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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 36, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/36\hfil Existence and uniqueness of solutions]
{Existence and uniqueness of solutions for  Caputo-Hadamard
sequential fractional order neutral functional differential
equations}

\author[B. Ahmad, S. K. Ntouyas \hfil EJDE-2017/36\hfilneg]
{Bashir Ahmad, Sotiris K. Ntouyas}

\address{Bashir Ahmad \newline
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}
\email{sntouyas@uoi.gr}

\thanks{Submitted November 11, 2016. Published February 2, 2017.}
\subjclass[2010]{34A08, 34K05}
\keywords{Fractional differential equations; existence; fixed point;
\hfill\break\indent  functional fractional differential
 equations; Caputo-Hadamard fractional differential equations}

\begin{abstract}
 In this article, we study the existence and uniqueness of solutions
 for Hadamard-type sequential fractional order neutral functional
 differential equations. The Banach fixed point theorem,  a nonlinear
 alternative of Leray-Schauder type and Krasnoselski fixed point theorem
 are used to obtain the desired results. Examples illustrating the main
 results are presented. An initial value integral condition case is also discussed.
\end{abstract}

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


\section{Introduction}

This work is concerned with the existence and uniqueness of
solutions to the following initial value problem (IVP) of
 Caputo-Hadamard sequential fractional order neutral functional
differential equations
\begin{gather}\label{e1}
D^{\alpha}[D^{\beta}y(t)-g(t,y_t)]=f(t,y_{t}),\quad  t\in J:=[1,b], \\
\label{e2-a}
y(t)=\phi(t),\quad t\in[1-r,1], \\
\label{e2}
D^{\beta}y(1)=\eta\in \mathbb{R},
\end{gather}
where $D^{\alpha}, D^{\beta}$ are the Caputo-Hadamard fractional
derivatives, $0<\alpha,\beta< 1$,
$f, g: J\times C([-r,0],\mathbb{R})\to\mathbb{R}$ are given functions and
$\phi\in C([1-r,1],\mathbb{R})$. For any function $y$
defined on $[1-r,b]$ and any $t\in J$,  we denote by $y_{t}$ the
element of $C_r:=C([-r,0],\mathbb{R})$   defined by
$$
y_{t}(\theta)=y(t+\theta), \quad  \theta\in [-r,0].
$$

Functional differential equations are found to be of central
importance in many disciplines such as control theory, neural
networks, epidemiology, etc. \cite{HaLu}. In analyzing the
behavior of real populations,  delay differential equations  are
regarded as effective tools. Since the  delay terms can be  finite
as well as infinite in nature, one needs to study these two cases
independently. Moreover, the delay terms may appear in the
derivatives involved in the given equation. As it is difficult to
formulate such a problem, an alternative approach is followed by
considering neutral functional differential equations. On the
other hand, fractional derivatives are capable to describe
hereditary and  memory effects in many processes and materials. So
the study of neutral functional differential equations in presence
of fractional derivatives constitutes an important area of
research. For more details, see the text \cite{YZ}.

In recent years, there has been a significant development  in
fractional calculus, and initial and boundary value problems of
fractional differential equations,
 see the monographs of Kilbas \emph{et al.} \cite{Kil}, Lakshmikantham \emph{et al.}
\cite{LLD}, Miller and Ross \cite{MiRo},
 Podlubny \cite{Pod},  Samko \emph{et al.} \cite{SaKiMa}, Diethelm \cite{Dieth}
and a series of  papers \cite{f1, f2, f3, f4, f5, f6, El2, f7, f8, f9}
and the references therein.
One can notice that much of the work on the topic involves
Riemann-Liouville and Caputo type fractional derivatives. Besides
these derivatives, there is an other fractional derivative
introduced by  Hadamard in 1892 \cite{Hadd}, which is known as
Hadamard derivative and  differs from aforementioned derivatives
in the sense that the kernel of the integral in its definition
 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.

 In \cite{BHNO}, the authors studied an initial value problem (IVP)
for Riemman-Liouville type  fractional functional and neutral functional
 differential equations with infinite delay. Recently, initial value problems
for fractional order Hadamard-type functional and neutral functional
differential equations and inclusions  were respectively investigated
in \cite{f3, FF1}, while an IVP for   retarded functional Caputo
type fractional impulsive differential equations with variable
moments was discussed in \cite{Erg}.

  In this paper, we investigate a new class of  Hadamard-type sequential
fractional neutral functional differential equations. Our study is based
on fixed point theorems due to Banach and
Krasnoselskii \cite{K}, 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
are presented in Section 4.  Examples are constructed in Section 5
for illustrating the obtained  results. Finally, a generalization
involving initial value integral condition is described in Section 6.


\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 $C_r$ is endowed with norm
$$
\|\phi\|_{C}:=\sup\{|\phi(\theta)|: -r\le \theta\le 0\}.
$$


 \begin{definition}[\cite{Kil}] \label{def2.1}  \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{Kil}] \label{def2.2} \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}

\begin{lemma} \label{l2}
 The function $y \in C^2([1-r, b], \mathbb{R})$ is a  solution of the problem
\begin{equation}\label{e-gr}
\begin{gathered}
 D^{\alpha}[D^{\beta}y(t)-g(t,y_t)]=f(t,y_t),\quad  t\in J:=[1,b],\\
y(t)=\phi(t),\quad t\in[1-r,1],\\
D^{\beta}y(1)=\eta\in \mathbb{R},
\end{gathered}
\end{equation}
 if and only if
 \begin{equation}\label{oper1}
    y(t)=\begin{cases}
\phi(t),&   \text{if } t\in [1-r,1], \\[4pt]
\phi(1)+(\eta-g(1,\phi(1)))\frac{(\log t)^{\beta}}{\Gamma(\beta+1)}\\
+\frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}
\frac{g(s,y_s)}{s}ds\\
+\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}
\frac{f(s,y_s)}{s}ds, & \text{if } t\in[1,b].
\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
The solution of Hadamard differential equation in \eqref{e-gr} can be written as
 \begin{equation}\label{g-1}
  D^{\beta}y(t)-g(t,y_t) = \frac{1}{\Gamma(\alpha)}\int_1^t
\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{f(s,y_s)}{s}ds+c_1,
 \end{equation}
where $c_1\in \mathbb{R}$ is arbitrary constant.
Using the  condition  $D^{\beta}y(1)=\eta$ we find that $c_1=\eta-g(1,\phi(1))$.
Then we obtain
\begin{align*}
y(t)&= (\eta-g(1,\phi))\frac{(\log
t)^{\beta}}{\Gamma(\beta+1)}+\frac{1}{\Gamma(\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\beta-1}\frac{g(s,y_s)}{s}ds  \\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{f(s,y_s)}{s}ds+c_2.
 \end{align*}
From the above equation we find $c_2=\phi(1)$ and  \eqref{oper1} is proved.
The converse follows by direct computation.
\end{proof}

\section{Existence and uniqueness result}

In this section, we establish the existence and uniqueness of a solution
for the IVP \eqref{e1}--\eqref{e2}.

\begin{definition} \label{d01} \rm
A function  $y\in C^2([1-r,b],\mathbb{R})$,
is said to be a solution of \eqref{e1}--\eqref{e2}  if $y$
satisfies the equation
$D^{\alpha}[D^{\beta}y(t)-g(t,y_t)]=f(t,y_{t})$
 on
$J$, the  condition  $y(t)=\phi(t)$ \ on \  $[1-r,1]$ and
$D^{\beta}y(1)=\eta$.
\end{definition}

The next theorem gives us a uniqueness result using the assumptions
\begin{itemize}
\item[(A1)] there exists $\ell>0$ such that
$$
|f(t,u)-f(t,v)|\le \ell\|u-v\|_{C}, \quad \text{for $t\in J$  and every }
 u, v \in C_r;
$$

\item[(A2)] there exists  a nonnegative constant $k$ such that
$$
|g(t,u)-g(t,v)|\le k\|u-v\|_{C}, \quad \text{for $t\in J$ and every }
  u, v \in C_r.$$
\end{itemize}

\begin{theorem} \label{con-2}
 Assume that {\rm (A1), (A2)} hold. If
\begin{equation}\label{cnf}
   \frac{k(\log b)^{\alpha}}{\Gamma(\alpha+1)}
+\frac{\ell(\log b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)} <1,
\end{equation}
then there exists a unique solution for IVP
\eqref{e1}--\eqref{e2} on the interval $[1-r,b]$.
\end{theorem}

\begin{proof} Consider the operator $N: C([1-r,b],\mathbb{R})\to
C([1-r,b],\mathbb{R})$ defined by
\begin{equation}\label{oper1a}
 N(y)(t)=\begin{cases}
\phi(t), &   \text{if  } t\in [1-r,1], \\[4pt]
\phi(1)+(\eta-g(1,\phi))\frac{(\log t)^{\beta}}{\Gamma(\beta+1)}\\
+\frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{g
(s,y_s)}{s}ds\\
+\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}
\frac{f(s,y_s)}{s}ds, & \text{if  } t\in J.
\end{cases}
\end{equation}
To show that the operator $N$ is a contraction, let
$y, z\in C([1-r,b],\mathbb{R})$. Then  we have
\begin{align*}
|N(y)(t)-N(z)(t)|
&\le  \frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}
\frac{|g(s,y_s)-g(s,z_s)|}{s}ds\\
&\quad+\frac{1}{\Gamma(\alpha+\beta)}\int_1^t\big(\log\frac{t}{s}
\big)^{\alpha+\beta-1}\frac{|f(s,y_s)-f(s,z_s)|}{s} ds\\
&\le \frac{k}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}
\frac{\|y_s-z_s\|_{C}}{s}ds\\
&\quad +\frac{\ell}{\Gamma(\alpha+\beta)}\int_1^t
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\|y_s-z_s\|_{C}\, ds\\
&\le \frac{k(\log t)^{\alpha}}{\Gamma(\alpha+1)}\|y-z\|_{[1-r,b]}
+\frac{\ell(\log t)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}\|y-z\|_{[1-r,b]}.
\end{align*}
Consequently we obtain
$$
\|N(y)-N(z)\|_{[1-r,b]}
\le \Big[\frac{k(\log b)^{\alpha}}{\Gamma(\alpha+1)}+\frac{\ell(\log
b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}\Big]\|y-z\|_{[1-r,b]},
$$
which, in view of \eqref{cnf}, implies that
 $N$ is a contraction. Hence $N$ has a unique fixed point by
Banach's contraction principle. This, in turn, shows that
problem \eqref{e1}--\eqref{e2} has a unique solution on
$[1-r,b]$.
\end{proof}

\section{Existence results}

In this section, we establish our existence  results  for the IVP
\eqref{e1}--\eqref{e2}. The first result is based on
Leray-Schauder nonlinear alternative.

 \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}

For the next theorem we need the following assumptions:
\begin{itemize}
\item[(A3)] $f, g: J\times C_r\to \mathbb{R}$  are continuous
functions;
\item[(A4)] there exist a continuous  nondecreasing
function $\psi : [0,\infty) \to (0,\infty)$ and a function
$p \in C(J,\mathbb{R}^+)$ such that
$$
|f(t,u)|\le p(t)\psi(\|u\|_C)  \text{for each } (t,u) \in J
\times C_r;
$$

\item[(A5)] there exist constants
$ d_1<\Gamma(\alpha+1)(\log b)^{-\alpha}$  and
$d_2\geq 0$ such that $$ |g(t,u)|\leq
d_1\|u\|_{C}+d_2,\ \ t\in J,\ u\in C_r.
$$

\item[(A6)] there exists a constant $M>0$ such that
$$
\frac{\Big(1-\frac{d_1(\log b)^{\alpha}}{\Gamma(\alpha+1)}\Big)M}{
M_0+\frac{d_2(\log b)^{\alpha}}{\Gamma(\alpha+1)}+\psi(M)\|p\|_{\infty}
\frac{1}{\Gamma(\alpha+\beta+1)}(\log b)^{\alpha+\beta}}> 1,
$$
where
$$
M_0=\|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
\frac{(\log b)^{\beta}}{\Gamma(\beta+1)}.
$$
\end{itemize}

\begin{theorem}\label{na} %\label{t1}
Under assumptions  {\rm (A3)--(A6)} hold, 
IVP \eqref{e1}--\eqref{e2} has at least one  solution on
$[1-r,b]$.
\end{theorem}

\begin{proof} We shall show that the operator
$N: C([1-r,b],\mathbb{R})\to C([1-r,b],\mathbb{R})$ defined by
\eqref{oper1a} 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
$C([1-r,b],\mathbb{R})$. 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}
\frac{|g(s,y_{ns})-g(s,y_s)|}{s}ds \\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}\big(\log\frac{t}{s}
 \big)^{\alpha+\beta-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]}|g(s,y_{ns})
 -g(s,y_s)|\frac{ds}{s}\\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{b}
 \big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\sup_{s\in[1,b]}|f(s,y_{ns})
 -f(s,y_s)|\frac{ds}{s}\\
&\leq \frac{\|g(\cdot,{y_n}_.)-g(\cdot,y_.)\|_{\infty}}{\Gamma(\alpha)}
 \int_1^{b}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{ds}{s}\\
&\quad +\frac{\|f(\cdot,{y_n}_.)-f(\cdot,y_.)\|_{\infty}}{\Gamma(\alpha+\beta)}
 \int_1^{b}\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{ds}{s}\\
&\leq \frac{(\log b)^{\alpha}\|g(\cdot,{y_n}_.)-g(\cdot,y_.)
 \|_{\infty}}{\Gamma(\alpha+1)} \\
&\quad +\frac{(\log b)^{\alpha+\beta}\|f(\cdot,{y_n}_.)-f(\cdot,y_.)
 \|_{\infty}}{\Gamma(\alpha+\beta+1)}.
\end{align*}

Since  $f, g$   are continuous functions,   we have
\begin{align*}
&\|N(y_{n})-N(y)\|_{\infty}\\
&\leq \frac{(\log b)^{\alpha}\|g(\cdot,{y_n}_.)
 -g(\cdot,y_.)\|_{\infty}}{\Gamma(\alpha+1)}
 + \frac{(\log b)^{\alpha+\beta}\|f(\cdot,{y_n}_.)-f(\cdot,y_.)\|_{\infty}
}{\Gamma(\alpha+\beta+1)} \to 0
\end{align*}
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 $\theta>0$ there exists a positive constant
$\tilde\ell$ such that for each
$y\in B_{\theta}=\{y\in C([1-r,b],\mathbb{R}): \|y\|_{\infty}\leq \theta \}$,
we have $\|N(y)\|_{\infty}\leq \tilde\ell$. By (A4) and (A5),
for each $t\in J$, we have
\begin{align*}
|N(y)(t)|
&\leq \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}\\
&\quad + \frac{1}{\Gamma(\alpha)}\int_1^{t}
 \big(\log\frac{t}{s}\big)^{\alpha-1}|g(s,y_s)|\frac{ds}{s}\\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
 \big(\log\frac{t}{s}\big)^{\alpha+\beta-1}|f(s,y_s)|\frac{ds}{s}\\
&\leq \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}\\
&\quad +\frac{d_1\|y\|_{[1-r,b]}+d_2}{\Gamma(\alpha)}
 \int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{ds}{s} \\
&\quad +\frac{\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}}{\Gamma(\alpha+\beta)}
\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{ds}{s}
\\
&\leq \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}\\
&\quad +\frac{d_1\|y\|_{[1-r,b]}+d_2}{\Gamma(\alpha+1)}(\log b)^{\alpha}
 +\frac{\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}}{\Gamma(\alpha+\beta+1)}
(\log b)^{\alpha+\beta}.
\end{align*}
Thus
 \begin{align*}
\|N(y)\|_{\infty}
&\leq  \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}\\
&\quad +\frac{d_1\theta+d_2}{\Gamma(\alpha+1)}(\log b)^{\alpha}+
\frac{\psi(\theta)\|p\|_{\infty}}{\Gamma(\alpha+\beta+1)}
(\log b)^{\alpha+\beta} :=\tilde\ell.
 \end{align*}
 \smallskip

\noindent\textbf{Step 3:} $N$ maps bounded sets  into
equicontinuous sets of $C([1-r,b],\mathbb{R})$.
Let $t_1, t_2\in J$, $ t_1<t_2$,
 $B_{\theta}$ be a
bounded set of $C([1-r,b],\mathbb{R})$ as in Step 2, and let
$y\in B_{\theta}$. Then
\begin{align*}
&|N(y)(t_2)-N(y)(t_1)|\\
&\le\frac{|\eta|+d_1\|\phi\|_{C}+d_2}{\Gamma(\beta+1)}[(\log t_2)^{\beta}-(\log t_1)^{\beta}]\\
&\quad+\Bigl|\frac{1}{\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] g(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}g(s,y_s)\frac{ds}{s}\Bigr|\\
&\quad +\Bigl|\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t_1}
 \Big[\Big(\log\frac{t_2}{s}\Big)^{
\alpha+\beta-1}-\Big(\log\frac{t_1}{s}\Big)^{\alpha+\beta-1}\Big]f(s,y_s)\frac{ds}{s}\\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_{t_1}^{t_2}
\Big(\log\frac{t_2}{s}\Big)^{\alpha+\beta-1}f(s,y_s)\frac{ds}{s}\Bigr|\\
&\leq \frac{|\eta|+d_1\|\phi\|_{C}+d_2}{\Gamma(\beta+1)}
 [(\log t_2)^{\beta}-(\log t_1)^{\beta}]\\
&\quad +\frac{d_1\theta+d_2}{\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{d_1\theta+d_2}{\Gamma(\alpha)} \int_{t_1}^{t_2}
\Big(\log\frac{t_2}{s}\Big)^{\alpha-1} \frac{ds}{s}\\
&\quad +\frac{\psi(\theta)\|p\|_{\infty}}{\Gamma(\alpha+\beta)}
\int_1^{t_1}\Big[\Big(\log\frac{t_2}{s}\Big)^{
\alpha+\beta-1}-\Big(\log\frac{t_1}{s}\Big)^{\alpha+\beta-1}\Big]\frac{ds}{s}\\
&\quad +\frac{\psi(\theta)\|p\|_{\infty}}{\Gamma(\alpha+\beta)} \int_{t_1}^{t_2}
 \Big(\log\frac{t_2}{s}\Big)^{\alpha+\beta-1}\frac{ds}{s}\\
&\leq  \frac{|\eta|+d_1\|\phi\|_{C}+d_2}{\Gamma(\beta+1)}
 [(\log t_2)^{\beta}-(\log t_1)^{\beta}]\\
&\quad +\frac{d_1\theta+d_2}{\Gamma(\alpha+1)}
[\big|(\log t_2)^{\alpha}-(\log t_1)^{\alpha}\big|+|\log  t_2/t_1|^{\alpha}]\\
&\quad  +\frac{\psi(\theta)\|p\|_{\infty}}{\Gamma(\alpha+\beta+1)}
[\big|(\log t_2)^{\alpha+\beta}-(\log t_1)^{\alpha+\beta}\big|
+|\log t_2/t_1|^{\alpha+\beta}].
\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.

As a 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
$0<\lambda<1$. Then, for each $t\in J$, we have
\begin{align*}
y(t)&= \lambda\Big(\phi(1)+(\eta-g(1,\phi(1)))\frac{(\log
t)^{\beta}}{\Gamma(\beta+1)}
+\frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{g
(s,y_s)}{s}ds\\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{f(s,y_s)}{s}ds\Big).
\end{align*}
 By our assumptions, for each $t\in J$, we obtain
\begin{align*}
|y(t)|
&\leq   \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}+d_2]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}\\
&\quad +\frac{d_1\|y\|_{[1-r,b]}+d_2}{\Gamma(\alpha)}
\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{ds}{s} \\
&\quad +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
 \big(\log\frac{t}{s}\big)^{\alpha+\beta-1}p(s)\psi(\|y_{s}\|_{C})\frac{ds}{s} \\
&\leq \|\phi\|_{C}+[|\eta|+d_1\|\phi\|_{C}
 +d_2]\frac{(\log b)^{\beta}}{\Gamma(\beta+1)}
 +\frac{d_1\|y\|_{[1-r,b]}+d_2}{\Gamma(\alpha+1)}(\log b)^{\alpha} \\
&\quad +\frac{\|p\|_{\infty}\psi(\|y\|_{[1-r,b]})}{\Gamma(\alpha+\beta+1)}
(\log b)^{\alpha+\beta},
\end{align*}
which can be expressed as
$$
\frac{\Big(1-\frac{d_1(\log b)^{\alpha}}{\Gamma(\alpha+1)}\Big)
\|y\|_{[1-r,b]}}{M_0+\frac{d_2(\log b)^{\alpha}}
{\Gamma(\alpha+1)}+\psi(\|y\|_{[1-r,b]})\|p\|_{\infty}
\frac{1}{\Gamma(\alpha+\beta+1)}(\log b)^{\alpha+\beta}}\le 1.
$$

In view of (A6), 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 the
problem \eqref{e1}-\eqref{e2}. This completes the proof.
\end{proof}

The second existence result is based on Krasnoselskii's fixed
point theorem.

\begin{lemma}[Krasnoselskii's fixed point theorem \cite{K}] \label{lem3-2}
Let $S$ be a closed, bounded, convex and nonempty subset of a
Banach space $X$. Let $A, B$ be the operators such that 
(a) $Ax+Bx\in S$ whenever $x,y\in S$; 
(b) $A$ is compact and continuous;
(c) $B$ is a contraction mapping. Then there exists
$z \in S$ such that $z = Az+Bz$.
\end{lemma}

\begin{theorem}\label{Krasno}
Assume that {\rm (A2)} and {\rm (A3)} hold.  In addition we assume that
\begin{itemize}
\item[(A7)]
 $|f(t,x)| \leq \mu(t)$, $|g(t,x)|\le \nu(t)$, for all $(t,x)\in J\times\mathbb{R}$, 
and $\mu, \nu\in C(J,\mathbb{R^+})$.
\end{itemize}
Then   problem \eqref{e1}-\eqref{e2} has at least one solution
on $[1-r,b]$, provided
\begin{equation}\label{Eq3-4}
 \frac{k(\log b)^{\alpha}}{\Gamma(\alpha+1)}<1.
\end{equation}
\end{theorem}

\begin{proof} 
We define the operators $\mathcal{G}_1$ and $\mathcal{G}_2$ by
\begin{gather}\label{oper2}
    \mathcal{G}_1y(t)
=\begin{cases}
0,  & \text{if } t\in [1-r,1],\\[4pt]
(\eta-g(1,\phi))\frac{(\log
t)^{\beta}}{\Gamma(\beta+1)} \\
+\frac{1}{\Gamma(\alpha)}\int_1^{t}\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{g
(s,y_s)}{s}ds,  &\text{if } t\in J.
\end{cases} \\
\label{oper3}
    \mathcal{G}_2y(t)
=\begin{cases}
\phi(t),&   \text{if } t\in [1-r,1], \\[4pt]
\phi(1) +\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{f
(s,y_s)}{s}ds, & \text{if } t\in J. 
\end{cases}
\end{gather}

Setting $\sup_{t\in[1,b]} \mu(t)= \|\mu\|_{\infty},
\sup_{t\in[1,b]} \nu(t)= \|\nu\|_{\infty}$ and choosing
\begin{equation} \label{Eq3-5}
\rho \geq \|\phi\|_{C} +[|\eta|+\|\nu\|_{\infty}]\frac{(\log
b)^{\beta}}{\Gamma(\beta+1)}+\frac{\|\nu\|(\log
b)^{\alpha}}{\Gamma(\alpha+1)}+\|\mu\|_{\infty}\frac{(\log
b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)},
\end{equation}
 we consider $B_{\rho}
= \{y \in C([1-r,b],\mathbb{R}):\|y\|_{\infty}\leq \rho\}$. For
any $y, z \in B_\rho$, we have
\begin{align*}
&|\mathcal{G}_1y(t)+\mathcal{G}_2z(t)| \\
& \leq  \sup_{t \in [1,b]}\Big\{
(\eta-g(1,\phi))\frac{(\log
t)^{\beta}}{\Gamma(\beta+1)}+\frac{1}{\Gamma(\alpha)}\int_1^{t}
\big(\log\frac{t}{s}\big)^{\alpha-1}\frac{g (s,y_s)}{s}ds\\
&\quad +\phi(1)
+\frac{1}{\Gamma(\alpha+\beta)}\int_1^{t}
 \big(\log\frac{t}{s}\big)^{\alpha+\beta-1}\frac{f(s,z_s)}{s}ds\Big\}\\
& \leq  \|\phi\|_{C} +[|\eta|+\|\nu\|_{\infty}]
 \frac{(\log b)^{\beta}}{\Gamma(\beta+1)}
 +\frac{\|\nu\|(\log b)^{\alpha}}{\Gamma(\alpha+1)}
 +\|\mu\|_{\infty}\frac{(\log b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}\\
& \le \rho.
\end{align*}
This shows that $\mathcal{G}_1y+\mathcal{G}_2z \in B_{\rho}$.
Using \eqref{Eq3-4} it is easy to see   that $\mathcal{G}_1$ is a
contraction mapping.

Continuity of $f$ implies that the operator $\mathcal{G}_2$ is
continuous. Also, $\mathcal{G}_2$ is uniformly bounded on
$B_{\rho}$ as
\[
\|\mathcal{G}_2y\| \leq  \|\phi\|_{C}+\|\mu\|_{\infty}\frac{(\log
b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}.
\]

Now we prove the compactness of the operator $\mathcal{G}_2$.
We define 
\[
\bar{f}= \sup_{(t,y)\in[1,b]\times B_{\rho}}|f(t,y)|<\infty,
\]
 and consequently, for
$t_1,t_2\in [1, b]$, $t_1<t_2$, we have
\begin{align*}
&|\mathcal{G}_2y(t_2)-\mathcal{G}_2y(t_1)| \\
& \le  \frac{\bar{f}}{\Gamma(\alpha+\beta)}
\int_1^{t_1}\Big|\Big(\log\frac{t_2}{s}\Big)^{
\alpha+\beta-1}-\Big(\log\frac{t_1}{s}\Big)^{\alpha+\beta-1}\Big|\frac{ds}{s}\\
&\quad +\frac{\bar{f}}{\Gamma(\alpha+\beta)}
\int_{t_1}^{t_2}\Big(\log\frac{t_2}{s}\Big)^{\alpha+\beta-1}\frac{ds}{s}\\
&\leq \frac{\bar{f}}{\Gamma(\alpha+\beta+1)}[\left|(\log
t_2)^{\alpha+\beta}-(\log t_1)^{\alpha+\beta}\right|+|\log
t_2/t_1|^{\alpha+\beta}],
\end{align*}
which is independent of $y$ and tends to zero as $t_2-t_1\to 0$.
Thus, $\mathcal{G}_2$ is equicontinuous. So $\mathcal{G}_2$ is
relatively compact on $B_{\rho}$. Hence, by the
Arzel$\acute{a}$-Ascoli theorem, $\mathcal{G}_2$ is compact on
$B_{\rho}$. Thus all the assumptions of Lemma $\ref{lem3-2}$ are
satisfied. So the conclusion of Lemma $\ref{lem3-2}$ implies that
the problem \eqref{e1}-\eqref{e2} has at least one solution on
$[1-r,b]$
\end{proof}

\section{Examples}

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}\Big[D^{3/4}y(t)-\frac{1+e^{-t}}{8+e^{t}}\frac{
\|y_t\|_C}{(1+\|y_t\|_C)}\Big]
=\frac{\|y_t\|_C}{2(1+\|y_t\|_C)}+e^{-t}, \\
 t\in J:=[1,e], \nonumber \\
\label{ex2}
y(t)=\phi(t), \quad t\in   [1-r,1], \\
\label{ex3}
D^{3/4}y(1)=1/2.
\end{gather}
Let
$$
f(t,x)=\frac{x}{2(1+x)}, \quad 
g(t,x)=\frac{1+e^{-t}}{8+e^{t}}\Big(\frac{x}{1+x}\Big),
\quad (t,x)\in[1,e]\times [0,\infty).
$$ 
For $x, y\in [0,\infty)$ and $t\in J$, we have
\[
|f(t,x)-f(t,y)|=\frac{1}{2}\Bigl|\frac{x}{1+x}-\frac{y}{1+y}\Bigr|
=\frac{|x-y|}{2(1+x)(1+y)} \leq\frac{1}{2}|x-y|,
\]
and
\begin{align*}
|g(t,x)-g(t,y)|
&= \frac{1+e^{-t}}{8+e^{t}}\Bigl|\frac{x}{1+x}-\frac{y}{1+y}\Bigr|
=\frac{1+e^{-t}}{8+e^{t}}\frac{|x-y|}{(1+x)(1+y)}\\
&\leq \frac{e+1}{e(e+8)}|x-y|.
\end{align*}

Hence  conditions (A1) and (A2) hold with $\ell=1/2$ and
$ k=\frac{e+1}{e(e+8)}$ respectively. Since 
$ \frac{k(\log b)^{\alpha}}{\Gamma(\alpha+1)}+\frac{\ell (\log
b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)}\approx 0.5853088<1$,
  therefore,  by Theorem  \ref{con-2},  problem
\eqref{ex1}-\eqref{ex3} has a unique solution on $[1-r,e]$.


Also $|f(t, x)| \le (1+2 e^{-t})/2=\mu(t)$, 
$|g(t, x)| \le (1+ e^{-t})/(8+e^t)=\nu(t)$ and 
$k(\log b)^{\alpha}/\Gamma(\alpha+1)=2(e+1)/\sqrt{\pi} e(e+8)\approx 0.144005 <1$.
Clearly the assumptions of Theorem  \ref{Krasno} are satisfied. Consequently,
by the conclusion  of Theorem  \ref{Krasno}, there exists a
solution of the problem \eqref{ex1}-\eqref{ex3}  on $[1-r,e]$.


\section{Initial value integral condition case}

 The results of this paper can be extended to
the case of an initial value integral condition of the form
\begin{equation}\label{e22} 
D^{\beta}y(1)=\int_1^b h(s, y_s)ds,
\end{equation}
where $h: J\times C([-r,0],\mathbb{R})\to\mathbb{R}$ is a
given function. In this case $\eta$ will be replaced with
$\int_1^b h(s, y_s)ds$ in \eqref{oper1a} and the statement of the
existence and uniqueness result for the problem
\eqref{e1}--\eqref{e2-a}--\eqref{e22} can be formulated as
follows.

\begin{theorem}\label{con-22}  
Assume that the conditions {\rm (A1)} and
{\rm (A2)} hold. Further, we suppose that
\begin{itemize}
\item[(A8)]
there exists  a nonnegative constant $m$ such that
$$
|h(t,u)-h(t,v)|\le m\|u-v\|_{C}, \quad \text{for } t\in J\text{ and every }
  u, v \in C_r.
$$
\end{itemize}
Then the problem \eqref{e1}-\eqref{e2-a}-\eqref{e22} has a unique solution 
on $[1-r,b]$ if
\[
  \frac{m(b-1)(\log b)^{\beta}}{\Gamma(\beta+1)}
+ \frac{k(\log b)^{\alpha}}{\Gamma(\alpha+1)}
  +\frac{\ell(\log b)^{\alpha+\beta}}{\Gamma(\alpha+\beta+1)} <1.
\]
\end{theorem}

We do not provide the proof of the above theorem as it is
similar to that of Theorem \ref{con-2}.

The analog form of the existence results: Theorems \ref{na} and
 \ref{Krasno} for the problem \eqref{e1}-\eqref{e2-a}-\eqref{e22} can be
 constructed in a similar manner.

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\end{document}