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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 161, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/161\hfil IVPs  for  hybrid Hadamard  fractional DEs]
{Initial-value problems for hybrid Hadamard fractional differential equations}

\author[B. Ahmad, S. K. Ntouyas \hfil EJDE-2014/161\hfilneg]
{Bashir Ahmad,  Sotiris K. Ntouyas}  % in alphabetical order

\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  May 17, 2014. Published July 24, 2014.}
\subjclass[2000]{34A08, 34B18}
\keywords{Hadamard fractional derivative;  initial value problem;
\hfill\break\indent fixed point theorem; existence}

\begin{abstract}
 In this article, we discuss  the existence of solutions for an initial-value
 problem of nonlinear hybrid differential equations of Hadamard type.
 The main result is proved  by  means of a fixed point theorem  due to Dhage.
 An example illustrating the existence result is also presented.
\end{abstract}

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


\section{Introduction}

In this article, we study the existence of solutions for an initial-value problem
of  hybrid  fractional differential equations of Hadamard type given by
\begin{equation}\label{e1}
\begin{gathered}
 _{H}D ^{\alpha}\Big(\frac{x(t)}{f(t,x(t))}\Big)=g(t,x(t)), \quad
 1\le  t \le T,\; 0<\alpha\le 1,\\
_{H}J^{1-\alpha}x(t)|_{t=1}=\eta,
\end{gathered}
\end{equation}
where $_{H}D^{\alpha}$ is the Hadamard  fractional derivative,
 $f\in C([1,T]\times {\mathbb R}, \mathbb{R}\setminus\{0\})$ and
$g: C([1,T]\times \mathbb{R},   \mathbb{R})$, $_{H}J^{(\cdot)}$
is the Hadamard fractional integral and $\eta\in {\mathbb R}$.

Fractional calculus has evolved into an important and interesting field of research
in view of its numerous applications in technical and applied sciences.
The mathematical modeling of many real world phenomena based on fractional-order
operators is regarded as better and improved than the one depending on integer-order
operators. In particular, fractional calculus has played a significant role
in the recent development of special functions and integral transforms,
signal processing, control theory, bioengineering and biomedical, viscoelasticity,
finance, stochastic processes, wave and diffusion phenomena, plasma physics,
social sciences, etc. For further details and applications, see \cite{b5,b2}.

Fractional differential equations involving  Riemann-Liouville and  Caputo type
fractional derivatives have extensively  been studied by several researchers.
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{B1, Had, Had1, Had2, b2,  Had3, Had4}.

Another interesting class of problems involves hybrid fractional differential
equations.  For some recent work on the topic , we refer to
\cite{B2, hb4, hb1, N1, hb3, hb2} and the references cited therein.

The article is organized as follows: Section 2 contains some preliminary facts
that we need in the sequel. In Section 3,  we present the main existence result
for the given problem whose proof  is based  on a fixed point theorem due to
Dhage \cite{Dh}.

\section{Preliminaries}

\begin{definition}[\cite{b2}]\rm
The Hadamard fractional integral of order $q$ for
a continuous function $g$ is defined as
$$
_{H}J^q g(t)=\frac{1}{\Gamma(q)}\int_1^t \Big(\log\frac{t}{s}\Big)^{q-1}
\frac{g(s)}{s}ds, ~~q>0.
$$
\end{definition}

\begin{definition}[\cite{b2}] \rm
The Hadamard derivative of fractional order $q$ for a continuous function
$g: [1, \infty)$ $\to \mathbb{R}$ is defined as
 $$
_{H}D^q g(t)=\frac{1}{\Gamma(n-q)}\Big(t\frac{d}{dt}\Big)^n
\int_{1}^t \Big(\log\frac{t}{s}\Big)^{n-q-1}\frac{g(s)}{s}ds, \quad
n-1 < q < n,\; n=[q]+1,
$$
where $[q]$ denotes the integer part of the real
 number $q$ and $\log (\cdot) =\log_e (\cdot)$.
\end{definition}

\begin{theorem}[{\cite[p. 213]{b2}}] \label{Ki}
Let $\alpha>0$, $n=-[-\alpha]$ and $0\le \gamma<1$. Let
$G$ be an open set in ${\mathbb R}$ and let $f: (a,b]\times G\to {\mathbb R}$  be a
function such that: $f(x,y)\in C_{\gamma,\log}[a,b]$  for any $y\in G$, then
the  problem
\begin{gather}\label{eKil}
 _HD^{\alpha}y(t)=f(t,y(t)), \quad \alpha>0, \\
\label{eKil-IC}
 _HJ^{\alpha-k}y(a+)=b_k, ~~b_k\in {\mathbb R}, \quad (k=1,\ldots, n, \; n=-[-\alpha]),
 \end{gather}
satisfies the  Volterra  integral equation
 \begin{equation}\label{eKi-V}
 y(t)=\sum_{j=1}^{n}\frac{b_j}{\Gamma(\alpha-j+1)}
\Big(\log\frac{t}{a}\Big)^{\alpha-j}+\frac{1}{\Gamma(\alpha)}
\int_a^t\Big(\log\frac{t}{s}\Big)^{\alpha-1}f(s,y(s))\frac{ds}{s}, 
\end{equation}
for $t>a>0$;  i.e., $y(t)\in C_{n-\alpha, \log}[a,b]$ satisfies
the relations \eqref{eKil}-\eqref{eKil-IC} if and
only if it satisfies the Volterra integral equation \eqref{eKi-V}.

In particular, if $0<\alpha\le 1$,  problem \eqref{eKil}-\eqref{eKil-IC}
is equivalent to the  equation
 \begin{equation}\label{ee}
 y(t)=\frac{b}{\Gamma(\alpha)}\Big(\log\frac{t}{a}\Big)^{\alpha-1}
+\frac{1}{\Gamma(\alpha)}\int_a^t\Big(\log\frac{t}{s}\Big)^{\alpha-1}
f(s,y(s))\frac{ds}{s}, \quad s>a>0.
 \end{equation}
\end{theorem}

Further details can be found in \cite{b2}.
From Theorem \ref{Ki} we have the following result.

\begin{lemma} \label{l2}
  Given  $y\in C([1,T], {\mathbb R})$, the integral solution of initial-value problem
\begin{equation}\label{e-gr}
\begin{gathered}
 _{H}D^{\alpha}\Big(\frac{x(t)}{f(t,x(t))}\Big) =y(t), \quad 0<t<1,  \\
_{H}J^{1-\alpha}x(t)|_{t=1}=\eta,
\end{gathered}
\end{equation}
 is given by
$$
 x(t)=f(t,x(t))\Big(\frac{\eta}{\Gamma(\alpha)} 
(\log t)^{\alpha-1}+\frac{1}{\Gamma(\alpha)}\int_1^t 
\Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{y(s)}{s}ds
 \Big),\quad t\in [1,T].
$$
 \end{lemma}


The following fixed point theorem due to Dhage \cite{Dh} 
is fundamental in the proof of our main result.

\begin{lemma}\label{Dh}
Let $S$ be a non-empty, closed convex and bounded subset of the Banach algebra 
$X$  let $A: X\to X$  and $B: S\to X$
be two   operators  such that:
\begin{itemize}
\item[(a)] $A$ is  Lipschitzian  with a Lipschitz constant $k$,
\item[(b)] $B$ is completely continuous,
\item[(c)] $x=AxBy\Rightarrow x\in S$ for all $y\in S$, and
\item[(d)] $Mk<1$, where $M=\|B(S)\|=\sup\{\|B(x)\|: x\in S\}$.
\end{itemize}
Then   the operator equation $x=AxBx$ has a solution.
\end{lemma}

\section{Existence result}

Let $C([1,T], {\mathbb R})$ denote the Banach space of all continuous real-valued 
functions defined on $[1,T]$ with the norm $\|x\|=\sup\{|x(t)|: t\in [1,T]\}$. 
For $t\in [1,T]$, we define $x_r(t)=(\log t)^r x(t), r\ge 0$. 
Let $C_r([1,T], {\mathbb R})$ be the space of all continuous functions 
$x$ such that $x_r\in C([1,T], {\mathbb R})$ which is indeed a Banach space 
endowed with the norm $\|x\|_C=\sup\{(\log t)^r|x(t)|: t\in [1,T]\}$.

Let $0\le \gamma<1$ and $C_{\gamma,\log}[1,T]$ denote the weighted space of continuous functions defined by
$$
C_{\gamma,\rm log}[1,T]=\{g(t): (\log t)^{\gamma}g(t)\in C[1,T],
 ~ \|y\|_{C_{\gamma,\log}}
=\|(\log t)^{\gamma}g(t)\|_{C}\}.
$$
In the following we denote $\|y\|_{C_{\gamma,\log}}$ by $\|y\|_C$.

\begin{theorem}\label{t1}
  Assume that:
\begin{itemize}
\item[(H1)] the function $f: [1,T]\times {\mathbb R}\to {\mathbb R}\setminus\{0\}$ 
is bounded continuous and there exists a positive bounded function $\phi$ with bound 
$\|\phi\|$ such that 
$$
|f(t,x(t))-f(t,y(t))|\le \phi(t)|x(t)-y(t)|,
$$ 
for $t\in [1,T]$ and for all $x,y\in {\mathbb R}$;

\item[(H2)] there exist a function $p\in C([1,T],{\mathbb R}^+)$ and a continuous
 nondecreasing function $\Omega: [0,\infty)\to (0,\infty)$ such that
$$
|g(t,x(t))|\le p(t)\Omega(|x|), \quad (t,x)\in [1,T]\times {\mathbb R};
$$

\item[(H3)] there exists a number $r>0$ such that
 \begin{equation}\label{e-r}
 r \ge  K\big[\frac{|\eta|}{\Gamma(\alpha)}
 +\log T\frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r)\big].
 \end{equation}
 where $|f(t,x)|\le K, ~~\forall (t,x)\in [1,T]\times {\mathbb R}$ and
 $$
\|\phi\|\big[\frac{|\eta|}{\Gamma(\alpha)}+\log T\frac{1}{\Gamma(\alpha+1)}
\|p\|\Omega(r)\big]<1.
$$
\end{itemize} 
Then the initial-value problem \eqref{e1} has at least one solution on
$[1,T]$.
\end{theorem}

\begin{proof} 
Set $X=C([1,T], {\mathbb R})$  and define a subset $S$ of $X$ as
$$
S=\{x\in X: \|x\|_C\le r\},
$$
where $r$ satisfies  inequality \eqref{e-r}.

Clearly $S$ is closed, convex and bounded subset of the Banach space $X$.
 By Lemma   \ref{l2},  the initial-value problem \eqref{e1} is equivalent to the 
integral equation
 \begin{equation}\label{op}
 x(t)=f(t,x(t))\Big(\frac{\eta}{\Gamma(\alpha)} (\log t)^{\alpha-1}
+\frac{1}{\Gamma(\alpha)}\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
\frac{g(s,x(s))}{s}ds
 \Big),
\end{equation}
for $ t\in [1,T]$.


Define two operators ${\mathcal A}: X\to X$ by
\begin{equation}\label{A}
{\mathcal A}x(t)=f(t,x(t)), \quad t\in [1,T],
\end{equation}
and ${\mathcal B}: S\to  X$ by
\begin{equation}\label{B}
{\mathcal B}x(t)=\frac{\eta}{\Gamma(\alpha)} (\log t)^{\alpha-1}
+\frac{1}{\Gamma(\alpha)}\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
\frac{g(s,x(s))}{s}ds,\quad t\in [1,T].
\end{equation}
Then $x={\mathcal A}x{\mathcal B}x$. We shall show that the operators 
${\mathcal A}$ and ${\mathcal B}$ satisfy all the conditions of Lemma \ref{Dh}. 
For the sake of clarity,  we split the  proof into a sequence of steps.
\smallskip

\noindent \textbf{Step 1.} 
We first show that ${\mathcal A}$ is a Lipschitz on $X$, i.e. (a) of 
Lemma \ref{Dh} holds.

Let $x,y\in X$. Then by (H1) we have
\begin{align*}
|(\log t)^{1-\alpha}{\mathcal A}x(t)-(\log t)^{1-\alpha}{\mathcal A}y(t)|
&= (\log t)^{1-\alpha}|f(t,x(t))-f(t,y(t))|\\
&\le \phi(t)(\log t)^{1-\alpha}|x(t)-y(t)|\\
&\le \|\phi\|\|x-y\|_C
\end{align*}
for all $t\in [1,T]$. Taking the supremum over the interval $[1,T]$, we obtain
$$
\|{\mathcal A}x-{\mathcal A}y\|_C\le \|\phi\| \|x-y\|_C
$$
for all $x, y\in  X$. So  ${\mathcal A}$ is a Lipschitz on $X$ 
with Lipschitz constant $\|\phi\|$.
\smallskip

\noindent \textbf{Step 2.} The operator ${\mathcal B}$ is completely
continuous on $S$, i.e. (b) of Lemma \ref{Dh} holds.

First we show that ${\mathcal B}$ is continuous  on $S$. Let $\{x_n\}$ 
be a sequence in $S$ converging to a point $x\in S$. Then by  Lebesque 
dominated convergence theorem,
\begin{align*}
&\lim_{n\to \infty}(\log t)^{1-\alpha}{\mathcal B}x_n(t)\\
&= \lim_{n\to \infty}\Big(\frac{\eta}{\Gamma(\alpha)}
  +(\log t)^{1-\alpha}\frac{1}{\Gamma(\alpha)}\int_1^t
  \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s,x_n(s))}{s}ds\Big)\\
&= \Big(\frac{\eta}{\Gamma(\alpha)}  
 +(\log t)^{1-\alpha}\frac{1}{\Gamma(\alpha)}\int_1^t 
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{\lim_{n\to \infty} g(s,x_n(s))}{s}ds\Big)\\
&=\Big(\frac{\eta}{\Gamma(\alpha)} +(\log t)^{1-\alpha}
 \frac{1}{\Gamma(\alpha)}\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
 \frac{ g(s,x(s))}{s}ds \Big)\\
&=(\log t)^{1-\alpha}{\mathcal B}x(t),
\end{align*}
for all $t\in [1,T]$. This shows that  ${\mathcal B}$ is continuous  os $S$. 
It is sufficient to show that ${\mathcal B}(S)$  is a uniformly bounded and 
equicontinuous set in $X$. First we note that
\begin{align*}
(\log t)^{1-\alpha}|{\mathcal B}x(t)|
&=\Big|\frac{\eta}{\Gamma(\alpha)} +(\log t)^{1-\alpha}\frac{1}{\Gamma(\alpha)}
 \int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s,x(s))}{s}ds\Big|\\
&\le \Big[\frac{|\eta|}{\Gamma(\alpha)} +\|p\|\Omega(r)(\log T)^{1-\alpha}
 \frac{1}{\Gamma(\alpha)}\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{1}{s}ds
\Big]\\
&= \frac{|\eta|}{\Gamma(\alpha)}+(\log T)\frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r),
\end{align*}
for all $t\in [1,T]$. Taking supremum over the interval $[1,T]$, the above 
inequality becomes
$$
\|{\mathcal B}x\|_C\le \frac{|\eta|}{\Gamma(\alpha)}  
 +(\log T)\frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r),
$$
for all $x\in S$. This shows that  ${\mathcal B}$ is uniformly bounded on $S$.

Next we show that ${\mathcal B}$ is an equicontinuous set in $X$. 
Let  $\tau_1, \tau_2 \in [1,T]$ with $\tau_1< \tau_2$ and  $x \in S$. Then we have
\begin{align*}
&\big|(\log \tau_2)^{1-\alpha}({\mathcal B}x)(\tau_2)
-(\log \tau_1)^{1-\alpha}({\mathcal B}x)(\tau_1)\big|\\
&\leq   \frac{\|p\|\Omega(r)}{\Gamma(\alpha)}
 \Big|\int_1^{\tau_2}(\log \tau_2)^{1-\alpha} 
 \left(\log\frac{\tau_2}{s}\right)^{\alpha-1}\frac{1}{s}ds
 -\int_1^{\tau_1} (\log \tau_1)^{1-\alpha}\left(\log\frac{\tau_1}{s}
 \right)^{\alpha-1}\frac{1}{s}ds\Big|\\
&\le \frac{\|p\|\Omega(r)}{\Gamma(\alpha)}\Big|\int_1^{\tau_1} 
 \Big[(\log \tau_2)^{1-\alpha}\left(\log\frac{\tau_2}{s}\right)
 ^{\alpha-1}-(\log \tau_1)^{1-\alpha}
 \left(\log\frac{\tau_1}{s}\right)^{\alpha-1}\Big]\frac{1}{s}ds\Big|\\
&\quad +\frac{\|p\|\Omega(r)}{\Gamma(\alpha)}
 \Big|\int_{\tau_1}^{\tau_2}(\log \tau_2)^{1-\alpha} 
 \left(\log\frac{\tau_2}{s}\right)^{\alpha-1}\frac{1}{s}ds\Big|.
\end{align*}
Obviously the right hand side of the above inequality tends to
zero independently of $x \in S$ as $t_2- t_1 \to 0$.
Therefore, it follows from the Arzel\'a-Ascoli theorem that ${\mathcal B}$ 
is  a completely continuous operator on $S$.
\smallskip

\noindent \textbf{Step 3.}  Next we show that hypothesis (c) of Lemma  \ref{Dh} 
is satisfied. Let $x\in X$ and $y\in S$ be arbitrary elements such that 
$x={\mathcal A}x {\mathcal B}y$. Then we have
\begin{align*}
(\log t)^{1-\alpha}|x(t)|
&=(\log t)^{1-\alpha}|{\mathcal A}x(t)| |{\mathcal B}y(t)|\\
&= |f(t,x(t))|\Big|\Big(\frac{\eta}{\Gamma(\alpha)} 
 +(\log t)^{1-\alpha}\frac{1}{\Gamma(\alpha)}\int_1^t 
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s,y(s))}{s}ds\Big)\Big|\\
&\le K\Big|\Big(\frac{\eta}{\Gamma(\alpha)} +(\log t)^{1-\alpha}
 \frac{1}{\Gamma(\alpha)}\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
 \frac{g(s,y(s))}{s}ds \Big)\Big|\\
&\le  K\Big[ \frac{|\eta|}{\Gamma(\alpha)} 
 +(\log T)^{1-\alpha}\|p\|\Omega(r)\frac{1}{\Gamma(\alpha)}\int_1^t 
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{1}{s}ds\Big]\\
&\le K\Big[\frac{|\eta|}{\Gamma(\alpha)} +(\log T)
 \frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r)\Big].
\end{align*}
Taking supremum for $t\in [1,T]$, we obtain
$$
\|x\|_C\le  K\big[\frac{|\eta|}{\Gamma(\alpha)} +(\log T)
\frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r)\big]\le r,
$$
that is, $x\in S$.
\smallskip

\noindent \textbf{Step 4.}  Now we show that $Mk<1$, that is, 
 (d) of Lemma \ref{Dh} holds.

This is obvious by $(H_4)$, since we have 
$M =\|B(S)\|=\sup\{\|{\mathcal B}x\|: x\in S\}
\le \frac{|\eta|}{\Gamma(\alpha)} +(\log T)\frac{1}{\Gamma(\alpha+1)}\|p\|\Omega(r)$ 
and $k=\|\phi\|$.

Thus all the conditions of Lemma \ref{Dh} are satisfied and hence the operator 
equation $x={\mathcal A}x{\mathcal B}x$ has a solution in $S$. In consequence, 
the problem \eqref{e1} has a solution on $[1,T]$. This completes the proof.
\end{proof}

\begin{example} \rm
Consider the initial-value problem
\begin{equation}\label{ex}
\begin{gathered}  
_{H}D^{1/2}\Big(\frac{x(t)}{f(t, x)}\Big) =g(t, x), \quad 1<t<e,  \\
 _{H}J^{1/2}x(t)|_{t=1}=1,
\end{gathered} 
\end{equation}
where 
\begin{gather*}
f(t,x)=\frac{1}{5 \sqrt{\pi}}(\sin t \tan^{-1}x+\pi/2), \\
g(t,x)= \frac{1}{10}\Big(\frac{1}{6} |x|+\frac{1}{8}\cos x 
+\frac{|x|}{4(1+|x|)}+\frac{1}{16}\Big).
\end{gather*}
 Obviously $ |f(t, x)|\le \frac{\sqrt{\pi}}{5}=K$, 
$\|\phi\| = \frac{1}{5\sqrt{\pi}}$, 
$ |g(t, x)| \le \frac{1}{10} \big(\frac{1}{6}|x|+\frac{7}{16}\big)$.
 We choose $  \|p\|=\frac{1}{10}$, $\Omega(r)=\frac{1}{6}r+\frac{7}{16}$. 
By the condition  (H3), it is found that
 $  \frac{261}{1192} \le r < \frac{3}{8}(400 \pi -87)$. 
Clearly all the conditions of Theorem \ref{t1} are satisfied. 
Hence, by the conclusion of Theorem \ref{t1}, it follows that    
problem \eqref{ex} has a solution.
\end{example}

\section{Discussion}

Operator equations such as  $x={\mathcal A}x{\mathcal B}x$ associated with 
problem  \eqref{e1},  are known as quadratic integral equations. 
Some recent works on these kinds of equations can be found in 
 \cite{Q1, Q2, Q3, Q4, Q5} and the references cited therein. 
It is interesting to note that the involvement of the term   
$\frac{\eta}{\Gamma(\alpha)} (\log t)^{\alpha-1}$ in the integral 
solution \eqref{op} of the problem \eqref{e1} makes it unbounded.
 In consequence, we have to consider an appropriate space to establish the 
existence of a solution to the given problem. In this scenario, 
Banach's fixed point theorem cannot be used in the weighted normed space.
 However, if we take $\eta=0$, then we can obtain an integral equation 
$x ={\mathcal{F}} x$, where the operator 
${\mathcal{F}}: C([1, T], \mathbb{R}) \to C([1, T], \mathbb{R})$ is
$$
({\mathcal{F}}x)(t)=\frac{1}{\Gamma(\alpha)}f(t,x(t))
\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s, x(s))}{s}ds,
$$ 
where $C([1,T], \mathbb{R})$ denote  the Banach space of all continuous functions 
from $[1,T] \to \mathbb{R}$ endowed with a topology of uniform convergence 
with the norm  defined by $\|x\|=\sup\{|x(t)|: t\in [1,T]\}$.
In this situation, we can apply Banach's contraction mapping principle.
 For that,  assuming that
$|f(t, x(t))| \le M_1$,  $|g(t, x(t))| \le M_2$, 
 $|f(t, x(t))-f(t, y(t))|\le L_1|x-y|$, $|g(t, x(t))-g(t, y(t))|\le L_2|x-y|$, 
for all $x, y \in \mathbb{R}$, we obtain
\begin{align*}
&|({\mathcal{F}}x)(t)-({\mathcal{F}}y)(t)| \\
&=  \frac{1}{\Gamma(\alpha)}\Big|f(t, x(t))\int_1^t 
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s, x(s))}{s}ds\\
&\quad - f(t, y(t))\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s, y(s))}{s}ds
 \Big|\\
&=  \frac{1}{\Gamma(\alpha)}\Big|[f(t, x(t))-f(t, y(t))]\int_1^t 
 \Big(\log\frac{t}{s}\Big)^{\alpha-1}\frac{g(s, x(s))}{s}ds\\
&\quad + f(t, y(t))\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
 \frac{[g(s, x(s))-g(s, y(s))]}{s}ds\Big|\\
&\le  \max_{t \in [1, T]}\Big[\frac{1}{\Gamma(\alpha)}
 \Big\{(L_1M_2+M_1L_2)\int_1^t \Big(\log\frac{t}{s}\Big)^{\alpha-1}
  \frac{1}{s}ds\Big\}\Big]\|x-y\|\\
&\le \frac{(\log T)^{\alpha}}{\Gamma(\alpha+1)}(L_1M_2+M_1L_2)\|x-y\|.
\end{align*}
Letting $ \frac{(\log T)^{\alpha}}{\Gamma(\alpha+1)}(L_1M_2+M_1L_2)<1$, 
the operator ${\mathcal{F}}$ is a contraction. Thus, by Banach's contraction 
principle, the problem \eqref{e1} with $\eta=0$ has a unique solution.

\subsection*{Acknowledgments}
The author wants to thank the anonymous  referees
for their useful comments.

S. K. Ntouyas is a Member of Nonlinear Analysis and Applied 
Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, 
Saudi Arabia.


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