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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 77, pp. 1--9.\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-2015/77\hfil Initial value problems of fractional order]
{Initial value problems of fractional order Hadamard-type  functional
differential equations}

\author[B. Ahmad, S. K. Ntouyas \hfil EJDE-2015/77\hfilneg]
{Bashir Ahmad, Sotiris K. Ntouyas}

\address{Bashir Ahmad \newline
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group,
Department of Mathematics, Faculty of Science,
 King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{bashirahmad\_qau@yahoo.com}

\address{Sotiris K. Ntouyas \newline
Department of Mathematics, 
University of Ioannina, 
451 10 Ioannina, Greece.\newline
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group,
Department of Mathematics, Faculty of Science,
 King Abdulaziz University,
P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{sntouyas@uoi.gr}

\thanks{Submitted January 10, 2015. Published March 26, 2015.}
\subjclass[2000]{34A08, 34K05}
\keywords{Fractional differential equation;
functional  differential equation; \hfill\break\indent
 Hadamard fractional differential equation; existence; fixed point}

\begin{abstract}
 The Banach fixed point theorem and a nonlinear alternative of
 Leray-Schauder type are used to investigate the existence and uniqueness of
 solutions for  fractional order Hadamard-type functional and neutral 
 functional differential equations.
\end{abstract}

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

\section{Introduction}

This article concerns the existence of solutions for initial value
problems (IVP for short) of fractional order functional and neutral functional
differential equations. In the first problem, we consider fractional order
 functional differential equations:
\begin{gather}\label{e1}
D^{\alpha}y(t)=f(t,y_{t}), \quad \text{for each } t\in J=[1,b],\; 0<\alpha<1,\\
\label{e2}
y(t)=\phi(t), \quad t\in  [1-r,1],
\end{gather}
where $D^{\alpha}$ is the Hadamard fractional
derivative,  $f: J\times C([-r,0],\mathbb{R})\to\mathbb{R}$ is a given function
and $\phi\in C([1-r,1],\mathbb{R})$ with $\phi(1)=0$. For any function $y$
defined on $[1-r,b]$ and any $t\in J$,  we denote by $y_{t}$ the
element of $C([-r,0],\mathbb{R})$ and is defined by 
\[
 y_{t}(\theta)=y(t+\theta), \quad 
\theta\in [-r,0].
\]
 Here $y_{t}(\cdot)$ represents the history of
the state from time $t-r$ up to the present time $t$. 

The second problem is devoted to the study of  fractional neutral 
functional differential equation:
\begin{gather}\label{e3}
D^{\alpha}[y(t)-g(t,y_t)]=f(t,y_{t}),\quad  t\in J, \\
\label{e4}
y(t)=\phi(t),\quad t\in[1-r,1],
\end{gather}
where  $f$ and $\phi$ are as in problem \eqref{e1}--\eqref{e2},
and   $ g: J\times C([-r,0],\mathbb{R})\to\mathbb{R}$ is a given
function such that $g(1,\phi)=0$.

 Functional and neutral functional differential equations arise in a
variety of areas of biological, physical, and engineering
applications, see, for example, the books  \cite{HaLu,KoMy} and the
references therein. Differential equations of fractional order have
recently proved to be valuable tools in the modeling of many
phenomena in various fields of science and engineering. Indeed, we
can find numerous applications in various fields such as viscoelasticity, 
electrochemistry, control, porous media, electromagnetic, etc. 
(see  \cite{KST, MiRo,  Pod, SaKiMa}).

Fractional differential equations involving  Riemann-Liouville and  
Caputo type fractional derivatives have extensively  been studied by 
several researchers \cite{AZH, AZWL, fp16,   fp23,  pf3,   fp1, El2, YG1, YuGa}. 
However, the literature on Hadamard type fractional differential equations 
is not enriched yet. The fractional derivative due to Hadamard, introduced 
in 1892 \cite{Hadd}, differs from the aforementioned derivatives in the 
sense that the kernel of the integral in the definition of Hadamard derivative  
contains logarithmic function of arbitrary exponent. A detailed description 
of Hadamard fractional  derivative and integral can be found 
in \cite{Had, Had1, Had2} and references cited therein.


The IVPs \eqref{e1}--\eqref{e2} and \eqref{e3}--\eqref{e4} in the case of 
infinite delay and Riemann-Liouville fractional derivative was studied 
in \cite{BHNO}. IVP for   hybrid Hadamard fractional differential equations 
was studied in \cite{AN-H}.  Here we study
 the problems involving  Hadamard-type fractional derivatives.
Our approach is based on the Banach fixed point theorem and nonlinear 
alternative of Leray-Schauder type \cite{GrDu}.
The rest of this paper is organized as follows: in Section 2 we recall some 
useful preliminaries. In Section 3
we discuss the  existence and uniqueness of solutions  for the problem  
\eqref{e1}--\eqref{e2}, while the
existence results for the problem  \eqref{e3}--\eqref{e4} are presented 
in Section 4.  Finally, an example is given in Section 5 for illustration 
of  the  results.


\section{Preliminaries}

 In this section, we introduce notation, definitions,
and preliminary facts that we need in the sequel.

By $C(J,\mathbb{R})$ we denote the Banach space  of all continuous functions
from $J$ into $\mathbb{R}$ with the norm 
$$ 
\|y\|_{\infty}:=\sup\{|y(t)|:t\in J\}.
 $$
Also $C([-r,0],\mathbb{R})$ is endowed with the norm 
$$
 \|\phi\|_{C}:=\sup\{|\phi(\theta)|: -r\le \theta\le 0\}.
$$


 \begin{definition}[\cite{KST}] \rm
The Hadamard derivative of fractional order $q$ for a function 
$g: [1, \infty)$ $\to \mathbb{R}$ is
defined as
 $$
D^q g(t)=\frac{1}{\Gamma(n-q)}\Big(t\frac{d}{dt}\Big)^n\int_1^t
\Big(\log\frac{t}{s}\Big)^{n-q-1}\frac{g(s)}{s}ds, \quad n-1 < q < n,
n=[q]+1,
$$   
where $[q]$ denotes the integer part of the real
 number $q$ and $\log (\cdot) =\log_e (\cdot)$. 
\end{definition}

\begin{definition}[\cite{KST}] \rm
The Hadamard fractional integral of order $q$ for a function $g$ is defined as
$$
I^q g(t)=\frac{1}{\Gamma(q)}\int_1^t 
\Big(\log\frac{t}{s}\Big)^{q-1}\frac{g(s)}{s}ds, \quad q>0, 
$$
provided the integral exists.
\end{definition}

\section{Functional differential equations}

\begin{definition}\label{d1}\rm
 A function  $y\in C([1-r,b],\mathbb{R})$, 
is said to be a solution of \eqref{e1}--\eqref{e2}  if $y$
satisfies the equation $D^{\alpha}y(t)=f(t,y_{t})$ on
$J$,  and
the condition $y(t)=\phi(t)$  on   $[1-r,1]$.
\end{definition}

 Our first existence result for  \eqref{e1}--\eqref{e2} is
based on the Banach contraction principle.

\begin{theorem}\label{con}
Let $f: J\times C([-r,0], \mathbb{R})\to \mathbb{R}$. Assume that
\begin{itemize}
\item[(H0)] there exists $\ell>0$ such that 
$$
|f(t,u)-f(t,v)|\le \ell\|u-v\|_{C},
\quad \text{for $t\in J$ and every $ u, v \in C([-r,0],\mathbb{R})$}.
$$
\end{itemize}
If $\frac{\ell (\log b)^{\alpha}}{\Gamma(\alpha+1)}<1$, then there exists a
unique solution for  \eqref{e1}--\eqref{e2}
on the interval $[1-r,b]$.
\end{theorem}

\begin{proof} 
Transform problem \eqref{e1}--\eqref{e2} into a
fixed point problem. Consider the operator 
$N: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined by
\begin{equation}\label{oper}
    N(y)(t)=\begin{cases}
\phi(t),& \text{if $t\in [1-r,1]$, }\\
\frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}
\frac{f(s,y_s)}{s}ds,
&\text{if $t\in[1,b]$}.
\end{cases}
\end{equation}

Let $y, z\in C([1-r,b],\mathbb{R})$. Then, for $t \in J$,
\begin{align*}
|N(y)(t)-N(z)(t\|
&\leq \frac{1}{\Gamma(\alpha)}\int_1^t\Big(\log\frac{t}{s}\Big)^{\alpha-1}
|f(s,y_s)-f(s,z_s)|\frac{ds}{s}\\
&\leq \frac{\ell}{\Gamma(\alpha)}\int_1^t
\Big(\log\frac{t}{s}\Big)^{\alpha-1}\|y_s-z_s\|_{C}\, \frac{ds}{s}\\
&\leq \frac{\ell}{\Gamma(\alpha)}\|y-z\|_{[1-r,b]}\int_1^t
\Big(\log\frac{t}{s}\Big)^{\alpha-1} \frac{ds}{s}\\ 
&\leq \frac{\ell (\log t)^{\alpha}}{\Gamma(\alpha+1)}
\|y-z\|_{[1-r,b]}.
\end{align*}
Consequently, 
$$
\|N(y)-N(z)\|_{[1-r,b]}\le \frac{\ell
(\log b)^{\alpha}}{\Gamma(\alpha+1)}\|y-z\|_{[1-r,b]},
$$ 
which implies that $N$ is a contraction, and hence $N$ has a unique 
fixed point by Banach's contraction principle.
\end{proof}

Our second existence result  for  \eqref{e1}--\eqref{e2} is based
on the nonlinear alternative of Leray-Schauder.

 \begin{lemma}[Nonlinear alternative for single valued maps \cite{GrDu}] 
\label{NAK}
 Let $E$ be a Banach space,
$C$ a closed, convex subset of $E$, $U$ an open subset of $C$ and $0\in U$. 
Suppose that $F:\overline{U}\to C$ is a continuous, compact (that is, 
$F(\overline{U})$ is a relatively compact subset of $C$) map. Then either
\begin{itemize}
\item[(i)] $F$ has a fixed point in $\overline{U}$, or
\item[(ii)] there is a $u\in \partial U$ (the boundary of $U$ in $C$) and
$\lambda\in(0,1)$ with $u=\lambda F(u)$.
\end{itemize}
\end{lemma}

\begin{theorem}\label{t1} %\label{na}
 Assume that the following hypotheses hold:
\begin{itemize}
\item[(H1)] $f: J\times C([-r,0], \mathbb{R})\to \mathbb{R}$ is a continuous
function;
\item[(H2)] there exist a continuous  nondecreasing function 
$\psi : [0,\infty) \to (0,\infty)$ and a function 
$p \in C([1,b],\mathbb{R}^+)$ such that
$$
|f(t,u)|\le p(t)\psi(\|u\|_C) \quad \text{for each } (t,u) \in [1,b]
\times C([-r,0],{\mathbb R});
$$

\item[(H3)] there exists a constant $M>0$ such that
$$
\frac{M}{\psi(M)\|p\|_{\infty} \frac{(\log b)^\alpha}{\Gamma(\alpha+1)} }>1.
$$
\end{itemize}
Then \eqref{e1}--\eqref{e2} has at least one  solution on
$[1-r,b]$.
\end{theorem}

\begin{proof} 
We consider the operator $N: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined 
by \eqref{oper}. We shall show that the operator $N$ is continuous and
completely continuous.  
\smallskip

\noindent\textbf{Step 1:}
 $N$ is continuous.  Let $\{y_n\}$ be a sequence such that
$y_n\to y$ in $C([1-r,b],\mathbb{R})$. Let $\eta>0$ such that  
$\|y_n\|_{\infty}\leq \eta$.  Then
\begin{align*}
|N(y_n)(t)-N(y)(t)|
&\leq \frac{1}{\Gamma(\alpha)}
\int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}|f(s,y_{ns})-f(s,y_s)|\frac{ds}{s}\\
&\leq \frac{1}{\Gamma(\alpha)}\int_1^{b}\Big(\log\frac{t}{s}\Big)^{\alpha-1}
\sup_{s\in[1,b]}|f(s,y_{ns})
-f(s,y_s)|\frac{ds}{s}\\ 
&\leq \frac{\|f(\cdot,{y_n}_.)-f(\cdot,y_.)
\|_{\infty}}{\Gamma(\alpha)}\int_1^{b}\Big(\log\frac{t}{s}\Big)^{\alpha-1}
\frac{ds}{s}\\
&\leq
\frac{(\log b)^{\alpha}\|f(\cdot,{y_n}_.)-f(\cdot,y_.)\|_{\infty}}
{\alpha\Gamma(\alpha)}.
\end{align*}
Since  $f$ is a continuous function,   we have 
$$
\|N(y_n)-N(y)\|_{\infty}\leq\frac{(\log b)^{\alpha}\|f(\cdot,{y_n}_.)
-f(\cdot,y_.)\|_{\infty} }{\Gamma(\alpha+1)} \to 0 \quad
\text{as } n\to\infty. 
$$  
\smallskip

\noindent\textbf{Step 2:} $N$ maps bounded sets into bounded sets in
$C([1-r,b],\mathbb{R})$. 
  Indeed, it is sufficient to show that for
any $\eta^*>0$ there exists a positive constant $\tilde\ell$ such
that for each $y\in B_{\eta^*}=\{y\in C([1-r,b],\mathbb{R}):
\|y\|_{\infty}\leq \eta^* \}$, we have $\|N(y)\|_{\infty}\leq
\tilde\ell$. By (H2), for each $t\in[1,b]$, we have
\begin{align*}
|N(y)(t)|
&\leq\frac{1}{\Gamma(\alpha)}\int_1^{t}
\Big(\log\frac{t}{s}\Big)^{\alpha-1}|f(s,y_s)|\frac{ds}{s}\\
&\leq  \frac{\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}}{\Gamma(\alpha)} 
\int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{ds}{s}\\
&\leq  \frac{\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}}{\Gamma(\alpha+1)} 
(\log b)^{\alpha}.
\end{align*}
Thus
 $$
\|N(y)\|_{\infty} \leq  \frac{\psi(\eta^*)\|p\|_{\infty}}{\Gamma(\alpha+1)} 
(\log b)^{\alpha}:=\tilde\ell.
 $$
\smallskip

\noindent\textbf{Step 3:}:  $N$ maps bounded sets  into
equicontinuous sets of $C([1-r,b],\mathbb{R})$. 
Let $t_1, t_2\in [1,b]$, $t_1<t_2$, $B_{\eta^*}$ be a bounded set
of $C([1-r,b],\mathbb{R})$ as in Step 2, and let $y\in B_{\eta^*}$. Then
\begin{align*}
|N(y)(t_2)-N(y)(t_1)|
&\leq \frac{1}{\Gamma(\alpha)}\Bigl|\int_1^{t_1}
\Big[\Big(\log\frac{t_2}{s}\Big)^{\alpha-1}
 -\Big(\log\frac{t_1}{s}\Big)^{\alpha-1}\Big]f(s,y_s)\frac{ds}{s}\\
&\quad +\frac{1}{\Gamma(\alpha)}\int_{t_1}^{t_2}\Big(\log\frac{t_2}{s}\Big)^{\alpha-1}
 f(s,y_s)\frac{ds}{s}\Bigr|\\
&\leq \frac{\psi(\eta^*)\|p\|_{\infty}}{\Gamma(\alpha)} \int_1^{t_1}
\Big[\Big(\log\frac{t_2}{s}\Big)^{\alpha-1}
-\Big(\log\frac{t_1}{s}\Big)^{\alpha-1}\Big]\frac{ds}{s}\\
&\quad +\frac{\psi(\eta^*)\|p\|_{\infty}}{\Gamma(\alpha)} 
\int_{t_1}^{t_2}\Big(\log\frac{t_2}{s}\Big)^{\alpha-1}\frac{ds}{s}.
\end{align*}
 As $t_1\to t_2$ the right-hand side of the
above inequality tends to zero. The equicontinuity for the cases
$t_1<t_2\leq 0$ and $t_1\leq 0\leq t_2$ is obvious.

In consequence of Steps 1 to 3, it follows by  the Arzel\'a-Ascoli
theorem that  $N:C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ is continuous and
completely continuous. 
\smallskip 

\noindent\textbf{Step 4:}  We show that there exists an open set $U\subseteq
C([1-r,b],\mathbb{R})$ with $y\ne \lambda N(y)$ for $\lambda\in (0,1)$ and
$y\in \partial U$.
Let $y\in C([1-r,b],\mathbb{R})$ and  $y=\lambda N(y)$ for some
$0<\lambda<1$. Thus, for each $t\in[1,b]$
$$
y(t)=\lambda\Big(\frac{1}{\Gamma(\alpha)}\int_1^{t}
\Big(\log\frac{t}{s}\Big)^{\alpha-1}f(s,y_{s})\,
\frac{ds}{s}\Big).
$$ 
By  assumption (H2), for each $t\in J$, we obtain
\begin{align*}
|y(t)|
&\leq   \frac{1}{\Gamma(\alpha)}\int_1^{t}
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}p(s)\psi(\|y_{s}\|_{C})\frac{ds}{s} \\
&\leq \frac{\|p\|_{\infty}\psi(\|y\|_{[1-r,b]})}{\Gamma(\alpha+1)}
(\log b)^\alpha ,
\end{align*}
which can be expressed as
$$
\frac{\|y\|_{[1-r,b]}}{\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}  
\frac{(\log b)^\alpha}{\Gamma(\alpha+1)}}\le 1.
$$
In view of (H4), there exists $M$ such that
$\|y\|_{[1-r,b]} \ne M$. Let us set
$$
U = \{y \in  C([1-r,b], \mathbb{R}) : \|y\|_{[1-r,b]} < M\}.
$$ 
Note that the operator $N:\overline{U} \to  C([1-r,b],\mathbb{R})$ is
continuous and completely continuous. From the choice of $U$,
there is no $y \in \partial U$ such that $y=\lambda Ny$ for some 
$\lambda \in (0,1)$. Consequently, by the nonlinear
alternative of Leray-Schauder type (Lemma \ref{NAK}), we deduce
that $N$ has a fixed point $y \in \overline{U}$ which is a
solution of  \eqref{e1}-\eqref{e2}. This completes the proof.  
\end{proof}

\section{Neutral functional differential equations}

In this section, we establish the existence results for 
\eqref{e3}--\eqref{e4}.

\begin{definition}\label{d01} \rm
A function  $y\in C([1-r,b],\mathbb{R})$,
is said to be a solution of \eqref{e3}--\eqref{e4}  if $y$
satisfies the equation $D^{\alpha}[y(t)-g(t,y_t)]=f(t,y_{t})$
 on $J$, and the  condition $y(t)=\phi(t)$  on   $[1-r,1]$.
\end{definition}


\begin{theorem}[Uniqueness result] \label{con-2}   
Assume that {\rm (H0)} and the following condition hold:
\begin{itemize}
\item[(A1)] there exists  a nonnegative constant $c_1$
such that 
$$
|g(t,u)-g(t,v)|\le c_1\|u-v\|_{C}, \quad \text{for every } u, v \in C([-r,0],\mathbb{R}).
$$
\end{itemize} 
If
\begin{equation}\label{cnf}
   c_1+\frac{\ell (\log b)^{\alpha}}{\Gamma(\alpha+1)} <1,
\end{equation}
then there exists a unique solution for \eqref{e3}--\eqref{e4} on the 
interval $[1-r,b]$.
\end{theorem}

\begin{proof}
 Consider the operator $N_1: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined by:
\begin{equation}\label{oper1}
    N_1(y)(t)=\begin{cases}
\phi(t),&   \text{if  $t\in [1-r,1]$, }\\
g(t,y_t)+\frac{1}{\Gamma(\alpha)}\int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}f
(s,y_s)ds, & \text{if  $t\in[1,b]$}.
\end{cases}
\end{equation}
To show that the operator $N_1$ is a contraction, let $y, z\in C([1-r,b],\mathbb{R})$. 
Then  we have
\begin{align*}
&|N_1(y)(t)-N_1(z)(t)| \\
&\leq  |g(t,y_t)-g(t,z_t)|
+\frac{1}{\Gamma(\alpha)}\int_1^t|f(s,y_s)-f(s,z_s)|
\Big(\log\frac{t}{s}\Big)^{\alpha-1}ds\\
&\leq c_1\|y_t-z_t\|_{C}+\frac{\ell}{\Gamma(\alpha)}\int_1^t
\Big(\log\frac{t}{s}\Big)^{\alpha-1}\|y_s-z_s\|_{C}\, ds\\
&\leq c_1\|y-z\|_{[1-r,b]}+\frac{\ell}{\Gamma(\alpha)}\|y-z\|_{[1-r,b]}\int_1^t
 \Big(\log\frac{t}{s}\Big)^{\alpha-1} ds\\
&\leq c_1\|y-z\|_{[1-r,b]}+\frac{\ell (\log t)^{\alpha}}{\Gamma(\alpha+1)}
\|y-z\|_{[1-r,b]}.
\end{align*}
Consequently we obtain 
$$
\|N_1(y)-N_1(z)\|_{[1-r,b]}\le \big[c_1+\frac{\ell
(\log b)^{\alpha}}{\Gamma(\alpha+1)}\big]\|y-z\|_{[1-r,b]},
$$ 
which, in view of \eqref{cnf}, implies that
 $N_1$ is a contraction. Hence $N_1$ has a unique fixed point by
Banach's contraction principle. This, in turn, shows that  the problem 
\eqref{e3}--\eqref{e4} has a unique solution on $[1-r,b]$.
\end{proof}

\begin{theorem} \label{t2} 
Assume that {\rm (H1)--(H2)} hold. Further we suppose that
\begin{itemize}
\item[(H4)]  the function $g$ is  continuous and
completely continuous, and for any bounded set $B$ in
$C([1-r,b],\mathbb{R})$, the set $\{t\to g(t,y_t): y\in B\}$ is
equicontinuous in $C([1,b],\mathbb{R})$, and there exist constants $0\leq
d_1<1$, $d_2\geq 0$ such that 
$$ 
|g(t,u)|\leq d_1\|u\|_{C}+d_2,\quad t\in[1,b],\; u\in C([-r,0],\mathbb{R}).
 $$

\item[(H5)] there exists a constant $M>0$ such that
$$
\frac{(1-d_1)M}{ d_2+\frac{\|p\|_{\infty}\psi(M)}{\Gamma(\alpha+1)}
(\log b)^\alpha}> 1\,.
$$
\end{itemize}

Then \eqref{e3}--\eqref{e4} has at least one solution on
$[1-r,b]$.
\end{theorem}

\begin{proof} 
We consider the operator $N_1: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined by  
\eqref{oper1} and show that the operator $N_1$ is  continuous
and completely continuous.
Using (H3), it suffices to show that the operator 
$N_2: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined by
 $$
N_2(y)(t)=\begin{cases} 
\phi(t), & t\in [1-r,1],\\
\frac{1}{\Gamma(\alpha)}\int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}f(s,
y_{s})\, ds, &t\in [1,b],
\end{cases}
$$ 
is continuous and completely continuous.
 The proof is similar to that of Theorem \ref{t1}. 
So we omit the details.

\emph{We now show that there exists an open set $U\subseteq C([1-r,b],\mathbb{R})$
with $y\ne \lambda N_1(y)$ for $\lambda\in (0,1)$ and $y\in
\partial U$.}
Let $y\in C([1-r,b],\mathbb{R})$ and  $y=\lambda N_1(y)$ for some
$0<\lambda<1$. Thus, for each $t\in[1,b]$, we have
\[
y(t)=\lambda\Big(g(t,y_t)+
\frac{1}{\Gamma(\alpha)}\int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}f(s,y_{s})\,
ds\Big).
\]
 For each $t\in J$, it follows by (H2) and (H3) that
\begin{align*}
|y(t)|
&\leq  d_1\|y_t\|_{C}+d_2+\frac{1}{\Gamma(\alpha)}
 \int_1^{t}\Big(\log\frac{t}{s}\Big)^{\alpha-1}p(s)\psi(\|y_{s}\|_{C})\frac{ds}{s} \\
&\leq d_1\|y_t\|_{C}+d_2+\frac{\|p\|_{\infty}
\psi(\|y\|_{[1-r,b]})}{\Gamma(\alpha+1)} (\log b)^\alpha\,,
\end{align*}
which yields
$$
(1-d_1)\|y\|_{[1-r,b]}\le d_2+\frac{\|p\|_{\infty}\psi(\|y\|_{[1-r,b]})}
{\Gamma(\alpha+1)}(\log b)^\alpha\,.
$$
In consequence, we obtain
 $$
\frac{(1-d_1)\|y\|_{[1-r,b]}}{ d_2+\frac{\|p\|_{\infty}
\psi(\|y\|_{[1-r,b]})}{\Gamma(\alpha+1)} (\log b)^\alpha }\le 1\,.
$$
In view of (H4), there exists $M$ such that
$\|y\|_{[1-r,b]} \ne M$. Let us set
$$
U = \{y \in  C([1-r,b], \mathbb{R}) : \|y\|_{[1-r,b]} < M\}.
$$ 
Note that the operator $N_1:\overline{U} \to  C([1-r,b],\mathbb{R})$ is
continuous and completely continuous. From the choice of $U$,
there is no $u \in \partial U$ such that $y=\lambda N_1y$ for some 
$\lambda \in (0,1)$. Thus, by the nonlinear
alternative of Leray-Schauder type (Lemma \ref{NAK}), we deduce
that $N_1$ has a fixed point $y \in \overline{U}$ which is a
solution of  problem \eqref{e3}-\eqref{e4}. This completes the proof. 
\end{proof}

\section{An example}

In this section we give an example to illustrate the usefulness of
our main results. Let us consider the fractional functional
differential equation,
\begin{gather}\label{ex1}
D^{1/2}y(t)=\frac{ \|y_t\|_C}{2(1+\|y_t\|_C)}, \quad t\in J:=[1,e], \\
\label{ex2}
y(t)=\phi(t), \quad  t\in   [1-r,1].
\end{gather}
Let
$$
f(t,x)=\frac{x}{2(1+x)}, \quad (t,x)\in[1,e]\times
[0,\infty).
$$ 
For $x, y\in [0,\infty)$ and $t\in J$, we have
\begin{align*}
|f(t,x)-f(t,y)|=\frac{1}{2}\Bigl|\frac{x}{1+x}-\frac{y}{1+y}\Bigr|
=\frac{|x-y|}{2(1+x)(1+y)}
\leq\frac{1}{2}|x-y|,
\end{align*}
Hence the condition (H0) holds with $\ell=1/2$. Since
$ \frac{\ell (\log b)^{\alpha}}{\Gamma(\alpha+1)}=\frac{1}{\sqrt{\pi}}<1$,  
by Theorem \ref{con},  problem
\eqref{ex1}-\eqref{ex2} has a unique solution on $[1-r,e]$.



\begin{thebibliography}{99}

\bibitem{AZH} R. P. Agarwal, Y. Zhou, Y. He;
 Existence of fractional neutral functional differential equations, 
\emph{Comput. Math. Appl.}   \textbf{59}   (2010),   1095-1100.

\bibitem{AZWL} R. P. Agarwal, Y. Zhou, J. R. Wang, X. Luo;
 Fractional functional differential equations with causal operators in
 Banach spaces,   \emph{Math. Comput. Modelling}   \textbf{54}  (2011),  1440-1452.

\bibitem{fp16}	B. Ahmad, S. K. Ntouyas;
 Nonlinear fractional differential equations and
inclusions of arbitrary order and multi-strip boundary conditions, 
\emph{Electron. J. Diff. Eqns.}, Vol. 2012 (2012), No. 98, pp. 1-22.


\bibitem{fp23}	B. Ahmad, S. K. Ntouyas;
 Integro-differential equations of fractional order  with  nonlocal 
fractional boundary conditions associated with financial asset model,
 \emph{Electron. J. Diff. Eqns.},  Vol. 2013 (2013), No. 60, pp. 1-10.

  \bibitem{pf3} B. Ahmad, S. K. Ntouyas,  A. Alsaedi;
 New existence results for nonlinear fractional
differential equations with three-point integral boundary
conditions, \emph{Adv. Differ. Equ.} (2011) Art. ID 107384, 11 pp.

\bibitem{fp1} B. Ahmad, S. Sivasundaram;
 On four-point nonlocal boundary value problems of nonlinear 
integro-differential equations of fractional order,  
\emph{Appl. Math. Comput.} \textbf{217} (2010), 480-487.

 \bibitem{AN-H}  B. Ahmad, S. K. Ntouyas;
 Initial value problems for   hybrid Hadamard fractional differential equations,
\emph{Electron. J. Diff. Equ.}, Vol. 2014 (2014), No. 161, pp. 1-8

\bibitem{BHNO} M.  Benchohra, J.  Henderson, S.  K. Ntouyas, A.  Ouahab;
 Existence results for fractional order functional differential equations 
with infinite delay, \emph{J. Math. Anal. Appl.} \textbf{338} (2008),
1340-1350.

\bibitem{Had} P. L. Butzer, A. A. Kilbas,  J. J. Trujillo;
Compositions of Hadamard-type fractional integration operators and
the semigroup property, \emph{J. Math. Anal. Appl.} \textbf{269} (2002), 387-400.

\bibitem{Had1} P. L. Butzer, A. A. Kilbas,  J. J. Trujillo;
 Fractional calculus in the Mellin setting and Hadamard-type fractional
integrals, \emph{J. Math. Anal. Appl.} \textbf{269} (2002), 1-27.

\bibitem{Had2} P. L. Butzer, A. A. Kilbas,  J. J. Trujillo;
 Mellin transform analysis and integration by parts for Hadamard-type
fractional integrals, \emph{J. Math. Anal. Appl.} \textbf{270} (2002), 1-15.

\bibitem{El2} A. M. A. El-Sayed;
 Nonlinear functional differential equations of arbitrary orders,
 \emph{Nonlin. Anal.} \textbf{33} (1998), 181-186.

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

\bibitem{Hadd} J. Hadamard;
 Essai sur l'etude des fonctions donnees par leur developpment de Taylor, 
\emph{J. Mat. Pure Appl. Ser.} \textbf{8} (1892) 101-186.

\bibitem{HaLu} J. Hale, S. M. Verduyn Lunel;
\emph{Introduction to Functional Differential Equations}, Applied Mathematical 
Sciences, 99, Springer-Verlag, New York, 1993.

\bibitem{KST} A. A. Kilbas, Hari M. Srivastava,  J. J. Trujillo;
\emph{Theory and Applications of Fractional Differential Equations}.
North-Holland Mathematics Studies, 204. Elsevier Science B.V.,
Amsterdam, 2006.

\bibitem{KoMy} V. Kolmanovskii,  A. Myshkis;
\emph{Introduction to the Theory and Applications of Functional-Differential 
Equations}. Mathematics and its Applications, 463.
 Kluwer Academic Publishers, Dordrecht, 1999.

\bibitem{MiRo} K. S. Miller, B. Ross;
 \emph{An Introduction to the Fractional Calculus and Differential Equations},
 John Wiley, New York, 1993.

\bibitem{MoHa} S. M. Momani, S. B. Hadid;
 Some comparison results for integro-fractional differential inequalities, 
\emph{J. Fract. Calc.} \textbf{24} (2003), 37-44.

\bibitem{Pod} I. Podlubny;
\emph{Fractional Differential Equations.} Academic Press, San Diego, 1999.

\bibitem{SaKiMa} S. G. Samko, A. A. Kilbas, O. I. Marichev;
\emph{Fractional Integrals and Derivatives. Theory and Applications}, 
Gordon and Breach, Yverdon, 1993.

\bibitem{YG1} C. Yu,  G. Gao;
Some results on a class of fractional functional differential equations, 
\emph{Commun. Appl. Nonlinear Anal.} \textbf{11} (2004), 67-75.

\bibitem{YuGa} C. Yu,  G. Gao;
 Existence of fractional differential equations, 
\emph{J. Math. Anal. Appl.} \textbf{310} (2005), 26-29.

\end{thebibliography}

\end{document}