\beta}\| L_G+K(\alpha)L_F(2+\mathcal{L } L_b)
\frac{T_0^{1-\alpha}}{(1-\alpha)}
+K(\alpha)\sum_{i=1}^\delta L_i\Big{]}\|z_1-z_2\|_{\mathcal{PC},\alpha}.
\end{align*}
From inequality \eqref{thmeq1}, we get that $Q$ is a contraction map.
Since $\mathcal{S}_\alpha$ is a closed subset of Banach space
 $Y=PC([0,T_0]; E_\alpha )$, therefore $\mathcal{S}_\alpha$ is a
complete metric space. Thus, by Banach fixed point theorem, there exists a
unique fixed point $u\in \mathcal{S}_\alpha$ of map $Q$ which is unique
fixed point, i.e., $Q u(t)=u(t)$. From the Theorem \eqref{thm1},
we conclude that $u$ is a solution for system \eqref{geq1}-\eqref{geq2}
on $[0,T_0]$.
\end{proof}

\section{Example}
In this section, we consider an example to illustrate the discussed theory.
We study the following differential equation with deviated argument
\begin{gather} \label{example1}
\begin{aligned}
&\partial_t[v(t,x)+\partial_x \mathcal{F}_1(t,v(b(t),x))]
-\partial_x(p(t,x)\partial_x)[v(t,x)+\partial_x \mathcal{F}_1(t,v(b(t),x))], \\
&= \widetilde{H}(x, v(t,x))+\widetilde{G}(t, x, v(t,x)); \quad
0<x<1,\; t\in(0,\frac{1}{2})\cup(\frac{1}{2},1)
\end{aligned} \\
v(t,0)= v(t,1)=0, \quad t>0,\\
v(0,x)= u_0(x),\quad x\in(0,1),\\
\label{example2}
\Delta v|_{t=1/2}= \frac{ v(\frac{1}{2})^{-}}{5+v(\frac{1}{2})^{-}},
\end{gather}
where
\begin{gather*}
\widetilde{H}(x,v(x,t))=\int^x_0 \mathcal{K}(x,y)v(y, N(t))dy, \\
N(t)=g_1(t)|v(x, g_2(t)|v(x, \dots g_\delta(t)| v(x,t)|)|)|,
\end{gather*}
 and the map $\widetilde{G}\in C(\mathbb{R}_{+}\times [0,1]\times \mathbb{R};
\mathbb{R})$ is locally Lipschitz continuous in $v$, locally H\"older continuous
in $t$, measurable and uniformly continuous in $x$. Here, we assume that
functions $g_i:\mathbb{R}_{+}\to \mathbb{R}_{+},\;(i=1,2,\dots,\delta)$
are locally H\"older continuous in $t$ such that $g_i(0)=0$ and
$\mathcal{K}:[0,1]\times[0,1]\to \mathbb{R}$ is continuously differentiable
function i.e., $\mathcal{K}\in C^1([0,1]\times[0,1],\mathbb{R})$.

We assume that $p$ is a function which is positive and has continuous partial
derivative $p_x$ such that for each $\tau\in[0,\infty)$ and $0<x<1$, we have
\begin{itemize}
\item[(i)] $0<p_0\leq p(\tau,x)<p'_0$,
\item[(ii)] $ |p_x(\tau,x)|\leq p_1$,
\item[(iii)] $|p(\tau,x)-p(s,x)|\leq C|\tau-s|^\epsilon$,
\item[(iv)] $|p_x(\tau,x)-p_x(s,x)|\leq C|\tau-s|^\epsilon$,
\end{itemize}
for some $\epsilon\in(0,1]$ and some constants $p_0, p_0', p_1, C>0$.
Let us consider $E=L^2((0,1);\mathbb{R})$ and
$$
-\frac{\partial}{\partial x}(p(t,x)\frac{\partial}{\partial x}u(t,x))=A(t)u(t,x),
$$
with $E_1=D(A(0))=H^2(0,1)\cap H^1_0(0,1),\;E_{1/2}=D((A(0))^{1/2})=H^1_0(0,1)$.
Clearly, the family $\{A(t):t>0\}$ satisfies the hypotheses (P1)--(P3)
 on each bounded interval $[0,T]$.

Now, we define the function $f:\mathbb{R}_{+}\times H^2(0,1)\times E_{-1/2}\to E$ as
\begin{equation}
f(t,\xi,\zeta)(x)=\widetilde{H}(x,\zeta)+\widetilde{G}(t,x,\xi),\quad
\text{for }x\in(0,1),
\end{equation}
where $\widetilde{H}:[0,1]\times E_{-1/2}\to E$ is defined as
\begin{equation}
\widetilde{H}(x,\zeta)=\int^x_0 \mathcal{K}(x, y)\zeta(y)dy,
\end{equation}
 and
$\widetilde{G}:\mathbb{R}_+\times[0,1]\times E_{1/2}\to E$
satisfies following condition
\begin{equation}
\|\widetilde{G}(t, x,\xi)\|\leq W(x,t)(1+\|\xi\|_{1/2}),
\end{equation}
where $Q$ is continuous in $t$ and $Q(\cdot, t)\in X$.
Also, we assume that the map $G:\mathbb{R}_+\times H^1_0(0,1)\to L^2(0,1)$
is such that
$$
G(t, v(b(t)))(x)=\partial_x \mathcal{F}_1(t, v(b(t),x))
$$
and satisfies the assumption (P7). There are some possibilities of the map $b$
as follows:
\begin{itemize}
\item[(i)] $b(t)=l t$ for $t\in[0,T]$ and $0<l\leq 1$;

\item[(ii)] $b(t)=l t^n$ for $t\in[0,1]$, $n\in \mathbb{N}$ and $0<l\leq 1$;

\item[(iii)] $b(t)=l\sin( t)$ for $t\in[0,\pi/2]$ and $0<l\leq 1 $.
\end{itemize}
For $v\in D(A)$ and $\lambda\in \mathbb{R}$, with $Av=\lambda v$, we have that
$$
-\frac{\partial}{\partial x}(p(t,x)\frac{\partial}{\partial x}v(x))
=\lambda v(x),
$$
which is the standard Strum-Liouville problem having real eigenvalues.
For $v\in D(A)$ and $\lambda\in\mathbb{R}$, with
$Av=\lambda v=-\frac{\partial}{\partial x}(p(t,x)\frac{\partial}{\partial x}v(x))$,
we have that $\langle Av,v\rangle =\langle \lambda v, v\rangle$; that is,
\begin{equation}
\langle -\frac{d}{dx}(p(t,x)v'), v\rangle
=\langle p(t,x)v', v'\rangle \geq p_0\|v'\|^2_{L^2}
\end{equation}
Since we assume that $p$ is a positive function with $p'_0>p(t,x)>p_0>0$,
where $p_0$ is constant. Thus, we get $\lambda|y|^2_{L^2}\geq p_0\|v'\|^2_{L^2}>0$.
So $\lambda>0$.
In particular case for $p(t,x)=1$, we have
\begin{equation}
v''+\lambda v=0.\label{FQ1}
\end{equation}
\smallskip

\noindent\textbf{Case 1} $\lambda=0$. Then solution of above equation is
$v=C_1 x+C_2$.
Using boundary condition $v(0)=v(1)=0$, we get $C_1=C_2=0$. Thus, $v(x)=0$
be the solution of $v''=0$, which is not an eigenfunction.
\smallskip

\noindent\textbf{Case 2} Let $\lambda=-\mu^2$ and $\mu\ne 0$.
Then equation \eqref{FQ1} reduce to
\begin{equation}
[D^2-\mu^2]v=0\label{FQ2}
\end{equation}
whose auxiliary equation is $D^2-\mu^2=0$ i.e. $D=\pm\mu$.
Thus solution of \eqref{FQ2} is
\begin{equation}
v(x)=C_1 e^{\mu x}+C_2 e^{-\mu x}, \label{FQ3}
\end{equation}
Using the boundary conditions, we get $C_1=C_2=0$.
Thus, \eqref{FQ3} gives $v=0$ which is not an eigenfunction.
\smallskip

\noindent\textbf{Cases 3}
 Let $\lambda=\mu^2$ with $\mu\ne 0$. Thus, equation \eqref{FQ1} reduce to
\begin{equation}
[D^2+\mu^2]v=0.\label{FQ4}
\end{equation}
Therefore, the solution of \eqref{FQ4} is
\begin{equation}
y=C_1\sin(\mu x)+C_2\cos(\mu x).\label{FQ5}
\end{equation}
Using the condition $v(0)=v(1)=0$, we get $C_2=0$ and $C_1\sin(\mu)=0$.
For the non-trivial solution, we have $C_1\ne 0$ and $\sin(\mu)=0$.
Thus, $\mu=n\pi$. Therefore $\lambda_n=\mu^2=n^2\pi^2,\;n\in \mathbb{N}$.
 Hence, \eqref{FQ5} reduces to $v(x)=C_1\sin(n\pi x)$ for $n=1,\dots$,
 and then $\lambda=\mu^2=n^2\pi^2,\;n=1,2,\dots,$. Hence the required
eigenfunction $v_n(x)$ with the corresponding eigenfunction $\lambda_n$
are given by
$$
v_n=C_1\sin(\sqrt{\lambda_n} x),\quad
\lambda_n=n^2\pi^2,\quad n=1,2,\dots.
$$
Next, we show that $\widetilde{H}:[0,1]\times E_{-1/2}\to E$ is defined as
\begin{equation}
\widetilde{H}(x, \zeta(x,t))=\int^x_0 \mathcal{K}(x,y)\zeta(y,t)dy,
\end{equation}
where $\zeta(x,t)=v(x, h_1(t,v(x,t)))$. It is easy to verify that
$f=\widetilde{H}+\widetilde{G}$ satisfies the assumption $(P4)$.
Similarly, we show that the maps $b_i:[0,T]\times E_{-1/2}\to[0,T]$ defined
as $b_i(t)=g_i(t)|\xi(x,\cdot)|$ for $i=1,2,\dots, \delta$ and satisfies
the assumption (P5). For each $t\in[0,T]$, we get
$$
|b_i(t,\xi)|=|g_i(t)||\xi(x,\cdot)|
\leq |g_i|_{\infty}\|\xi\|_{L^\infty(0,1) }\leq N\|\xi\|_{-1/2},
$$
where $N$ is a positive constant, depending on the bounds on $g_i$'s and
we use the embedding $H^1_0(0,1)\subset C[0,1]$. Since we have that
$g_i$ satisfies the condition
\begin{equation}
|g_i(t)-g_i(s)|\leq L_{g_i}|t-s|^{\mu} ,\quad t,s\in[0,T],
\end{equation}
where $L_{g_i}$ is a positive constant and $\mu\in(0,1]$.
For $z_1,z_2\in X_{-1/2}$ and $t\in[0,T]$
\begin{align*}
|b_i(t,z_1)-b_i(t,z_2)|
&\leq \|g_i\|_{\infty}\|z_1-z_2\|_{L^\infty(0,1)}+L_{g_i}|t-s|^{\mu}
 \|z_2\|_{L^\infty(0,1)}, \\
&\leq N\|g_i\|_{\infty}\|z_1-z_2\|_{-1/2}+L_{g_i}|t-s|^{\mu}
 \|z_2\|_{-1/2}, \\
&\leq \max\{N\|g_i\|_{\infty}, L_{g_i\|z_2\|_{\infty}}\}
 (\|z_1-z_2\|_{-1/2}+|t-s|^\mu).
\end{align*}
For $z_1, z_2\in D((-A)^{-1/2})$, then
\begin{equation}
\|I_i(z_1)-I_i(z_2)\|_{1/2}
 \leq\frac{\|z_1-z_2\|_{1/2}}{\|(5+z_1)(5+z_2)\|_{1/2}}
 \leq\frac{1}{25}\|z_1-z_2\|_{1/2}.
\end{equation}
Thus, we can apply the results of previous sections to obtain the existence
 result of the solution for \eqref{example1}-\eqref{example2}.

\subsection*{Acknowledgments}
The authors would like to thank the referee for the valuable comments and suggestions.
The work of the first author is supported by the University Grants
 Commission (UGC), Government of India, New Delhi and Indian Institute
of Technology, Roorkee.


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