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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 38, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/38\hfil Semilinear functional differential equations]
{Semilinear functional differential equations of
fractional order with state-dependent delay}

\author[M. A. Darwish, S. K. Ntouyas\hfil EJDE-2009/38\hfilneg]
{Mohamed Abdalla Darwish, Sotiris K. Ntouyas}  % in alphabetical order


\address{Mohamed Abdalla Darwish\newline
Department of Mathematics, Faculty of Science\\
Alexandria University at Damanhour, 22511 Damanhour, Egypt}
\email{mdarwish@ictp.it, darwishma@yahoo.com}

\address{Sotiris K. Ntouyas\newline
Department of Mathematics, University of Ioannina,
451 10 Ioannina, Greece}
\email{sntouyas@cc.uoi.gr}

\thanks{Submitted January 8, 2009. Published March 10, 2009.}
\subjclass[2000]{26A33, 26A42, 34K30}
\keywords{Functional differential
equations; fractional derivative; \hfill\break\indent
fractional integral; existence; state-dependent delay;
infinite delay; fixed point}

\begin{abstract}
 In this paper we study the existence of solutions for the initial
 value problem  for semilinear functional differential equations
 of fractional order with state-dependent delay.
 The nonlinear alternative of Leray-Schauder type is the main tool
 in our analysis.
\end{abstract}

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


\section{Introduction}

Recently in \cite{DaNt2}, existence results were proved for an initial
value problem for functional
differential equations of fractional order with state-dependent delay
\begin{gather}\label{ee1}
D^{\beta}y(t)=f(t,y_{\rho(t,y_{t})}), \quad  t\in J=[0,b],\; 0<\beta<1, \\
\label{ee2}
y(t)=\varphi(t), \quad t\in   (-\infty,0]
\end{gather}
as well as for  neutral functional
differential equations of fractional order with state-dependent delay
\begin{gather}\label{ee3}
D^{\beta}[y(t)-g(t,y_{\rho(t,y_{t})})]=f(t,y_{\rho(t,y_{t})}), \quad
 \hbox{for }  t\in J, \\
\label{ee4}
y(t)=\varphi(t),\quad  t\in(-\infty,0],
\end{gather}
where $D^\beta$ is the standard Riemman-Liouville fractional
derivative, $f:J \times \mathcal{B}\to \mathbb{R}$,
$ g: J\times \mathcal{B}\to \mathbb{R}$ and
$\rho:J\times \mathcal{B}\to (-\infty, b]$ are appropriate
given functions, $\varphi\in \mathcal{B}$, $\varphi(0)=0$,
$g(0,\varphi)=0$ and $\mathcal{B}$
is called a {\em phase space}.


The purpose of this paper is to extend the results of \cite{DaNt2}
by studying  the existence of solutions
for initial value problems for a functional
semilinear differential equations of fractional order with state-dependent
delay, as well as, for a neutral functional semilinear
differential equations of fractional order with state-dependent delay.
In particular, in Section 3, we consider the following initial value
problem  for a functional semilinear differential equations of fractional
order with state-dependent delay
\begin{gather}\label{e1}
D^{\beta}y(t)=Ay(t)+f(t,y_{\rho(t,y_{t})}), \quad  t\in J=[0,b],\quad 0<\beta<1, \\
\label{e2}
y(t)=\varphi(t), \quad t\in   (-\infty,0],
\end{gather}
while in Section 4, we consider the following initial value problem
for a neutral functional semilinear
differential equations of fractional order with state-dependent delay,
\begin{gather}\label{e3}
D^{\beta}[y(t)-g(t,y_{\rho(t,y_{t})})]=A[y(t)-g(t,y_{\rho(t,y_{t})})]
+f(t,y_{\rho(t,y_{t})}), 
\quad    t\in J, \\
\label{e4}
y(t)=\varphi(t),\quad  t\in(-\infty,0],
\end{gather}
where $D^\beta$ is the standard Riemman-Liouville fractional derivative.

Here, $f:J \times \mathcal{B}\to E$,
$ g: J\times \mathcal{B}\to E$ and
$\rho:J\times \mathcal{B}\to (-\infty, b]$ are appropriate
given functions, $\varphi\in \mathcal{B}$, $\varphi(0)=0$,
$g(0,\varphi)=0$, $A:D(A)\subseteq E\to E$ is the infinitesimal
generator of a strongly continuous semigroup $\{T(t)\}_{t\ge 0}$,
and $\mathcal{B}$
is called a {\em phase space} that will be defined
later (see Section 2).

The notion of the phase space $\mathcal{B}$ plays an important
role in the
study of both qualitative and quantitative theory
for functional differential equations.  A usual choice
is a semi-normed space satisfying suitable axioms,
which was introduced by Hale and Kato \cite{HaKa} (see also
Kappel and Schappacher \cite{KaSc} and Schumacher \cite{Sch}).
For a detailed
discussion on this topic we refer the reader to the
  book by Hino {\em et al} \cite{HiMuNa}.


While functional differential equations have been used in modelling a
panorama of natural phenomena as discussed in the books by Kolmanovskii and
Myshkis \cite{KoMy} and Hale and Lunel \cite{HaLu}, it has been only recently that
fractional differential equations have begun to see extensive utilization in
modelling problems that arise in engineering and other sciences, including
viscoelasticity, electrochemistry, control, porous media flow, physics,
mechanics and others \cite{GaRaKa,Hi,KiSrTr,MiRo,Pod,SaKiOl,YuGa}.
On the other hand, functional differential equations with state-dependent
delay appear frequently in applications as model of equations and for
this reason the study of this type of equation has received a significant
amount of attention in the last years, we refer to
\cite{AiFrWu,Ba,CaFaGa,DoDrLi,He1,He2,He3} and the references therein.

In part, differential equations of fractional order play a very important role in describing some real world problems.
For example some problems in physics, mechanics and other fields can be described with the help of fractional differential
equations, see \cite{GaRaKa,GlNo,Hi,Mai,SaKiOl,SaKa,SaMaHa} and references therein.
The theory of differential equations of fractional order has recently received a lot of attention and now
constitutes a significant branch of nonlinear analysis. Numerous research papers and monographs have appeared devoted to
fractional differential equations, for example see
\cite{Ag,BeHeNtOu,DaNt1,DaNt3,DeRo,KiSrTr,La,LaDe,LaVa1,LaVa2,LaVa3,Pod,YuGa}.

Our approach is based on the
nonlinear alternative of Leray-Schauder
type \cite{DuGr}. These results can be
considered as a contribution to this emerging field.

\section{Preliminaries}

In this section, we introduce notation,
definitions, and preliminary facts which are used throughout
this paper.

By $C(J,E)$ we denote the Banach space  of continuous functions from
$J$ into $E$ with the norm
$$
\|y\|_{\infty}:=\sup\{|y(t)|: t\in J\}.
$$

Now, we recall some definitions and facts about fractional derivatives
and fractional integrals of arbitrary orders, see
\cite{KiSrTr,MiRo,Pod,SaKiOl}.

\begin{definition}  \rm
The fractional  primitive of order $\beta>0$ of a function
$h: (0,b]\to E$  is defined by
$$
I^{\beta}_0h(t)=\int_0^t\frac{(t-s)^{\beta-1}}{\Gamma(\beta)}h(s)ds,
$$
provided the right hand side exists pointwise on $(0,b]$, where
$\Gamma$ is the gamma function.
\end{definition}

For instance, $I^{\beta}h$ exists for all $\beta>0$,
when $h\in C((0,b],E)\cap L^1((0,b],E);$ note also that when $h\in
C([0,b],E)$ then $I^{\beta}h\in C([0,b],E)$ and moreover
$I^{\beta}h(0)=0$.

\begin{definition} \rm
The fractional derivative of order $\beta>0$ of a continuous
function $h: (o,b]\to E$ is given by
\[
\frac{d^{\beta}h(t)}{dt^{\beta}}
= \frac{1}{\Gamma(1-\beta)}\frac
{d}{dt}\int_a^t(t-s)^{-\beta}h(s)ds
= \frac{d}{dt}I_a^{1-\beta}h(t).
\]
\end{definition}

In this paper, we will employ an axiomatic definition for the phase
space $\mathcal{B}$ which is similar to
those introduced in \cite{HiMuNa}. More precisely, $\mathcal{B}$ will
be a linear space of all functions from $(-\infty,0]$ to
$E$ endowed with a seminorm $\|\cdot\|_{\mathcal{B}}$ satisfying the
 following axioms:
\begin{itemize}
\item[(A)] If  $y:(-\infty,b]\to E$, $b>0$, is continuous on $J$ and
$y_{0}\in\mathcal{B}$, then for every $t\in J$ the
following conditions hold:
\begin{itemize}
\item[(i)]  $y_{t}\in\mathcal{B}$,
\item[(ii)]
$\|y_{t}\|_{\mathcal{B}}\le K(t)\sup \{|y(s)| : 0 \le s\le t\}+
M(t)\|
y_{0}\|_{\mathcal{B}}$,
\item[(iii)] $|y(t)|\leq H\|y_t\|_{\mathcal{B}}$,
\end{itemize}
where $H>0$ is a constant, $K:[0,\infty)\to [1,
\infty)$ is
continuous,   $M:[0,\infty)\to [1,
\infty)$ is
locally bounded
and $H,\;K,\;M$  are independent of $y(\cdot)$.
 \item[(A1)] For the function $y(\cdot)$ in  $(A)$,
$y_{t}$ is a
 $\mathcal{B}$-valued continuous
function on $[0, b]$.
\item[(A2)] The space $\mathcal{B}$ is complete.
\end{itemize}
The next lemma is a consequence of the phase space axioms and is
proved in \cite{He1}.

\begin{lemma}\label{lh1}
Let $\varphi\in \mathcal{B}$ and $I=(\gamma,0]$ be such that
$\varphi_t\in \mathcal{B}$ for every $t\in I$. Assume
that there exists a locally bounded function $J^{\varphi}:I\to [0,\infty)$
such that $\|\varphi_t\|_{\mathcal{B}}\le
J^{\varphi}(t)\|\varphi\|_{\mathcal{B}}$ for every $t\in I$.
If $y:(\infty, b]\to \mathbb{R}$ is continuous on $J$ and
$y_0=\varphi$, then
$$
\|y_t\|_{\mathcal{B}}\le (M_b+J^{\varphi}(\max\{\gamma,-|s|\})\|\varphi\|_{\mathcal{B}}+K_b
\sup\{|y(\theta)|:\theta\in [0,\max\{0,s\}]\},
$$
for $s\in (\gamma,b]$, where we denoted $K_b=\sup_{t\in J}K(t)$
and $M_b=\sup_{t\in J}M(t)$.
\end{lemma}

\section{Main Result}

In this section, the nonlinear alternative of Leray-Schauder type is
used to investigate the existence of solutions of problem
\eqref{e1}-\eqref{e2}.

Let us start by defining what we mean by a solution
of problem \eqref{e1}-\eqref{e2}.

\begin{definition}\label{d1} \rm
A function $y:(-\infty,b]\to E$ is said to be a solution of
\eqref{e1}-\eqref{e2} if $y_0=\varphi, y_{\rho(s,y_s)}\in \mathcal{B}$
for every $s\in J$ and
$$
y(t)=\frac{1}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s)
f(s,y_{\rho(s,y_{s})})\,ds, \quad t\in J.
$$
\end{definition}

In what follows we assume that $\rho:J\times \mathcal{B}\to (-\infty,b]$
is continuous and $\varphi\in \mathcal{B}$ and   the following hypotheses
are satisfied
\begin{itemize}
\item[(H1)] $A$ is the infinitesimal generator of a strongly continuous
semigroup of bound\-ed linear operators $T(t), t\ge 0$ in $E$, which
is compact for $t>0$, and there exist constant  $M\ge 1$
 such that $\|T(t)\|_{B(E)}\le M, t\ge 0$;

\item[(H2)] $f:J\times\mathcal{B}\to E$ is a continuous function;

\item[(H3)]  there exists $p\in C([0,b],\mathbb{R}^+) $ and
$\Omega:[0,\infty)\to (0,\infty)$ continuous and nondecreasing such
that
$$
|f(t,u)|\leq p(t)\Omega(\|u\|_{\mathcal{B}})
$$
for $ t\in [0,b]$ and each $u\in \mathcal{B}$;

\item[(H4)]
there exists a number $K_0>0$ such that
$$
\frac{K_0}{(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
+MK_b\Omega(K_0)\|I^{\beta}p\|_{\infty}}> 1;
$$

\item[(H5)] the function $t\to\varphi_t$ is well defined and continuous
from the set $\mathcal{R}(\rho^-)=\{\rho(s,\psi):(s,\psi)\in J\times B,
\rho(s,\psi)\leq 0\}$ into $\mathcal{B}$. Moreover, there exists a
continuous and bounded function
$J^{\varphi}:\mathcal{R}(\rho^-)\to (0,\infty)$ such that
$\|\varphi_t\|_{\mathcal{B}}\le J^{\varphi}(t)\|\varphi\|_{\mathcal{B}}$
for every $t\in \mathcal{R}(\rho^-)$.
\end{itemize}

\begin{remark} \rm
The hypothesis (H5) is adapted from \cite{He1}, where we refer for
remarks concerning this hypothesis.
\end{remark}

\begin{theorem}\label{t1}
Assume that the     hypotheses {\rm (H1)--(H5)} hold.
If $\rho(t,\psi)\le t$ for every $(t,\psi)\in J\times\mathcal{B}$, then
the \eqref{e1}-\eqref{e2} has at least one  solution on $(-\infty,b]$.
\end{theorem}

\begin{proof}
 Let $Y=\{u\in C(J, E):u(0)=\varphi(0)=0\}$ endowed with the
uniform convergence topology and $N:Y\to Y$ be the operator defined by
$$
Ny(t)=\frac{1}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s)f(s, \bar
y_{\rho(s,\bar{y}_{s})})\,ds, \quad t\in J,
$$
where $\bar y:(-\infty,b]\to E$  is such that
$\bar y_0=\varphi$ and $\bar y=y$ on $J$. From axiom (A) and our
assumption on $\varphi$, we infer that $Ny(\cdot)$ is well defined
and continuous.

Let $\bar \varphi:(-\infty,b]\to E$ be the extension of
$\varphi$ to $(-\infty, b]$ such that $\bar{\varphi}(\theta)=\varphi(0)=0$
on $J$ and
$\tilde J^{\varphi}=\sup\{J^{\varphi}:s\in \mathcal{R}(\rho^{-})\}$.

We will prove that $N(\cdot)$ is completely continuous from
$B_r(\bar\varphi|_J,Y)$ to $B_r(\bar\varphi|_J,Y)$.

\noindent{\bf Step 1:}
 $N$ is continuous on  $B_r(\bar\varphi|_J,Y)$.
This was proved in \cite[p. 515, Step 3]{He1}.

\noindent{\bf Step 2:}  $N$ maps bounded sets into bounded sets.
  If $y\in B_r(\bar\varphi|_J,Y)$, from Lemma \ref{lh1} follows that
$$
\|\bar y_{\rho(t,\bar y_{t})}\|_{\mathcal{B}}
\le r^*:=(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_br
$$
and so
\begin{align*}
|(Ny)(t)|&= \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}T(t-s)f(s, \bar
y_{\rho(s,\bar{y}_{s})})\,ds\\
&\leq \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}
p(s)\Omega(\|\bar{y}_{\rho(s,\bar{y}_{s})}\|_{\mathcal{B}}) \,ds\\
&\leq  \frac{M}{\Gamma(\beta)}\|p\|_{\infty}\Omega(r^*)\int_0^t(t-s)^{\beta-1}\, ds\\
&\leq  \frac{Mb^{\beta}}{\Gamma(\beta+1)}\|p\|_{\infty}\Omega(r^*).
\end{align*}
Thus
$$
\|Ny\|_{\infty}\le \frac{Mb^{\beta}}{\Gamma(\beta+1)}\|p\|_{\infty}\Omega(r^*):=\ell.$$

\noindent{\bf Step $3$:}
 $N$ maps bounded sets into equicontinuous sets of $B$.
Let $t_1,\;t_2\in(0,b]$ with $t_1<t_2$ and $B_\alpha$ be a bounded set
 as in Step $2$. Let
$\epsilon>0$ be given. Now let
$\tau_1,\tau_2\in J$ with
$\tau_2>\tau_1$. We consider two cases $\tau_1>\epsilon$ and $\tau_1\le
\epsilon$.

\noindent{\bf Case 1.} If $\tau_1>\epsilon$  then
\begin{align*}
&|({N}y)(t_2)-({N}y)(t_1)|\\
&\leq \frac{1}{\Gamma(\beta)}\int_0^{t_1-\epsilon}
[(t_2-s)^{\beta-1}T(t_2-s)-(t_1-s)^{\beta-1}T(t_1-s)]|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\
&\quad +\frac{1}{\Gamma(\beta)}\int_{t_1-\epsilon}^{t_1}\left[(t_2-s)^{\beta-1}T(t_2-s)-(t_1-s)^{\beta-1}T(t_1-s)\right]|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\
&\quad +\frac{1}{\Gamma(\beta)}\int_{t_1}^{t_2}(t_2-s)^{\beta-1}T(t_2-s)|f(s,\bar{y}_{\rho(s,\bar{y}_{s})})|\,ds\\
&\leq \frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)}
\Big(\Big|\int_0^{t_1-\epsilon}[(t_2-s)^{\beta-1}
-(t_1-s)^{\beta-1}]T(t_2-s)\,ds\Big|\\
&\quad +\Big|\int_0^{t_1-\epsilon}(t_1-s)^{\beta-1}T(t_1-\epsilon-s)
[T(t_2-t_1-\epsilon)-T(\epsilon)]ds\Big|\\
&\quad +\Big|\int_{t_1-\epsilon}^{t_1}
[(t_2-s)^{\beta-1}-(t_1-s)^{\beta-1}]T(t_2-s)\,ds\Big|\\
&\quad +\Big|\int_{t_1-\epsilon}^{t_1}
(t_1-s)^{\beta-1}T(t_1-\epsilon-s)[T(t_2-t_1-\epsilon)-T(\epsilon)]\,ds\Big|
+\int_{t_1}^{t_2} \! (t_2-s)^{\beta-1}\, ds\Big)\\
&\leq \frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)}
 \Big(M\int_0^{t_1-\epsilon}[(t_2-s)^{\beta-1}
-(t_1-s)^{\beta-1}]\, ds \\
&\quad +M\|T(t_2-t_1-\epsilon)-T(\epsilon)\|_{B(E)}
 \int_0^{t_1-\epsilon}(t_2-s)^{\beta-1}\, ds\\
&\quad +M\int_{t_1-\epsilon}^{t_1}
 [(t_2-s)^{\beta-1}-(t_1-s)^{\beta-1}]ds\\
&\quad +M\|T(t_2-t_1-\epsilon)-T(\epsilon)\|_{B(E)}
 \int_{t_1-\epsilon}^{t_1}(t_2-s)^{\beta-1}\,
ds+M\int_{t_1}^{t_2}(t_2-s)^{\beta-1}\, ds\Big),
\end{align*}
where  we have used the semigroup identities
$$
T(\tau_2-s)=T(\tau_2-\tau_1+\epsilon)T(\tau_1-s-\epsilon),
\quad T(\tau_1-s)=T(\tau_1-s-\epsilon)T(\epsilon).
$$


\noindent{\bf Case 2.} Let  $\tau_1\le \epsilon$.
For $\tau_2-\tau_1<\epsilon$ we get
\begin{align*}
|({N}y)(t_2)-({N}y)(t_1)|
&\leq \frac{1}{\Gamma(\beta)}\Big|\int_0^{t_2}
(t_2-s)^{\beta-1}T(t_2-s)  f(s,\bar{y}_{\rho(s,\bar{y}_{s})}) \,ds\\
&\quad -\int_0^{t_1}(t_2-s)^{\beta-1}T(t_2-s)  f(s,\bar{y}_{\rho(s,\bar{y}_{s})})
 \,ds\Big|\\
&\leq M\frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)}
\Big(\int_0^{2\epsilon}(t_2-s)^{\beta-1}\,ds
+\int_0^{\epsilon}(t_1-s)^{\beta-1}\, ds\Big).
\end{align*}
Note equicontinuity follows since (i). $T(t), t \geq 0$ is a strongly
continuous semigroup and (ii).
$T(t)$ is compact for $t>0$ (so $T(t)$ is continuous in
the uniform operator topology for $t>0$) \cite{Pa}.

From the steps $1$ to $3$, together with   Arzel\'a-Ascoli theorem,
it suffices to show that $N$ maps $B_{\alpha}$ into a precompact
set in $E$.

Let $0<t<b$ be fixed and let $\epsilon$ be a real number satisfying
$0<\epsilon<t$. For $y\in B_{\alpha}$ we define
$$
{N}_{\epsilon}(y)(t)=\frac{T(\epsilon)}{\Gamma(\beta)}
\int_0^{t-\epsilon}(t-s-\epsilon)^{\beta-1}
T(t-s-\epsilon)f(s,\bar{y}_{\rho(s,\bar{y}_{s})})\, ds.
$$
Since $T(t)$ is a compact operator for $t>0$, the set
$Y_{\epsilon}(t)=\{{N}_{\epsilon}(y)(t):y\in B_{\alpha}\}$
is precompact in $E$ for
every $\epsilon$, $0<\epsilon<t$. Moreover
\begin{align*}
&|{N}(y)(t)-{N}_{\epsilon}(y)(t)|\\
&\leq  M\frac{\|p\|_{\infty}\Omega(r^*)}{\Gamma(\beta)}
\Big(\int_0^{t-\epsilon}
[(t-s)^{\beta-1}-(t-s-\epsilon)^{\beta-1}]\, ds
+\int_{t-\epsilon}^t(t-s)^{\beta-1}\, ds\Big).
\end{align*}
Therefore, the set $Y(t)=\{{N}(y)(t):y\in B_{\alpha}\}$ is precompact
in $E$. Hence the operator ${N}$ is completely continuous.


\noindent{\bf Step 4:}  (A priori bounds).
 We now show there exists an open set $U\subseteq Y$ with
$y\ne \lambda N(y)$ for $\lambda\in (0,1)$ and
$y\in \partial U$.
Let $y\in Y$ and  $y=\lambda N(y)$ for
some $0<\lambda<1$. Then for each $t\in[0,b]$  we have
$$
y(t)=\lambda\Bigl[\frac{1}{\Gamma(\beta)}
\int_{0}^{t}(t-s)^{\beta-1} T(t-s) f(s,
\bar{y}_{\rho(s,\bar{y}_{s})})\,ds\Bigr].
$$
This implies by (H3) and lemma \ref{lh1} that
\begin{align*}
|y(t)|&\leq \frac{1}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}||T(t-s)|
 f(s,\bar{y}_{\rho(s,\bar{y}_{s})})| \,ds\\
&\leq \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}p(s)
 \Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
 +K_b\sup\{|\bar{y}(s)|:s\in [0,t]\}) \,ds,
\end{align*}
since $\rho(s,\bar{y}_{s})\le s$ for every $s\in J$.
Here $\bar J^{\phi}=\sup\{J^{\phi}(s): s\in \mathcal{R}(\rho^{-})\}$.

Set $\mu(t)=\sup\{|y(s)|:0\le s\le t\}\; t\in [0,b]$. Then we have
$$
\mu(t)\le \frac{M}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}p(s)
\Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s)) \,ds.
$$
If $\xi(t)=(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(t)$
then we obtain
\begin{align*}
\xi(t)&\leq  (M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
+\frac{MK_b}{\Gamma(\beta)}\int_0^t (t-s)^{\beta-1}p(s)\Omega(\xi(s))\, ds\\
&\leq  (M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
+MK_b\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}.
\end{align*}
Then
$$
\frac{\|\xi\|_{\infty}}{(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
+MK_b\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}}\le 1\,.
$$
By (H4), there exists  $M_*$ such that $\|y\|_{\infty}\not=M_*$.
Set
$$
U=\{y\in Y: \|y\|_{\infty}<M^*+1\}.
$$
Then $N : \overline U\to Y$ is continuous and completely continuous.
 From the choice of $U$, there is no $y\in \partial U$ such that
$y=\lambda N(y)$,  for   $\lambda\in(0,1)$. As a consequence of the
nonlinear alternative of Leray-Schauder type \cite{DuGr}, we
deduce that $N$   has a fixed point $y$ in $U$, which is a solution
of \eqref{e1}-\eqref{e2}.
\end{proof}

\section{NFDEs  of Fractional Order}

In this section we give an  existence result  for \eqref{e3}-\eqref{e4}.

\begin{definition}\label{d01} \rm
A function $y:(-\infty,b]\to E$ is said to be a solution of
\eqref{e3}-\eqref{e4}  if $y_0=\varphi, y_{\rho(s,y_s)}\in \mathcal{B}$
for every $s\in J$ and
$$
y(t)=g(s, y_{\rho(s,y_{s})})+\frac{1}{\Gamma(\beta)}
\int_0^t(t-s)^{\beta-1}T(t-s)f(s,y_{\rho(s,y_{s})})ds,
\quad t\in J.
$$
\end{definition}

\begin{theorem} \label{t2}
Assume {\rm (H1)-(H3),  (H5)} are satisfied. In addition we suppose
that  the following two conditions hold:
\begin{itemize}
\item[(H6)]  the function $g$ is  continuous and
completely continuous, and for
any bounded set $Q$ in $\mathcal{B}\cap C([0,b],E)$, the set
$\{t\to g(t,y_t):
y\in Q\}$ is equicontinuous in $C([0,b],E)$, and
there exist
constants  $0\leq d_1<1/K_b, \ d_2\geq 0$ such that
$$
|g(t,u)|\leq d_1\|u\|_\mathcal{B}+d_2,\ \    t\in[0,b],\  u\in
\mathcal{B};
$$
\item[(H7)]  there exists a number $K_0>0$ such that
$$
\frac{\|\xi\|_{\infty}}
{(M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}+\frac{K_b}{1-K_b d_1}
\big\{d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{
B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\big\}}>1.
$$
\end{itemize}
If $\rho(t,\psi)\le t$ for every $(t,\psi)\in J\times\mathcal{B}$, then
the \eqref{e3}-\eqref{e4} has at least one solution on $(-\infty,b]$.
\end{theorem}

\begin{proof}
Consider the operator
$N_0: C((-\infty,b],E)\to C((-\infty,b],E)$
defined by,
$$
N_0(y)(t)=\begin{cases}
\varphi(t),&   \text{if } t\in (-\infty,0], \\
g(t,y_{\rho(t,y_{t})})\\
+\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^
{\beta-1}T(t-s)f(s, y_{\rho(s,y_{s})})ds,
&\text{if  } t\in[0,b].
\end{cases}
$$
In analogy to Theorem \ref{t1}, we consider the
operator $N_1:Y\to Y$ defined by
$$
(N_1y)(t)=\begin{cases}
0, & t\le 0\\
g(t, \bar{y}_{\rho(s,\bar{y}_{s})})+\frac{1}{\Gamma(\beta)}
\int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s, \bar{y}_{\rho(s,\bar{y}_{s})})ds,
& t\in [0,b].
\end{cases}
$$

 We shall show that the operator $N_1$ is  continuous
and completely continuous.
Using (H6) it suffices to show that the operator $N_2:
Y\to Y$, defined by
$$
N_2(y)(t)=\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s,
\bar{y}_{\rho(s,\bar{y}_{s})})\, ds,\quad t\in [0,b],
$$
is continuous and completely continuous. This was proved  in
Theorem \ref{t1}.

{\em We now show there exists an open set $U\subseteq
Y$ with $y\ne \lambda N_1(y)$ for $\lambda\in (0,1)$ and $y\in
\partial U.$}
Let $y\in Y$ and  $y=\lambda N_1(y)$ for some
$0<\lambda<1$. Then
\[
y(t)=\lambda\Big[g(s, \bar
y_{\rho(s,\bar{y}_{s})})\,+\frac{1}{\Gamma(\beta)}\int_{0}^{t}(t-s)^{\beta-1}T(t-s)f(s, \bar
y_{\rho(s,\bar{y}_{s})})\,ds\Big],\quad t\in[0,b],
\]
and
\begin{align*}
|y(t)|
&\leq d_1\Big[(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\sup\{|y(s)|:s\in [0,t]\}\Big]+d_2
\\
&\quad+\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)
\Omega((M_b+\tilde J^{\varphi})\|\varphi\|_{\mathcal{B}}
+K_b\sup\{|\bar y(s)|:s\in [0,t]\})ds,
\end{align*}
for $t\in(0,b]$.
If $\mu(t)=\sup\{|y(s)|:s\in [0,t]\}$ then
\begin{align*}
\mu(t)&\leq d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+d_1K_b\mu(t)+d_2 \\
&\quad +\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega((M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s))\, ds,
\end{align*}
or
\begin{align*}
\mu(t)
&\leq  \frac{1}{1-K_b d_1}\big[d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{
B}}+d_2\big]\\
&\quad +\frac{1}{1-K_b d_1}\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega((M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s))\, ds,
\end{align*}
for $t\in (0,b]$.
If $\xi(t)=(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+K_b\mu(s)$ then we have
\begin{align*}
\xi(t)&\leq (M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}\\
&\quad +\frac{K_b}{1-K_b d_1}\Big\{d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{
B}}+d_2+\frac{M}{\Gamma(\beta)}\int_0^t(t-s)^{\beta-1}p(s)\Omega(\xi(s))\,
 ds\Big\}\\
&\leq (M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}\\
&\quad +\frac{K_b}{1-K_b d_1}\Big\{d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{
B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\Big\}.
\end{align*}
Consequently,
$$
\frac{\|\xi\|_{\infty}}{(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{B}}+\frac{K_b}{1-K_b d_1}
\big\{d_1(M_b+\tilde
J^{\varphi})\|\varphi\|_{\mathcal{
B}}+d_2+M\Omega(\|\xi\|_{\infty})\|I^{\beta}p\|_{\infty}\big\}}\le 1.
$$
By (H7), there exists $L^*$ such that $\|y\|_{\infty}\ne L^*$.
Set
$$
U_1=\{y\in Y: \|y\|_{\infty}<L^*+1\}.
$$
 From the choice of $U_1$ there is no $y\in
\partial U_1$ such that $y=\lambda N_1(y)$ for
$\lambda\in(0,1)$. As a consequence of the nonlinear
alternative of Leray-Schauder type \cite{DuGr}, we deduce that
$N_1$ has a fixed point $y$ in $U_1$. Then $N_1$ has a fixed
point, which is a solution
of \eqref{e3}-\eqref{e4}.
\end{proof}


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\end{document}