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\fancyhead[CE]{  M. SHABIBI } 
\fancyhead[CO]{  BI-SINULAR TYPE OF A  FRACTIONAL-ORDER ...}



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{\noindent Journal of Mathematical Extension \\
%Journal Pre-proof }\\
ISSN: 1735-8299\\
URL: http://www.ijmex.com\\
\vspace*{9mm}

\begin{center}

{\Large \bf 
 Bi-singular type of a  fractional-order  multi-points boundary value condition problem\\} 
\vspace{2mm}

%\let\thefootnote\relax\footnote{\scriptsize Received:  March 2019; Accepted: December 2019}


{\bf M. Shabibi$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small Department of Mathematics, Meharn Branch, Islamic Azad University, Mehran, Iran} \vspace{2mm}


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\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this article, by using control functions, the existence of a  solution for a bi-singular fractional differential equation with multi-point initial value conditions is considered. In the following, an example elucidates our main result.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 34A08; 37C25; 46F30

\noindent{\bf Keywords:}  Bi-singular, Caputo derivation, Control functions, Fractional differential equation 
\end{quotation}}

\section{Introduction}
Altough it is awhile that definitions for the fractional derivatives have been provided, differential equations with fractional order have played a prominent role in the researches of mathematicians  (see, for example,  \cite{5ab}- \cite{d}), which among which singular ones are more significant.(see  \cite{m6}- \cite{ms10}).\\ 
In 2013, the fractional problem $\mathcal{D}^{r} \nu(\xi) + y(t,\nu(\xi))=0$ with boundary conditions 
$\nu'(0)=\nu''(0)=\dots=\nu^{(k_0-1)}(0)=0$ and $\nu(1)= \int_{0}^{1} \nu(s)d \gamma(s)$ investigated, where $0<\xi<1$, $n\geq2$, $r \in(k_0-1,k_0)$, $\gamma(s)$ is a function of bounded variation, $y$ could be singular at $\xi=1$ and $\int_{0}^{1} d \gamma(s)<1$ (\cite{5}). \\  In 2015, 
the fractional problem
$\mathcal{D}^{\rho}y(\zeta)=\psi(\zeta,y(\zeta), \mathcal{D}^{\sigma}y(\zeta))$ with boundary conditions $y(0)+y'(0)=g(x)$, $\int_{0}^{1} y(\zeta) dt=m_0$ and
$y''(0)=y^{(3)}(0)=\dots=y^{(n_{\rho}-1)}(0)=0$ was studied where, $0<\zeta<1$, $m_0$ is a real number, $n_{\rho}\geq2$, $\rho \in(n_{\rho}-1,n_{\rho})$, $0< \sigma <1$, $\mathcal{D}^{\rho}$ and $D^{\sigma}$ is the Caputo fractional derivatives,
$g \in C([0,1],\mathbb{R}) \to \mathbb{R}$ and $\psi: (0,1] \times \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is a continuous function in which $\psi(\zeta,u,v)$ could  has singularity at $\zeta=0$ (\cite{a}).\\
In 2018, the existence of a solution for  the following three steps crisis problem was investigated: 
\begin{eqnarray}
\mathcal{D}^{\eta} z(\tau)+ \psi(\tau , z(\tau), z'(\tau), \mathcal{D}^{\sigma}z(\tau), \int_{0}^{\tau} \Omega(\xi) z(\xi) d\xi , \omega(x(\tau)))=0 \nonumber
\end{eqnarray}
with boundary conditions 
$z(1)=z(0)=z''(0)=z^{n_\eta}(0)=0$, where $\eta \geq 2$, $\lambda, \mu, \sigma \in (0,1)$, $\Omega \in L^1[0,1]$,  $\omega: C^1[0,1] \rightarrow C^1[0,1]$ is a mapping such that $\| \omega(x_1) - \omega(x_2)\| \leq \iota_0 \|x_1- x_2\| + \iota_1 \|x_1'- x_2' \|$ for some $\iota_0$, $\iota_1 \in [0,\infty)$ and all $x_1, x_2 \in C^1[0,1]$,
$\mathcal{D}^{\eta}$ is the $\eta$-order Caputo fractional derivative, 
$\psi(\tau,z_1(\tau),..., z_5(\tau))=\psi_1(\tau,z_1(\tau),..., z_5(\tau))$ for all $\tau \in[0,\lambda)$,
$\psi(\tau,z_1(\tau),..., z_5(\tau))=\psi_2(\tau,z_1(\tau),..., z_5(\tau))$ for all $\tau \in[\lambda ,\mu]$ and
$\psi(\tau,z_1(\tau),..., z_5(\tau))=\psi_3(\tau,z_1(\tau),..., z_5(\tau))$ for all $\tau \in(\mu,1]$, $\psi_1(\tau,.,.,.,.,.)$ and $\psi_3(\tau,.,.,.,.,.)$ are continuous on $[0,\lambda)$ and $(\mu,1]$ and $\psi_2(\tau,.,.,.,.,.)$ is multi-singular (\cite{kh}).
\\
Motivated by the mentioned articles, we  investigate the non-controlled bi-singular fractional differential  equation
\begin{eqnarray}
\label{prb1}
\mathcal{D}^\mathfrak{a} ( g(t) \mathcal{D}^\mathfrak{r}(\nu (t)))  = \Theta(t, \nu(t), \nu'(t), \phi_{\nu}(t))
\end{eqnarray}
with boundary conditions 
$\mathcal{D}^{(\mathfrak{r} +j)}\nu (0)= \nu^{(j^*)}(0)=0$ for all  $ 1 \leq j^* \leq k-1, 0 \leq j \leq n-1$ and $\nu'(\eta) = \sum_{i=1}^{k_0} \lambda_i \nu(\gamma_i)$, for some $k_0 \in \mathbb{N}$, where $n=[\mathfrak{a}]+1$,   $k=[\mathfrak{r}]+1$, $\mathfrak{a}, \mathfrak{r} \geq 1$, $\mathfrak{a} + \mathfrak{r} \geq 3$, $ \lambda_i \in \mathbb{R}$, $\sum_{i=1}^{k_0} \lambda_i \neq 0$, $ \eta, \gamma_i \in (0,1)$, $g: [0,1] \to \mathbb{R}$ is a function which can be zero at some points $t \in [0,1]$, $\phi: X \to \mathbb{R}$ is a function such that for all $u,v \in X$ and $ t \in [0,1]$, satisfies the following inequality:
$$ |\phi_u(t) -\phi_v(t)| \leq \omega_1 |u(t) - v(t)| +\omega_2 |u'(t) - v'(t)|,$$
$\omega_1, \omega_2 \in [0, \infty)$  and $X= C^1[0,1]$. 
$\mathcal{D}^{\mathfrak{a}}$ is the Caputo fractional derivative of order $\mathfrak{a}$ and
$f:[0,1] \times \mathbb{R}^{3} \to \mathbb{R}$ is a function such that
$\Theta(t,.,.,.)$ is singular at some points $t\in [0,1]$. In fact,  $\Theta$ is stated to be multi-sigular when  it is singular at more than one point $t$. Note that the differential equation  $\mathcal{D}^{\mathfrak{a}}(g(t) \mathcal{D}^{\mathfrak{r}}w(t))= \mathcal{U}(t, w(t))$ is  sigular when  $\mathcal{U}$ is singular or $g(t)=0 $ at some points $t \in [0,1]$. When  $\mathcal{U}$ is singular and $g(t)=0$, we call the equation  $\mathcal{D}^{\mathfrak{a}}(g(t) \mathcal{D}^{\mathfrak{r}}w(t))= \mathcal{U}(t, w(t))$ to be bi-singular.
Likewise, $\mathcal{D}^{\mathfrak{a}}w(t)+\mathcal{U}(t)=0$ is pointwise defined equation on $[0,1]$ if there is the set $E \subset [0,1]$ such that its measure of complenment  $E^c$ is zero and equation on $E$ is being hold. 
It is obvious that each equation  is a pointwisly defined equation. In this paper, we use $\| .\|_1$ as the norm of $ L ^{1} [0,1]$, $\|.\|$ as the sup norm $Y=C[0,1]$ and $\left \|w \right\|_{*} = \max \{\| w\|, \|w'\| \} $ as the norm of $X=C^{1}[0,1]$.\\

	The Riemann-Liouville integral of order $r$ with the lower limit $\nu \geq 0$ for a function $ \mathcal{Y} : (\nu,\infty)\rightarrow \mathbb{R} $ is defined by
	$\mathcal{I}^{r}_{\nu^+} \mathcal{Y}(x)=\frac{1}{\Gamma(r)} \int_{\nu}^{x} (x-\zeta)^{r-1}  \mathcal{Y}(\zeta)d\zeta$
	provided that the right-hand side is  pointwise defined on $(\nu,\infty)$. we denote $\mathcal{I}^{r} \mathcal{Y}(x)$ for $\mathcal{I}^{r}_{0^+} \mathcal{Y}(x)$. Also, The Caputo fractional derivative of order $r>0$ of a function $  \mathcal{Y}:(0,\infty)\to \mathbb{R}$ is defined by $^{c}\mathcal{D}^{r} \mathcal{Y}(x)=\frac{1}{\Gamma(n_r-r)}\displaystyle\int_{0}^{x}\!\frac{ \mathcal{Y}^{n_r}(\zeta)}{(x-\zeta)^{r+1-n_r}}d\zeta$, 
where $n_r=[r]+1$ (\cite{4}). \\ Let $\Psi$ be the family of nondecreasing functions $\psi :[0,\infty) \to [0,\infty)$ such that $\sum_{j=1}^{\infty} \psi^{j}(\zeta)<\infty$ for all $\zeta> 0$ (\cite{9}).
It is easy to see that $\psi(\zeta)<\zeta$ is held for all $\zeta>0$ (\cite{9}). Let $\mathcal{T}: \mathcal{E} \to \mathcal{E}$ and $\mathcal{A} :\mathcal{E} \times \mathcal{E} \to [0,\infty)$ be two maps. Then $\mathcal{T}$ is called an $\mathcal{A}$-admissible map whenever $\mathcal{A}(x,y) \geq 1$ implies $\mathcal{A}(\mathcal{T}x,\mathcal{T}y) \geq 1$ (\cite{10}). Let $(\mathcal{E},d)$ be a complete metric space, $\psi \in \Psi$ and $\mathcal{A} :\mathcal{E} \times \mathcal{E} \to [0,\infty)$ a map. A self-map $\mathcal{T}:\mathcal{E} \to \mathcal{E}$ is called an $\mathcal{A}$-$\psi$-contraction whenever $\mathcal{A}(x,y) d(\mathcal{T}x,\mathcal{T}y) \leq \psi(d(x,y))$ for all $x,y \in \mathcal{E}$ (\cite{10}).
We need the following results.
\begin{lemma}\label{l1.22}(\cite{5a}) Assume that $0<n-1\leq r <n$ and $v \in C[0,1] \cap L^{1}[0,1]$. Then $\mathcal{I}^{r} \mathcal{D}^{r}v(\xi)=v(\xi)+ \sum_{i=0}^{n-1} \iota_{i}\xi^{i}$ for some constants \\ $\iota_0,\dots,\iota_{n-1}\in\mathbb{R}$.
\end{lemma}
\begin{lemma} \label{l1.1}(\cite{7}) Consider a complete metric space $(\mathcal{E},d)$, a map $\mathcal{A} :\mathcal{E} \times \mathcal{E} \to [0,\infty)$, $\psi \in \Psi$, and $\mathcal{L} \to \mathcal{E}$ an $\mathcal{A}$-admissible $\mathcal{A}$-$\psi$-contraction. If $\mathcal{L}$ is continuous and there exists $u_{0} \in \mathcal{E}$ such that $\mathcal{A}(u_{0}, \mathcal{L}u_{0}) \geq 1$, then $\mathcal{L}$ has a fixed point.
\end{lemma}
\begin{lemma} \label{beta} (\cite{13}) For all $\zeta > -1$ and $w >0,$ we have \\
	$\int^t_0 (t-s)^{w - 1} s^{\zeta} ds =\mathcal{B}(\zeta + 1, w) t^{w + \zeta}$,
	where $\mathcal{B}(\zeta, w) = \frac{\Gamma(w) \Gamma(\zeta)}{\Gamma(w+\zeta)}$.
\end{lemma}
\section{Main Results}
\begin{lemma} Let $\mathfrak{a} \geq 1$, $\mathfrak{r} \geq 2$, $\lambda_i \in \mathbb{R}$, $\eta \in (0,1)$ for $1 \leq i \leq k_0$, $k_0 \in \mathbb{N}$,  $n=[\mathfrak{a}]+1$,   $k=[\mathfrak{r}]+1$
 and $f \in L^{1}[0,1]$. Then $x_0 \in X$ is a solution for the problem
\begin{eqnarray}
\mathcal{D}^\mathfrak{a} ( g(t) \mathcal{D}^\mathfrak{r}(\nu (t)))  = f(t)  \label{p2}
\end{eqnarray}
 with boundary conditions 
$\mathcal{D}^{(\mathfrak{r} +j)}\nu (0)= \nu^{(j^*)}(0)=0$ for all  $ 1 \leq j^* \leq k-1, 0 \leq j \leq n-1$ and $\nu'(\eta) = \sum_{i=1}^{k_0} \lambda_i \nu(\gamma_i)$, if and only if $x_0$ is given as follow
\begin{eqnarray}
x_0(t)=\frac{1}{\Gamma(\mathfrak{a})\Gamma(\mathfrak{r})} \int_0^t f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)d\zeta \nonumber\\
+ \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} -1)} \int_0^{\eta} f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)d\zeta \nonumber\\
- \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} \lambda_i \int_0^{\gamma_i} f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta) d\zeta, \nonumber
\end{eqnarray}
where
$$\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)= \int_s^t \frac{(t- \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{g(\xi)} d\xi$$
and $\Delta = \sum_{i=1}^{k_0} \lambda_i$.

 
\end{lemma}
\begin{proof} Let $x_0$ be a solution for the problem (\ref{p2}), then regarding Lemma (\ref{l1.22}), it is evinced that
$$ g(t) \mathcal{D}^{\mathfrak{r}} x_0(t) =  \frac{1}{\Gamma (\mathfrak{a})} \int^t_0 (t-\zeta)^{\mathfrak{a} - 1}f(\zeta) d\zeta + m_0 + m_1 t+ ...+ m_{n-1}t^{n-1}.$$
Since $\mathcal{D}^{\mathfrak{r}} x_0 (0)=0$, we $m_0=0$. Also we have $(g(t) \mathcal{D}^{\mathfrak{r}} x_0 (t))' \bigg|_{t=0}=m_1$, hence 
$$g'(0) \mathcal{D}^{\mathfrak{r}} x_0 (0) + g(0) \mathcal{D}^{\mathfrak{r} +1} x_0 (0) =m_1.$$ 
Since for $0 \leq j \leq n-1$,
$\mathcal{D}^{\mathfrak{r} +j} x_0 (0) =0$,  we conclude that $m_1=0$. Using the same argument, it is concluded $m_2=...=m_{n-1}=0$. So
$$ \mathcal{D}^{\mathfrak{a}} x_0(t) =  \frac{1}{ g(t) \Gamma (\mathfrak{a})} \int^t_0 (t-\zeta)^{\mathfrak{a} - 1}f(\zeta) d\zeta.$$
 Using Lemma (\ref{l1.22}) again, it is resulted
 \begin{eqnarray}
 x_0(t)& =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^t_0 \frac{(t-\xi)^{\mathfrak{r} - 1}}{g(\xi)}(\int_0^{\xi} (\xi -\zeta)^{\mathfrak{a} - 1} f(\zeta) d\zeta) d\xi \nonumber\\
 & &+ \iota_0 +  \iota_1 t+ ...+  \iota_{k-1} t^{k-1}. \nonumber
 \end{eqnarray}
As regarded $x^{(j^*)}(0)=0$ for $ 1 \leq j^* \leq k-1$ then $\iota_1= \iota_2= ...=\iota_{k-1}=0$. Replacing in the above equality, we have
 \begin{eqnarray} \label{ex0}
 x_0(t)& =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^t_0 \frac{(t-\xi)^{\mathfrak{r} - 1}}{g(\xi)}(\int_0^{\xi} (\xi -\zeta)^{\mathfrak{a} - 1} f(\zeta) d\zeta) d\xi + \iota_0. 
 \end{eqnarray}
 By derivation from the last equality, it is deduced that
 \begin{eqnarray}
 x'_0(t)& =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^t_0 \frac{(t-\xi)^{\mathfrak{r} - 2}}{g(\xi)}(\int_0^{\xi} (\xi -\zeta)^{\mathfrak{a} - 1} f(\zeta) d\zeta) d\xi, \nonumber
 \end{eqnarray}
so 
 \begin{eqnarray}
 x'_0(\eta)& =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0 \frac{(\eta-\xi)^{\mathfrak{r} - 2}}{g(\xi)}(\int_0^{\xi} (\xi -\zeta)^{\mathfrak{a} - 1} f(\zeta) d\zeta) d\xi \nonumber\\
 & =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0 \int_0^{\xi} \frac{(\eta-\xi)^{\mathfrak{r} - 2} (\xi -\zeta)^{\mathfrak{a} - 1}}{g(\xi)}  f(\zeta) d\zeta d\xi \nonumber\\
  & =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0 \int_\zeta^{\eta} \frac{(\eta-\xi)^{\mathfrak{r} - 2} (\xi -\zeta)^{\mathfrak{a} - 1}}{g(\xi)}  f(\zeta) d\xi d\zeta \nonumber\\
  & =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0  f(\zeta) (\int_\zeta^{\eta} \frac{(\eta-\xi)^{\mathfrak{r} - 2} (\xi -\zeta)^{\mathfrak{a} - 1}}{g(\xi)}  d\xi) d\zeta. \nonumber
 \end{eqnarray}
Put 
$$\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)= \int_\zeta^t \frac{(t- \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{g(\xi)} d\xi,$$
then, it is obtained that
$$ x'_0(\eta)= \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0  f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta, \zeta) d\zeta. $$
Also by (\ref{ex0}), we induce that
 \begin{eqnarray}
 x_0(t)& =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^\zeta_0 \int_0^{\xi} \frac{(t-\xi)^{\mathfrak{r} - 1} (\xi -\zeta)^{\mathfrak{a} - 1}}{g(\xi)} f(\zeta) d\zeta d\xi + \iota_0 \nonumber\\
 & =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^t_0 \int_\zeta^{t} \frac{(t-\xi)^{\mathfrak{r} - 1} (\xi -\zeta)^{\mathfrak{a} - 1}}{g(\xi)} f(\zeta) d\xi  d\zeta+ \iota_0 \nonumber\\
 & =&  \frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^t_0 \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)  f(\zeta)  d\zeta+ \iota_0. \nonumber
 \end{eqnarray}
Hence, for $1 \leq i \leq k_0$, we have 
$$\lambda_i  x_0(\gamma_i)=\frac{\lambda_i}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})} \int^{\gamma_i}_0 \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i, \zeta)  f(\zeta)  d\zeta+\lambda_i \iota_0.$$
Therefore 
$$\sum_{i=1}^{k_0} \lambda_i  x_0(\gamma_i)=\frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})}\sum_{i=1}^{k_0} \lambda_i \int^{\gamma_i}_0 \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i, \zeta)  f(\zeta)  d\zeta+ \iota_0 \sum_{i=1}^{k_0} \lambda_i.$$
By hypothesis $x'_0(\eta)=\sum_{i=1}^{k_0} \lambda_i  x_0(\gamma_i)$, so we have
\begin{eqnarray}
& &\frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0  f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta, \zeta) d\zeta \nonumber\\
&& =\frac{1}{\Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})}\sum_{i=1}^{k_0} \lambda_i \int^{\gamma_i}_0 \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i, \zeta)  f(\zeta)  d\zeta+ \iota_0 \sum_{i=1}^{k_0} \lambda_i, \nonumber
\end{eqnarray}
so $\iota_0$  is obtained as follow
\begin{eqnarray}
\iota_0 &= &\frac{1}{\Delta \Gamma (\mathfrak{a})  \Gamma (\mathfrak{r} -1)} \int^{\eta}_0  f(\zeta) \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta, \zeta) d\zeta \nonumber\\
&& - \frac{1}{\Delta \Gamma (\mathfrak{a})  \Gamma (\mathfrak{r})}\sum_{i=1}^{k_0} \lambda_i \int^{\gamma_i}_0 \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i, \zeta)  f(\zeta)  d\zeta, \nonumber
\end{eqnarray}
where $\Delta = \sum_{i=1}^{k_0} \lambda_i$. Hence $x_0(t)$ is given as
\begin{eqnarray}
x_0(t)=\frac{1}{\Gamma(\mathfrak{a})\Gamma(\mathfrak{r})} \int_0^t \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta) f(\zeta) d\zeta \nonumber\\
+ \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} -1)} \int_0^{\eta} \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta) f(\zeta) d\zeta \nonumber\\
- \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} \lambda_i \int_0^{\gamma_i} \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta) f(\zeta)  d\zeta. \nonumber
\end{eqnarray}
\end{proof}
\\Now, let $\mathcal{L}:X \rightarrow X$ be defined as
\begin{eqnarray}
 \mathcal{L}\mathfrak{u}(t) = \frac{1}{\Gamma(\mathfrak{a})\Gamma(\mathfrak{r})} \int_0^t \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta) \Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) d\zeta \nonumber\\
+ \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} -1)} \int_0^{\eta}  \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)  \Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) d\zeta \nonumber\\
- \frac{1}{\Delta \Gamma(\mathfrak{a})\Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} \lambda_i \int_0^{\gamma_i}  \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)  \Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) d\zeta. \nonumber
\end{eqnarray}
where $\phi: X \rightarrow X$ is a mapping such that 
$$ |\phi_\mathfrak{u}(t) -\phi_\mathfrak{v}(t)| \leq \omega_1 |\mathfrak{u}(t) - \mathfrak{v}(t)| +\omega_2 |\mathfrak{u}'(t) - \mathfrak{v}'(t)|,$$
for all $\mathfrak{u} ,\mathfrak{v} \in X$, $t \in [0,1]$ and some functions $\omega_1, \omega_2 \in [0, \infty)$. It easy to see that $ \mathcal{L}'$ is given as follow
\begin{eqnarray}
 \mathcal{L}'\mathfrak{u}(t) =\frac{\partial  \mathcal{L}\mathfrak{u}}{\partial t} =\frac{1}{\Gamma(\alpha)\Gamma(\mathfrak{r})} \int_0^t \frac{\partial   \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)}{\partial t}  \Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) d\zeta \nonumber\\
 = \frac{1}{\Gamma(\mathfrak{a})\Gamma(\mathfrak{r} -1)} \int_0^t  \mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta) \Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) d\zeta. \nonumber
\end{eqnarray}



Now, we investigate $\mathcal{L} : X \rightarrow X$, to prove the existence of a solution  in $X$ for the problem (\ref{prb1}). Applying lemma (\ref {l1.22}), it is indicated that $\mathcal{L}$ has a fixed point in $X$. In the next results, by using some functions which are called control functions,  we will control the singularity and then, inequalities help us to 
consider  a sloution  for the bi-singular fractional differential problem. 
\begin{theorem}\label{t2.3}
Let $\mathfrak{a}, \mathfrak{r} \geq 1$, $\mathfrak{a} + \mathfrak{r} \geq 3$,  $n=[\mathfrak{a}]+1$, $k=[\mathfrak{r}]+1$, $ \lambda_i \in \mathbb{R}$, $\sum_{i=1}^{k_0} \lambda_i \neq 0$, $ \eta, \gamma_i \in (0,1)$, $\phi: X \to \mathbb{R}$ is a function such that for all $u,v \in X$ and $ t \in [0,1]$, 
$$ |\phi_\mathfrak{u}(t) -\phi_\mathfrak{v}(t)| \leq \omega_1 |\mathfrak{u}(t) - \mathfrak{v}(t)| +\omega_2 |\mathfrak{u}'(t) - \mathfrak{v}'(t)|,$$
for some $\omega_1, \omega_2 \in [0, \infty)$, 
 $g: [0,1] \to  \mathbb{R}$ may be zero at some points $t_0 \in [0,1]$, $\|\bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\|:= \int_{0}^{1} \frac{(1-\zeta)^{\mathfrak{r} -2} \zeta^{\mathfrak{a}}}{|g(\zeta)|} d\zeta < \infty,$ 
$\Theta:[0,1]\times(C^{1}[0,1])^{3}\to \mathbb{R}$ be a singular function   at some points $t \in [0,1]$ such that
\begin{eqnarray}
|\Theta(t, w_1, w_2, w_3) - \Theta(t, z_1, z_2, z_3)|  \leq \sum_{j=1}^{k^*} \theta_{j}(t) \Lambda_j(|w_1 - z_1|, |w_2 - z_2|, |w_3 - z_3|), \nonumber
\end{eqnarray}
for all $w_1, w_2, w_3, z_1, z_2, z_3 \in X$, almost $ t \in [0,1]$ and some $k^* \in \mathbb{N}$, where 
$\Lambda_j : X^{3}\to  [0,\infty)$ for each $1 \leq j \leq k^*$, is a nondecreasing function with respect to  all their components, $\theta_j : [0,1] \to  [0,\infty),$
$\lim_{z \to 0^{+}} \frac{\Lambda(z,z,z)}{z} = q_j \in [0,  \infty)$ 
and $\|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,t]}:= \int_0^t \frac{\hat{\theta}^{\mathfrak{a}, \mathfrak{r}-1}_j(t,\xi)}{|g(\xi)|} d\xi \in L^{1}[0,1],$
where $\hat{\theta}^{\mathfrak{a}, \mathfrak{r}}_j(t,\xi) = \int_0^{\xi}(t-\zeta)^{\mathfrak{a} +\mathfrak{r} -2} \theta_j(\zeta) d\zeta$.
Also let $|\Theta(t, x_1,x_2,x_3)| \leq \sum_{i=1}^3 \mathcal{N}_i(t,x_i)$, where $\mathcal{N}_i: [0,1] \times X \to [0, \infty)$ for each $ 1 \leq i \leq 3$ is nondecreasing with respect to its second component and
$\lim_{z \to 0^{+}} \frac{ \mathcal{N}_i(t, z)}{z} = \mathcal{V}_i(t)$ a.e. $[0,1]$, such that $\|\hat{\mathcal{V}}_i^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,t]} \in L^1[0,1]$
and

\begin{eqnarray}
&&     \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg)  \in [0, \frac{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}{ \Xi}),  \nonumber
\end{eqnarray}
where 
$\Xi = max \{1, \omega_1 +\omega_2 \}$. If 
   \begin{eqnarray} 
& & \frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \bigg) <1, \nonumber
\end{eqnarray}
then the singular fractional differential equation
\begin{eqnarray}
\mathcal{D}^\mathfrak{a} ( g(t) \mathcal{D}^\mathfrak{r}(\nu (t)))  = \Theta(t, \nu(t), \nu'(t), \phi_{\nu}(t)) \nonumber
\end{eqnarray}
with boundary conditions 
$\mathcal{D}^{(\mathfrak{r} +j)}\nu (0)= \nu^{(j^*)}(0)=0$ for all  $ 1 \leq j^* \leq k-1, 0 \leq j \leq n-1$ and $\nu'(\eta) = \sum_{i=1}^{k_0} \lambda_i \nu(\gamma_i)$, for some $k_0 \in \mathbb{N}$.
\end{theorem}
\begin{proof}
Firstly, we prove that $\mathcal{L}$ is continuous on $X$. Let $\mathfrak{u},\mathfrak{v} \in X$, then for all $t \in [0,1]$ we have
\begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)|
\leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \bigg|\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) \nonumber\\ & & - \Theta(\zeta, \mathfrak{v}(\zeta), \mathfrak{v}'(\zeta), \phi_{\mathfrak{v}}(\zeta)) \bigg| d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \bigg|\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) \nonumber\\ & & - \Theta(\zeta, \mathfrak{v}(\zeta), \mathfrak{v}'(\zeta), \phi_{\mathfrak{v}}(\zeta)) \bigg| d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0}| \lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \bigg|\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) \nonumber\\ & & - \Theta(\zeta, \mathfrak{v}(\zeta), \mathfrak{v}'(\zeta), \phi_{\mathfrak{v}}(\zeta)) \bigg| d\zeta \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t \bigg( |\mathcal{H}_{\alpha, \mathfrak{r}}(t,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (|\mathfrak{u}(\zeta) -\mathfrak{v}(\zeta)|, |\mathfrak{u}'(\zeta)-\mathfrak{v}'(\zeta)|, |\phi_{\mathfrak{u}}(\zeta)- \phi_{\mathfrak{v}}(\zeta)|) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (|\mathfrak{u}(\zeta) -\mathfrak{v}(\zeta)|, |\mathfrak{u}'(\zeta)-\mathfrak{v}'(\zeta)|, |\phi_{\mathfrak{u}}(\zeta)- \phi_{\mathfrak{v}}(\zeta)|) \bigg)d\zeta \nonumber
\end{eqnarray}

\begin{eqnarray}
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (|\mathfrak{u}(\zeta) -\mathfrak{v}(\zeta)|, |\mathfrak{u}'(\zeta)-\mathfrak{v}'(\zeta)|, |\phi_{\mathfrak{u}}(\zeta)- \phi_{\mathfrak{v}}(\zeta)|) \bigg)d\zeta  \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t \bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|, \|\mathfrak{u}' -\mathfrak{v}' \|, \omega_1 \|\mathfrak{u}-\mathfrak{v}\|+\omega_2 \|\mathfrak{u}'-\mathfrak{v}'\|) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|, \|\mathfrak{u}' -\mathfrak{v}' \|, \omega_1 \|\mathfrak{u}-\mathfrak{v}\|+\omega_2 \|\mathfrak{u}'-\mathfrak{v}'\|) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|, \|\mathfrak{u}' -\mathfrak{v}' \|, \omega_1 \|\mathfrak{u}-\mathfrak{v}\|+\omega_2 \|\mathfrak{u}'-\mathfrak{v}'\|) \bigg)d\zeta \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t \bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|_{*}, \|\mathfrak{u} -\mathfrak{v}\|_{*}, (\omega_1 +\omega_2) \|\mathfrak{u} -\mathfrak{v}\|_{*}) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|_{*}, \|\mathfrak{u} -\mathfrak{v}\|_{*}, (\omega_1 +\omega_2) \|\mathfrak{u} -\mathfrak{v}\|_{*}) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i}\bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|_{*}, \|\mathfrak{u} -\mathfrak{v}\|_{*}, (\omega_1 +\omega_2) \|\mathfrak{u} -\mathfrak{v}\|_{*}) \bigg)d\zeta. \nonumber
\end{eqnarray}
Let $\Xi := max \{1, \omega_1 +\omega_2 \}$, then by the last equality, for all $t \in [0,1]$ it is concluded that

\begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \nonumber\\
&& \leq \sum_{j=1}^{k^*} \bigg(  \frac{ \Lambda_j ( \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*})}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)|  \theta_j(\zeta) d\zeta \bigg) \nonumber\\
& & + \sum_{j=1}^{k^*} \bigg(  \frac{ \Lambda_j ( \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*})}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \nonumber\\
& & \times \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)|  \theta_j(\zeta) d\zeta \bigg) \nonumber\\
& & + \sum_{j=1}^{k^*} \bigg(  \frac{ \Lambda_j ( \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*}, \Xi \|\mathfrak{u} -\mathfrak{v}\|_{*})}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \nonumber\\
& & \times ( \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)|  \theta_j(\zeta) d\zeta) \bigg). \nonumber
\end{eqnarray}
Regarding the properties $\lim_{z \to 0^{+}} \frac{\Lambda(\Xi z, \Xi z, \Xi z)}{\Xi z}=q_j$ for all $1 \leq j \leq k^*$, for $\epsilon >0$ there exists    $0<\delta(\epsilon) >0$ such that $z \in (0, \delta(\epsilon)]$ implies
 $0< |\frac{\Lambda(\Xi z, \Xi z, \Xi z)}{\Xi z}| \leq q_j+ \epsilon$, for all $1 \leq j \leq k^*$, so  
  $0< \Lambda(\Xi z, \Xi z, \Xi z) \leq (q_j+ \epsilon) \Xi z$,  for all $z \in (0, \delta(\epsilon)]$ and $1 \leq j \leq k^*$. Let $\delta_m(\epsilon) = \min \{ \epsilon, \delta(\epsilon) \}$, then $\|\mathfrak{u} -\mathfrak{v}\|_*< \delta_m(\epsilon) $ implies 
 \begin{eqnarray}
 \Lambda(\Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*) \leq \Xi  (q_j+ \epsilon) \|\mathfrak{u} -\mathfrak{v}\|_{*}, \label{str}
 \end{eqnarray}
so when    $\|\mathfrak{u} -\mathfrak{v}\|_*< \delta_m(\epsilon) $, then for all $t \in [0,1]$
  \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \nonumber\\
&& \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon) \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)|  \theta_j(\zeta) d\zeta \bigg] \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon) \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)|  \theta_j(\zeta) d\zeta \bigg] \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon) \bigg( \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)|  \theta_j(\zeta) d\zeta \bigg) \bigg]. \nonumber
\end{eqnarray}
On the other hand, 
    \begin{eqnarray}
& & \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)|  \theta_j(\zeta) d\zeta =  \int_0^t | \int_s^t  \frac{(t-\xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{g(\xi)}d\xi | \theta_j(\zeta) d\zeta \nonumber\\
& & \leq \int_0^t  \int_{\zeta}^t  \frac{(t-\xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} \theta_j(\zeta) d\xi d\zeta \nonumber\\
& & = \int_0^t  \int_0^{\xi}  \frac{(t-\xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|}  \theta_j(\zeta) d\zeta  d\xi. \nonumber
\end{eqnarray}
  When $\mathfrak{a}, \mathfrak{r} \geq 1$ and $\xi \in [\zeta, t]$, we have $(t-\zeta)^{\mathfrak{r} -1} \geq (t- \xi)^{\mathfrak{r} -1}$ and 
  $(t-\zeta)^{\mathfrak{a} -1} \geq (\xi -\zeta)^{\mathfrak{a} -1}$, so by the above inequality, we conclude that 
     \begin{eqnarray}
& & \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)|  \theta_j(\zeta) d\zeta \leq \int_0^t \frac{1}{|g(\xi)|} \bigg( \int_0^{\xi}   (t-\xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1} \theta_j(\zeta) d\zeta \bigg)  d\xi \nonumber\\
& & \leq \int_0^t \frac{1}{|g(\xi)|} \bigg( \int_0^{\xi}   (t- \zeta)^{\mathfrak{a} + \mathfrak{r} -2} \theta_j(\zeta) d\zeta \bigg)  d\xi =
 \int_0^t \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(t,\xi)}{|g(\xi)|}  d\xi,  \nonumber
\end{eqnarray}
  where $\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(t,\xi) = \int_0^{\xi}   (t- \zeta)^{\mathfrak{a} + \mathfrak{r} -2} \theta_j(\zeta) d\zeta   $.
It is evident that $\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(t,\xi)$ is nondecreasing with respect with their components, also $\hat{\theta}_{\mathfrak{a}, \mathfrak{r}} \leq \hat{\theta}_{\mathfrak{a}, \mathfrak{r}^*}$ when $\mathfrak{r} \geq \mathfrak{r}^*$. By the same manner, it is resulted that 
  $$  \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)|  \theta_j(\zeta) d\zeta \leq \int_0^\eta \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\xi)}{|g(\xi)|}  d\xi$$
  and for all $1 \leq i \leq k_0$, we have
    $$  \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)|  \theta_j(\zeta) d\zeta \leq \int_0^{\gamma_i} \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\xi)}{|g(\xi)|}  d\xi.$$
  Hence for all $ t \in [0,1]$ and $\mathfrak{u}, \mathfrak{v} \in X$ in which $\|\mathfrak{u} -\mathfrak{v}\|_{*} \leq \delta_m(\epsilon)$, the following inequality can be concluded
  \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \nonumber\\
&& \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \int_0^t \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(t,\xi)}{|g(\xi)|}  d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  (q_j+ \epsilon) \int_0^\eta \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\xi)}{|g(\xi)|}  d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon) \bigg(  \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} \frac{\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\xi)}{|g(\xi)|}  d\xi \bigg) \bigg]. \nonumber
\end{eqnarray}
Let 
$\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(t, \xi) :=  \frac{\hat{\theta}^{\mathfrak{a}, \mathfrak{r}}_j(t,\xi)}{|g(\xi)|} $
  and 
  $\|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}:= \int_0^1 \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(1, \xi) d\xi.$
  Since $\hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\xi)$ is nondecreasing  with respect to $t$, $\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(t, \xi)$ also is nondecreasing  with respect to $t$. Also since $\hat{\theta}_{\mathfrak{a}, \mathfrak{r} -1}(\gamma_i,\xi) \geq \hat{\theta}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\xi)$ for all $t, \xi \in [0,1]$, we conclude that $\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(t, \xi) \leq \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r} -1}(t, \xi)$ for all $t, \xi \in [0,1]$,
  hence $\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}(1, \xi) \in L^1[0,1]$ implies that  $\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r} -1} (1, \xi) \in L^1[0,1]$. 
So for all $ t \in [0,1]$ and $\mathfrak{u}, \mathfrak{ر} \in X$ in which $\|\mathfrak{u} -\mathfrak{v}\|_{*} \leq \delta_m(\epsilon)$, we have
    \begin{eqnarray} \label{hash}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \int_0^1 \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(1, \xi) d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  (q_j+ \epsilon)  \int_0^1 \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}(1, \xi) d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon) \bigg(  \sum_{i=1}^{k_0} |\lambda_i|  \int_0^1 \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}(1, \xi) d\xi \bigg) \bigg]. 
\end{eqnarray}
  Therefore 
      \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \leq \frac{ \Xi  \delta_m(\epsilon)}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}\nonumber\\
& & +   \frac{ \Xi  \delta_m(\epsilon)}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  (q_j+ \epsilon)   \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& & +   \frac{ \Xi \delta_m(\epsilon)}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*} \sum_{i=1}^{k_0}  |\lambda_i|  (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \nonumber\\
& &  \leq \frac{ \Xi  \epsilon}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( |\Delta| \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \nonumber\\
& & 
 +  (\mathfrak{r} -1) \sum_{j=1}^{k^*}  (q_j+ \epsilon)   \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} +   \sum_{j=1}^{k^*}( \sum_{i=1}^{k_0}  |\lambda_i| ) (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \bigg). \nonumber
\end{eqnarray}
  So
 \begin{eqnarray}
& &\|\mathcal{L}\mathfrak{u}-\mathcal{L}\mathfrak{v}\|  \leq \frac{ \Xi  \epsilon}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( |\Delta| \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \nonumber\\
& & 
 +  (\mathfrak{r} -1) \sum_{j=1}^{k^*}  (q_j+ \epsilon)   \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} +   \sum_{j=1}^{k^*}( \sum_{i=1}^{k_0}  |\lambda_i| ) (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \bigg). \nonumber
\end{eqnarray}
  Also for all $t \in [0,1]$ and $u, v \in X$, we have
  
  \begin{eqnarray}
& &|\mathcal{L'}\mathfrak{u}(t)-\mathcal{L'}\mathfrak{v}(t)|
\leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t |\frac{\partial \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)}{\partial t}| \bigg|\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) \nonumber\\ & & - \Theta(\zeta, \mathfrak{v}(\zeta), \mathfrak{v}'(\zeta), \phi_{\mathfrak{v}}(\zeta)) \bigg| d\zeta. \nonumber
\end{eqnarray}
  Note that for $\mathfrak{z} \in X$, we have 
    \begin{eqnarray}
& &\mathcal{L'}\mathfrak{z}(t)
=  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t \frac{\partial \mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)}{\partial t} \Theta(\zeta, \mathfrak{z}(\zeta), \mathfrak{z}'(\zeta), \phi_{\mathfrak{z}}(\zeta)) d\zeta \nonumber\\ & & = \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \int_0^t \mathcal{H}_{\mathfrak{a}, \mathfrak{r} -1}(t,\zeta) \Theta(\zeta, \mathfrak{z}(\zeta), \mathfrak{z}'(\zeta), \phi_{\mathfrak{z}}(\zeta)) d\zeta. \nonumber
\end{eqnarray}
For all $t \in [0,1]$ and $\mathfrak{u}, \mathfrak{v} \in X$, it is concluded that
    \begin{eqnarray}
& &|\mathcal{L'}\mathfrak{u}(t)-\mathcal{L'}\mathfrak{v}(t)|
\leq   \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \int_0^t \bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r} -1}(t,\zeta)|  \nonumber\\ & & \times |\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta))- \Theta(\zeta, \mathfrak{v}(\zeta), \mathfrak{v}'(\zeta), \phi_{\mathfrak{v}}(\zeta)) | \bigg) d\zeta \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \int_0^t \bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r} -1}(t,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (|\mathfrak{u}(\zeta) -\mathfrak{v}(\zeta)|, |\mathfrak{u}'(\zeta)-\mathfrak{v}'(\zeta)|, |\phi_{\mathfrak{u}}(\zeta)- \phi_{\mathfrak{v}}(\zeta)|) \bigg)d\zeta \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \int_0^t \bigg( |\mathcal{H}_{\mathfrak{a}, \mathfrak{r} -1}(t,\zeta)| \nonumber\\
&& \times  \sum_{j=1}^{k^*} \theta_j(\zeta) \Lambda_j (\|\mathfrak{u} -\mathfrak{v}\|, \|\mathfrak{u}'-\mathfrak{v}'\|, \omega_1 \|\mathfrak{u} -\mathfrak{v}\|+ \omega_2 \|\mathfrak{u}'-\mathfrak{v}'\|) \bigg)d\zeta \nonumber\\
&& \leq \sum_{j=1}^{k^*} \bigg[ \frac{ \Lambda_j (\Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*)}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}  \nonumber\\
&& \times \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r} -1}(t,\zeta)|  \theta_j(\zeta) d\zeta \bigg]. \nonumber
\end{eqnarray}
By (\ref{str}), when $\|\mathfrak{u} -\mathfrak{v}\|_* \leq \delta_m(\epsilon)$, for all $t \in [0,1]$ we infer that
\begin{eqnarray}
& &|\mathcal{L'}\mathfrak{u}(t)-\mathcal{L'}\mathfrak{v}(t)| \nonumber\\
&& \leq \sum_{j=1}^{k^*} \frac{ \Xi (q_j + \epsilon) \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)}   \int_0^t  |\mathcal{H}_{\alpha, \mathfrak{r} -1}(t,\zeta)|  \theta_j(\zeta) d\zeta  \nonumber\\
&& \leq  \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \int_0^t \int_{\zeta}^t  \frac{(t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi \theta_j(\zeta) d\zeta  \nonumber\\
&& \leq  \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \int_0^t \int_0^{\xi}  \frac{(t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} \theta_j(\zeta) d\zeta  d\xi  \nonumber
\end{eqnarray}

  \begin{eqnarray}
&& \leq  \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \int_0^t \frac{1}{|g(\xi)|} (\int_0^{\xi}   (t - \zeta)^{\mathfrak{a}+ \mathfrak{r} -3}  \theta_j(\zeta) d\zeta)  d\xi  \nonumber\\
&& =  \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \int_0^t \frac{\hat{\theta}_{j}^{\mathfrak{a}, \mathfrak{r} -1}(t, \xi) }{|g(\xi)|}  d\xi  \nonumber\\
&& \leq  \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \| \tilde{g}_{\theta_j}^{\mathfrak{a}, \mathfrak{r} -1}\|_{[0,1]}  \nonumber\\
&& \leq  \frac{ \Xi \delta_m(\epsilon)}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \| \tilde{g}_{\theta_j}^{\mathfrak{a}, \mathfrak{r} -1}\|_{[0,1]}  \nonumber\\
&& \leq  \frac{ \Xi  \epsilon}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \sum_{j=1}^{k^*}  (q_j + \epsilon) \| \tilde{g}_{\theta_j}^{\mathfrak{a}, \mathfrak{r} -1}\|_{[0,1]}  \nonumber\\
&& =  \frac{ \Xi  \epsilon (\mathfrak{r}-1)}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  (q_j + \epsilon) \| \tilde{g}_{\theta_j}^{\mathfrak{a}, \mathfrak{r} }\|_{[0,1]}.  \nonumber
\end{eqnarray}
Which leads to
  \begin{eqnarray}
& &||\mathcal{L'}\mathfrak{u} -\mathcal{L'}\mathfrak{v} \|  \leq  \frac{ \Xi  \epsilon (\mathfrak{r}-1)}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  (q_j + \epsilon) \| \tilde{g}_{\theta_j}^{\mathfrak{a}, \mathfrak{r} }\|_{[0,1]}.  \nonumber
\end{eqnarray}
  Therefore
\begin{eqnarray}
& &\|\mathcal{L}\mathfrak{u}-\mathcal{L}\mathfrak{v}\|_{*} = \max \{ \|\mathcal{L}\mathfrak{u}-\mathcal{L}\mathfrak{v}\|, \|\mathcal{L}'\mathfrak{u}-\mathcal{L}'\mathfrak{v}\| \} \nonumber\\
&& \leq \frac{ \Xi  \epsilon}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( |\Delta| \sum_{j=1}^{k^*}   (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \nonumber\\
& & 
 +  (\mathfrak{r} -1) \max\{1, |\Delta| \} \sum_{j=1}^{k^*}  (q_j+ \epsilon)   \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &  +   \sum_{j=1}^{k^*}( \sum_{i=1}^{k_0}  |\lambda_i| ) (q_j+ \epsilon)  \| \tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \bigg). \nonumber
\end{eqnarray}
  
  Since $\epsilon >0$ is arbitary,  $\mathfrak{u} \to \mathfrak{v}$ in $X$ implies $\mathcal{L}\mathfrak{u} \to \mathcal{L}\mathfrak{v}$  in $X$, therefore $\mathcal{L}$ is continuous on $X$. Now since 
$\lim_{\|z\| \to 0^{+}} \frac{ \mathcal{N}_i(t, z)}{\|z\|} = \mathcal{V}_i(t)$, for all $1 \leq i \leq 3$ and almost all $t \in [0,1]$,  $\lim_{z \to 0^{+}} \frac{ \mathcal{N}_i(t, \Xi z)}{\Xi z} = \mathcal{V}_i(t)$. Therefore for $\epsilon> 0$ there exists $\delta(\epsilon) >0$ such that $0 < z \leq \delta(\epsilon)$ implies $\frac{ \mathcal{N}_i(t, \Xi z)}{\Xi z} < \mathcal{V}_i(t) +\epsilon$ and thus $\mathcal{N}_i(t, \Xi z) < (\mathcal{V}_i(t) +\epsilon) \Xi z,$ for all $1 \leq i \leq 3$ and almost all  $t \in [0,1]$. Since
\begin{eqnarray}
&&  \frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg[  \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg) \bigg]<1  \nonumber
\end{eqnarray}
  then there exists $\epsilon_0 >0$ such that 
\begin{eqnarray}
&&  \bigg(\frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg[  \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg) \bigg]    
 + \frac{3 \Xi \epsilon_0}{|\Delta| \Gamma(\mathfrak{a}+1) \Gamma(\mathfrak{r})} 
 \bigg[ |\Delta| (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| 
\nonumber\\
&& 
 + (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| + (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg] \bigg)<1,  \nonumber
\end{eqnarray}
similarly since 
   \begin{eqnarray} 
& & \frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} \bigg) <1 \nonumber
\end{eqnarray}
  there exists $\epsilon_1 >0$, such that 
  \begin{eqnarray} 
& &  \bigg[ \frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}\bigg) \nonumber\\
& & +   \frac{ \Xi \epsilon_1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}    \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*} \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+   \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0}  |\lambda_i|   \|\tilde{g}_{\theta_ j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}\bigg) \bigg]   <1. \nonumber
\end{eqnarray}
  Let $R_0= \min \{ \epsilon_0, \frac{\delta_m(\epsilon_1)}{2} \}$, so  $\mathcal{N}_i(t, \rho z) < (\mathcal{V}_i(t) +\epsilon_0) \rho z,$ for all $0 < z \leq R_0$.   Put $\Omega = \{ \mathfrak{u} \in X : \|\mathfrak{u}\|_* \leq R_0 \}$. Define the  map $\mathcal{A} : X^2 \to [0, \infty)$ by
  $\mathcal{A}(\mathfrak{u},\mathfrak{v}) =1$ when $\mathfrak{u}, \mathfrak{v} \in \Omega$, otherwise let  $\mathcal{A}(\mathfrak{u},\mathfrak{v}) =0$. Suppose that $\mathfrak{u}, \mathfrak{v} \in X$ be such that  $\mathcal{A}(\mathfrak{u},\mathfrak{v}) \geq 1$, so $\mathfrak{u}, \mathfrak{v} \in \Omega$,    $\|\mathfrak{u}\|_* \leq R_0$ and $\|\mathfrak{v}\|_* \leq R_0$. Then for all $t \in [0,1]$, we have
  \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)|
\leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| |\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) | d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| |\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) | d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} \lambda_i \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| |\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) | d\zeta \nonumber
\end{eqnarray}

  \begin{eqnarray} 
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \mathfrak{u}(\zeta) ) +  \mathcal{N}_2 (\zeta, \mathfrak{u}'(\zeta) ) + \mathcal{N}_3 (\zeta, \phi_\mathfrak{u}(\zeta) ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
& & \times \bigg( \mathcal{N}_1 (\zeta, \mathfrak{u}(\zeta) ) +  \mathcal{N}_2 (\zeta, \mathfrak{u}'(\zeta) ) + \mathcal{N}_3 (\zeta, \phi_\mathfrak{u}(\zeta) ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
& & \times \bigg( \mathcal{N}_1 (\zeta, \mathfrak{u}(\zeta) ) +  \mathcal{N}_2 (\zeta, \mathfrak{u}'(\zeta) ) + \mathcal{N}_3 (\zeta, \phi_\mathfrak{u}(\zeta) ) \bigg)d\zeta. \nonumber
\end{eqnarray}
 Consequently, for $u \in \Omega$,  hence
    \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)|  \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \|\mathfrak{u}\| ) +  \mathcal{N}_2 (\zeta, \|\mathfrak{u}'\| ) + \mathcal{N}_3 (\zeta, \omega_1 \|\mathfrak{u}\| +\omega_2 \|\mathfrak{u}'\| ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \|\mathfrak{u}\| ) +  \mathcal{N}_2 (\zeta, \|\mathfrak{u}'\| ) + \mathcal{N}_3 (\zeta, \omega_1 \|\mathfrak{u}\| +\omega_2 \|\mathfrak{u}'\| ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \|\mathfrak{u}\| ) +  \mathcal{N}_2 (\zeta, \|\mathfrak{u}'\| ) + \mathcal{N}_3 (\zeta, \omega_1 \|\mathfrak{u}\| +\omega_2 \|\mathfrak{u}'\| ) \bigg)d\zeta \nonumber\\
&&  \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber
\end{eqnarray}
 
\begin{eqnarray} 
  & & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
  & & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
&&  \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_* ) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
&&  \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| \Xi \|\mathfrak{u}\|_* (\sum_{j=1}^{3} \mathcal{V}_j(\zeta)+\epsilon_0) d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )} \int_0^{\eta} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| \Xi \|\mathfrak{u}\|_* (\sum_{j=1}^{3} \mathcal{V}_j(\zeta)+\epsilon_0) d\zeta \nonumber\\
& & + \frac{1}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )} \sum_{i=1}^{k_0} |\lambda_i| \int_0^{\gamma_i} |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| \Xi \|\mathfrak{u}\|_* (\sum_{j=1}^{3} \mathcal{V}_j(\zeta)+\epsilon_0) d\zeta \nonumber\\
&&  =  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} (\int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)|  \mathcal{V}_j(\zeta)  d\zeta) +3\epsilon_0 \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(t,\zeta)| d\zeta \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )}  \bigg( \sum_{j=1}^{3} (\int_0^{\eta}  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)|  \mathcal{V}_j(\zeta)  d\zeta) \nonumber\\
& &+3\epsilon_0 \int_0^{\eta}  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(\eta,\zeta)| d\zeta \bigg) \nonumber
\end{eqnarray}
  
  \begin{eqnarray}
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )}  \sum_{i=1}^{k_0} |\lambda_i| \bigg( \sum_{j=1}^{3} (\int_0^{\gamma_i}  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)|  \mathcal{V}_j(\zeta)  d\zeta) \nonumber\\
& &+3\epsilon_0 \int_0^{\gamma_i}  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}}(\gamma_i,\zeta)| d\zeta \bigg) \nonumber\\
  &&  =  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} (\int_0^t  \int_{\zeta}^t \frac{(t - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi  \mathcal{V}_j(\zeta)  d\zeta) \nonumber\\
& & +3\epsilon_0 \int_0^t   \int_\zeta^t \frac{(t - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi  d\zeta \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )}  \bigg( \sum_{j=1}^{3} (\int_0^{\eta}   \int_\zeta^t \frac{(\eta - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi   \mathcal{V}_j(\zeta)  d\zeta) \nonumber\\
& &+3\epsilon_0 \int_0^{\eta}   \int_\zeta^{\eta} \frac{(\eta - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi  d\zeta \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )}  \sum_{i=1}^{k_0} |\lambda_i| \bigg( \sum_{j=1}^{3} (\int_0^{\gamma_i}    \int_\zeta^{\gamma_i} \frac{(\gamma_i - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi   \mathcal{V}_j(\zeta)  d\zeta) \nonumber\\
& &+3\epsilon_0 \int_0^{\gamma_i}    \int_\zeta^{\gamma_i} \frac{(\gamma_i - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi  d\zeta \bigg), \nonumber
\end{eqnarray}

thus, it is concluded that for all $\mathfrak{u} \in \Omega$ and $t \in [0,1]$
\begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)| 
  \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} (\int_0^t  \int_0^{\xi} \frac{(t - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|}  \mathcal{V}_j(\zeta)  d\zeta d\xi ) \nonumber\\
& & +3\epsilon_0 \int_0^t  \frac{1}{|g(\xi)|}  \int_0^{\xi}  (t - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}  d\zeta d\xi \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )}  \bigg( \sum_{j=1}^{3} (\int_0^{\eta}   \int_0^{\xi} \frac{(\eta - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|}   \mathcal{V}_j(\zeta)  d\zeta d\xi ) \nonumber\\
& &+3\epsilon_0 \int_0^{\eta}   \frac{1}{|g(\xi)|}  \int_0^{\xi}  (\eta - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1} d\zeta d\xi \bigg) \nonumber
\end{eqnarray}


\begin{eqnarray}
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )}  \sum_{i=1}^{k_0} |\lambda_i| \bigg( \sum_{j=1}^{3} (\int_0^{\gamma_i}    \int_0^{\xi} \frac{(\gamma_i - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} \mathcal{V}_j(\zeta)  d\zeta d\xi  ) \nonumber\\
& &+3\epsilon_0 \int_0^{\gamma_i}    \frac{1}{|g(\xi)|}  \int_0^{\xi}  (\gamma_i - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1}  d\zeta d\xi  \bigg). \nonumber
\end{eqnarray}
  Since $\xi \in [\zeta,t]$, then $(t- \xi)^{\mathfrak{r} -1}(\xi -\zeta)^{\mathfrak{a} -1} \leq (t-\zeta)^{\mathfrak{a}+ \mathfrak{r} -2}$, hence for  $\xi \in [\zeta,t]$ we have
  \begin{eqnarray}
\int_0^{\xi} \frac{(t - \xi)^{\mathfrak{r} -1} (\xi -s)^{\mathfrak{a} -1}}{|g(\xi)|} \mathcal{V}_j(\zeta)  d\zeta \leq \frac{1}{|g(\xi)|}  \int_0^{\xi}  (t - \zeta)^{\mathfrak{a}+ \mathfrak{r} -2}  \mathcal{V}_j(\zeta) d\zeta \nonumber\\
  = \frac{1}{|g(\xi)|} \hat{ \mathcal{V}}_j^{\mathfrak{a}, \mathfrak{r}}(t, \xi), \nonumber
  \end{eqnarray}
  also we have $\int_0^{\xi} (t - \xi)^{\mathfrak{r} -1} (\xi -\zeta)^{\mathfrak{a} -1} d\zeta = \frac{ (t - \xi)^{ \mathfrak{r} -1} \xi^{\mathfrak{a}}}{\mathfrak{a}}.$ 
 So  for all $\mathfrak{u} \in \Omega$ and $t \in [0,1]$ it is infered that
  \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)| \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} \int_0^t  \frac{\hat{ \mathcal{V}}_j^{\mathfrak{a}, \mathfrak{r}}(t, \xi)}{|g(\xi)|}  d\xi  +3\epsilon_0 \int_0^t  \frac{(t - \xi)^{ \mathfrak{r} -1} \xi^{\mathfrak{a}}}{\mathfrak{a} |g(\xi)|}  d\xi \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1 )}  \bigg( \sum_{j=1}^{3} \int_0^{\eta}   \frac{\hat{ \mathcal{V}}_j^{\mathfrak{a}, \mathfrak{r}}(\eta, \xi)}{|g(\xi)|}  d\xi  +3\epsilon_0 \int_0^{\eta}  \frac{(\eta - \xi)^{ \mathfrak{r} -2} \xi^{\mathfrak{a}}}{\mathfrak{a} |g(\xi)|}  d\xi \bigg) \nonumber\\
& & + \frac{\Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} )}  \sum_{i=1}^{k_0} |\lambda_i| \bigg( \sum_{j=1}^{3} \int_0^{\gamma_i}   \frac{\hat{ \mathcal{V}}_j^{\mathfrak{a}, \mathfrak{r}}(\gamma_i, \xi)}{|g(\xi)|}  d\xi  \nonumber\\
&& +3\epsilon_0 \int_0^{\gamma_i}  \frac{(\gamma_i - \xi)^{ \mathfrak{r} -1} \xi^{\mathfrak{a}}}{\mathfrak{a} |g(\xi)|}  d\xi \bigg), \nonumber
\end{eqnarray}
  therefore 
    \begin{eqnarray}
& &|\mathcal{L}\mathfrak{u}(t)| \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,t]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg) \nonumber\\
& & +  \frac{ \Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg) \nonumber\\
& & +  \frac{ \Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} \bigg[ \sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg]  + \frac{3\epsilon_0}{\mathfrak{a}} (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg). \nonumber
\end{eqnarray}
  Taking the supremum norm over $[0,1]$, we conclude that 
      \begin{eqnarray}
& &\|\mathcal{L}\mathfrak{u}| \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg) \nonumber\\
& & +  \frac{ \Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r} -1)} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg) \nonumber\\
& & +  \frac{ \Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( \sum_{j=1}^{3} \bigg[ \sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg]  + \frac{3\epsilon_0}{\mathfrak{a}} (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg) \nonumber\\
& &=  \frac{ \Xi \|\mathfrak{u}\|_* }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg[  \sum_{j=1}^{3} \bigg( |\Delta| \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg) \bigg]\nonumber\\
&& 
  + \frac{3 \Xi \epsilon_0 \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}+1) \Gamma(\mathfrak{r})} 
 \bigg[ |\Delta \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| + (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| + (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg] \nonumber\\
&& \leq  \bigg(\frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg[  \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\alpha, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg) \bigg]\nonumber\\
&& 
  + \frac{3 \Xi \epsilon_0}{|\Delta| \Gamma(\mathfrak{a}+1) \Gamma(\mathfrak{r})} 
 \bigg[ |\Delta| (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| + (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| + (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg] \bigg) R_0 \nonumber\\
&& \leq R_0. \nonumber
\end{eqnarray}
Likewise, for all $t \in [0,1]$ and $\mathfrak{u} \in \Omega$ we have
 \begin{eqnarray}
& &|\mathcal{L}'\mathfrak{u}(t)|
\leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \int_0^t |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)| |\Theta(\zeta, \mathfrak{u}(\zeta), \mathfrak{u}'(\zeta), \phi_{\mathfrak{u}}(\zeta)) | d\zeta \nonumber\\
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \|\mathfrak{u}\| ) +  \mathcal{N}_2 (\zeta, \|\mathfrak{u}'\| ) + \mathcal{N}_3 (\zeta, \omega_1\|\mathfrak{u}\|+\omega_2 \|\mathfrak{u}'\| ) \bigg)d\zeta \nonumber
\end{eqnarray}
  
        \begin{eqnarray}
&& \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)| \nonumber\\
&&\times \bigg( \mathcal{N}_1 (\zeta, \Xi \|\mathfrak{u}\|_* ) +  \mathcal{N}_2 (\zeta, \Xi \|\mathfrak{u}\|_*) + \mathcal{N}_3 (\zeta, \Xi \|\mathfrak{u}\|_* ) \bigg)d\zeta \nonumber\\
&&  \leq  \frac{1}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)| \Xi \|\mathfrak{u}\|_* (\sum_{j=1}^{3} \mathcal{V}_j(\zeta)+\epsilon_0) d\zeta \nonumber\\
 &&  =  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)|  \mathcal{V}_j(\zeta)  d\zeta +3\epsilon_0 \int_0^t  |\mathcal{H}_{\mathfrak{a}, \mathfrak{r}-1}(t,\zeta)| d\zeta \bigg) \nonumber\\
&&  =  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \int_0^t  \int_\zeta^t \frac{(t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} \mathcal{V}_j(\zeta)  d\xi  d\zeta \nonumber\\
& & +3\epsilon_0 \int_0^t   \int_\zeta^t \frac{(t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|} d\xi  d\zeta \bigg) \nonumber\\
&& =  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} (\int_0^t  \int_0^{\xi} \frac{(t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}}{|g(\xi)|}  \mathcal{V}_j(\zeta)  d\zeta d\xi ) \nonumber\\
& & +3\epsilon_0 \int_0^t  \frac{1}{|g(\xi)|}  \int_0^{\xi}  (t - \xi)^{\mathfrak{r} -2} (\xi -\zeta)^{\mathfrak{a} -1}  d\zeta d\xi \bigg) \nonumber\\
&& \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \int_0^t  \frac{1}{|g(\xi)|} (\int_0^{\xi} (t - \zeta)^{\mathfrak{a}+ \mathfrak{r} -3}  \mathcal{V}_j(\zeta)  d\zeta) d\xi  \nonumber\\
& & + \frac{3\epsilon_0}{\mathfrak{a}} \int_0^t  \frac{ (t - \xi)^{\mathfrak{r} -2} \xi ^{\mathfrak{a} }}{|g(\xi)|}    d\xi \bigg) \nonumber\\
& & \leq  \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \int_0^t  \frac{\hat{ \mathcal{V}}_j^{\mathfrak{a}, \mathfrak{r}-1}(t, \xi)}{|g(\xi)|}  d\xi  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg) \nonumber\\
& &= \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,t]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg). \nonumber
\end{eqnarray}
Therefore
 \begin{eqnarray}
& &\|\mathcal{L}'\mathfrak{u}\|
\leq \frac{ \Xi \|\mathfrak{u}\|_*}{\Gamma(\mathfrak{a}) \Gamma(\mathfrak{r}-1)} \bigg( \sum_{j=1}^{3} \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]}  + \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg) \nonumber
\end{eqnarray}
  
        \begin{eqnarray}
&&= \frac{ \Xi \|\mathfrak{u}\|_*}{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg( |\Delta| (\mathfrak{r} -1) \sum_{j=1}^{3} \|  \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]}  + |\Delta| (\mathfrak{r} -1)  \frac{3\epsilon_0}{\mathfrak{a}} \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| \bigg) \nonumber\\
&& \leq  \bigg(\frac{ \Xi }{|\Delta| \Gamma(\mathfrak{a}) \Gamma(\mathfrak{r})} \bigg[  \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\mathfrak{a}, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg) \bigg]   
  + \frac{3 \Xi \epsilon_0}{|\Delta| \Gamma(\mathfrak{r}+1) \Gamma(\mathfrak{r})} 
 \bigg[ |\Delta| (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| 
\nonumber\\
&& 
+ (\mathfrak{r}-1) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}-1}\| + (\sum_{i=1}^{k_0} |\lambda_i| ) \| \bar{g}_{\mathfrak{a}, \mathfrak{r}}\| \bigg] \bigg) R_0 \leq R_0. \nonumber
\end{eqnarray}
Thus, we conclude that
$$ \| \mathcal{L}\mathfrak{u}\|_* =\max \{ \|\mathcal{L}\mathfrak{u}\|, \|\mathcal{L}'\mathfrak{u}\| \} \leq R_0,$$
hence $\mathcal{L}\mathfrak{u} \in \Omega$. By a similar way, it is resulted in $\mathcal{L}\mathfrak{v} \in \Omega$, this implies that $\mathcal{A}(\mathcal{L}\mathfrak{u}, \mathcal{L}\mathfrak{v}) \geq 1,$ therefore $\mathcal{L}$ is $\mathcal{A}$- admissible. Evidently $\Omega$ is nonempty, so there exists $\mathfrak{u}_0 \in \Omega$, we further proved that $\mathcal{L}\mathfrak{u}_0 \in \Omega$, which leads to $\mathcal{A}(\mathfrak{u}_0, \mathcal{L}\mathfrak{u}_0) \geq 1.$ Let $\mathfrak{u}, \mathfrak{v} \in X$, if  $\mathcal{A}(\mathfrak{u}, \mathfrak{v}) \neq 0$, then $\mathfrak{u}, \mathfrak{v} \in \Omega$, therefore
$$d(\mathfrak{u},\mathfrak{v}) \leq \| \mathfrak{u}\|_* +\|\mathfrak{v}\|_* \leq 2 \frac{\delta_m(\epsilon_1)}{2} = \delta_m(\epsilon_1).$$
By (\ref{str}), the following inequality is held
 \begin{eqnarray}
 \Lambda(\Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*, \Xi \|\mathfrak{u} -\mathfrak{v}\|_*) \leq \Xi  (q_j+ \epsilon_1) \|\mathfrak{u} -\mathfrak{v}\|_{*}, \nonumber
 \end{eqnarray}
so for all $t \in [0,1]$, (\ref{hash}) implies that

    \begin{eqnarray} 
& &|\mathcal{L}\mathfrak{u}(t)-\mathcal{L}\mathfrak{v}(t)| \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{\Gamma(\alpha) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}   (q_j+ \epsilon_1)  \int_0^1 \tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}(1, \xi) d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  (q_j+ \epsilon_1)  \int_0^1 \tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}(1, \xi) d\xi \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  \bigg[ (q_j+ \epsilon_1) \bigg(  \sum_{i=1}^{k_0} |\lambda_i|  \int_0^1 \tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}(1, \xi) d\xi \bigg) \bigg] \nonumber\\
 & & \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{\Gamma(\alpha) \Gamma(\mathfrak{r})}  \sum_{j=1}^{k^*}   (q_j+ \epsilon_1)  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]} \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r} -1)}  \sum_{j=1}^{k^*}  (q_j+ \epsilon_1)  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& & +   \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r} )} \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} (q_j+ \epsilon_1)    |\lambda_i| \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  \nonumber\\
 & & = \bigg[ \frac{ \Xi }{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}\bigg) \nonumber\\
& & +   \frac{ \Xi \epsilon_1}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}    \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*} \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+   \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0}  |\lambda_i|   \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}\bigg) \bigg]   \|\mathfrak{u} -\mathfrak{v}\|_{*}. \nonumber
\end{eqnarray}
Let 

 \begin{eqnarray} 
& & \lambda:= \bigg[ \frac{ \Xi }{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \beta-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}\bigg) +   \frac{ \Xi \epsilon_1}{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}    \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  \nonumber\\
& & + (\mathfrak{r}-1) \sum_{j=1}^{k^*} \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} +   \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0}  |\lambda_i|   \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}\bigg) \bigg]   <1. \nonumber
\end{eqnarray}
So $\|\mathcal{L}\mathfrak{u}-\mathcal{L}\mathfrak{v}\| \leq \lambda \|\mathfrak{u} -\mathfrak{v}\|_{*}.$
By similar way, for $u,v \in X$ in which $\mathcal{A}(\mathfrak{u},\mathfrak{v}) \neq 0$,  it follows that
  \begin{eqnarray} 
& &|\mathcal{L}'\mathfrak{u}(t)-\mathcal{L}'\mathfrak{v}(t)| 
 \leq \frac{ \Xi  \|\mathfrak{u} -\mathfrak{v}\|_{*}}{ \Gamma(\alpha) \Gamma(\mathfrak{r}-1)}  \sum_{j=1}^{k^*}   (q_j+ \epsilon_1)  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
  & & = \bigg[ \frac{ \Xi }{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta| (\mathfrak{r}-1) \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \bigg)  \nonumber\\ 
 & &+ 
 \frac{ \Xi \epsilon_0 }{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta| (\mathfrak{r}-1) \sum_{j=1}^{k^*}   \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \bigg) \bigg] \|\mathfrak{u} -\mathfrak{v}\|_{*} \nonumber\\
 && \leq \lambda \|\mathfrak{u} -\mathfrak{v}\|_{*}. \nonumber
\end{eqnarray}
So $\|\mathcal{L}'\mathfrak{u}-\mathcal{L}'\mathfrak{v}\| \leq \lambda \|\mathfrak{u} -\mathfrak{v}\|_{*}.$
and $\|\mathcal{L}\mathfrak{u}-\mathcal{L}\mathfrak{v}\|_* \leq \lambda \|\mathfrak{u} -\mathfrak{v}\|_{*}.$
Define $\psi :[0, \infty) \to [0, \infty)$ as $\psi(t) = \lambda t$, then $\sum_{i=1}^{\infty} \psi^{i}(t) = \frac{\lambda}{1 - \lambda} t < \infty$ for all $ t \in [0, \infty)$, so $\psi \in \Psi$. Therefore we have proved $\mathfrak{u}, \mathfrak{v} \in X$ in which $\mathcal{A}(\mathfrak{u}, \mathfrak{v}) \neq 0$, $\mathcal{A}(\mathfrak{u}, \mathfrak{v}) d(\mathcal{L}\mathfrak{u}, \mathcal{L}\mathfrak{v}) \leq  \psi( d(\mathfrak{u},\mathfrak{v}) )$. In the case $\mathcal{A}(\mathfrak{u}, \mathfrak{v}) = 0$, the inequality is obvious. So for all $\mathfrak{u}, \mathfrak{v} \in X$, the inequality 
 $\mathcal{A}(\mathfrak{u}, \mathfrak{v}) d(\mathcal{L}\mathfrak{u}, \mathcal{L}\mathfrak{v}) \leq  \psi( d(\mathfrak{u},\mathfrak{v}) )$ is held. 
Now, regarding to  lemma (\ref{l1.1}), $\mathcal{L}:X \to X$ has a fixed point in $X$, so  the singular problem (\ref{prb1}) has a solution.
\end{proof}
\\ The following example demonstrates the main result.
\begin{example}
Let 
$$ c(t) = \left\{ \begin{array}{ll}
0 \ \ \ \ \ \ \ \ \ t \in [0,1] \cap Q
\\
\\
1 \ \ \ \ \ \ \ \ \ t \in (0,1) \cap Q^{c}. 
\ \ \ \end{array}\right.$$
and
$$\Theta(t,x_1, x_2, x_3)= \frac{1}{c(t)}(\|x_1\|+\|x_2\|+\|x_3\|).$$
Consider the following pointwise defined bi-singular equation
\begin{eqnarray}
\label{ex1} \mathcal{D}^{\frac{3}{2}}\bigg(3\sqrt{t}  \mathcal{D}^{\frac{5}{2}}  \mathfrak{u}(t) \bigg) =\Theta(t, \mathfrak{u}(t), \mathfrak{u}'(t), \mathcal{D}^{\frac{1}{2}} \mathfrak{u}(t))
\end{eqnarray}
with boundary condition 
$ \mathcal{D}^{\frac{5}{2} +j}\mathfrak{u}(0)=\mathfrak{u}'(0)=0$ for $0 \leq j \leq 2$ and $\mathfrak{u}'(\frac{1}{2})=2 \mathfrak{u}(\frac{1}{2})$.
Put $k_0 =1$, $\gamma_1 =\frac{1}{2}$, $\eta = \frac{1}{2}$, $g(t) =3t$  and $\phi_\mathfrak{u}(t)= \mathcal{D}^{\frac{1}{2}} \mathfrak{u}(t)$,  then 
$$\| \phi_\mathfrak{u} - \phi_\mathfrak{v} \| \leq \frac{1}{\Gamma(2 - \frac{1}{2})} \|\mathfrak{u}' - \mathfrak{v}'\| = \frac{2}{\sqrt{\pi}} \|\mathfrak{u}' - \mathfrak{v}'\|, $$
so $\omega_1 =0$, $\omega_2=  \frac{2}{\sqrt{\pi}}$ and $\Xi= \max \{1, \omega_1+\omega_2\} =1$.
Regarding to lemma (\ref{beta}), it is resulted in 
\begin{eqnarray}
\|\bar{g}_{\alpha, \mathfrak{r}-1}\|= \int_0^1 \frac{(1-\zeta)^{\frac{1}{2}}\zeta^{\frac{3}{2}}}{3 \sqrt{\zeta}} = \frac{1}{3}  \int_0^1 (1-\zeta)^{\frac{1}{2}} \zeta= \frac{1}{3}  \mathcal{B}(2, \frac{3}{2}) \nonumber\\
= \frac{\Gamma(\frac{3}{2})  \Gamma(2)}{ 3 \Gamma(\frac{5}{2})} = \frac{2}{9} < \infty. \nonumber
\end{eqnarray}
Let $k^* =1$, $\theta_1(t) = \frac{1}{c(t)}$, $\mathcal{N}_i(t, x_i) = \frac{1}{c(t)} x_i$, and $\Lambda_1( x_1, x_2, x_3)= x_1 +x_2 +x_3$, then
$$|\Theta(t, \omega_1, \omega_2, \omega_3) - \Theta(t, z_1, z_2, z_3)| \leq \theta_1 \Lambda_1( |\omega_1 - z_1|, |\omega_2 - z_2|, |\omega_3 - z_3|),$$
$\Lambda_1$ is nondecreasing with respect to all their components, $q_i := \frac{\Lambda(z, z, z)}{z}= 3 \in [0, \infty)$,
$$\hat{\theta}_{1}^{\alpha, \mathfrak{r}}(t, \xi) = \int_0^{\xi} \frac{(t-\zeta)^2}{c(\zeta)} d\zeta = \frac{1}{3} [t^3- (t- \xi)^3  ],$$
\begin{eqnarray}
\|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]}:= \frac{1}{3} \int_0^1  [1- (1- \xi)^3] \xi^{\frac{-1}{2}} d\xi =\frac{38}{105}, \nonumber
\end{eqnarray}
$|\Theta(t, x_1,x_2,x_3)| \leq \sum_{i=11}^3 \mathcal{N}_i(t,x_i)$,  $\mathcal{N}_i: [0,1] \times X \to [0, \infty)$ for each $ 1 \leq i \leq 3$ is nondecreasing with respect to its second component,
$\mathcal{V}_i(t) = \lim_{z \to 0^+} \frac{\mathcal{N}_i(t, z)}{z}= \frac{1}{c(t)},$
$$\hat{\mathcal{V}}_i^{\alpha, \mathfrak{r}-1}= \int_0^\xi (t - s)^{\alpha+ \mathfrak{r} -3} \mathcal{V}_i(s) ds = \frac{1}{2} [t^2- (t- \xi)^2  ]$$
and
$$\|\hat{\mathcal{V}}_i^{\alpha, \mathfrak{r}-1}\|_{[0,1]} =\frac{1}{2} \int_0^1  [1- (1- \xi)^2] \xi^{\frac{-1}{2}} d\xi = \frac{7}{15}.$$
It is easy to see the other properties in Theorem (\ref{t2.3}) are held and 
\begin{eqnarray}
&&     \sum_{j=1}^{3} \bigg( |\Delta| (\mathfrak{r}-1) \| \hat{g}_{\mathcal{V}_j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} + (\mathfrak{r} -1)  \| \hat{g}_{\mathcal{V}_j}^{\alpha, \mathfrak{r}}\|_{[0,\eta]}\nonumber\\
&& +\sum_{i=1}^{k_0} |\lambda_i|  \| \hat{g}_{\mathcal{V}_j}^{\alpha, \mathfrak{r}}\|_{[0,\gamma_i]} \bigg)  \in [0, \frac{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}{ \Xi})  \nonumber
\end{eqnarray}
and 
   \begin{eqnarray} 
& & \frac{ \Xi }{|\Delta| \Gamma(\alpha) \Gamma(\mathfrak{r})}  \bigg( |\Delta|  \sum_{j=1}^{k^*}   q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]}  + (\mathfrak{r}-1) \sum_{j=1}^{k^*}  q_j  \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}-1}\|_{[0,1]} \nonumber\\
& &+  \sum_{j=1}^{k^*}  \sum_{i=1}^{k_0} q_j  |\lambda_i| \|\tilde{g}_{\theta_ j}^{\alpha, \mathfrak{r}}\|_{[0,1]} \bigg) <1. \nonumber
\end{eqnarray}
Therefore, by using Theorem  (\ref{t2.3}) , the bi-singular problem (\ref{ex1}) has a solution.
\end{example}
\section{Conclusion}
Various fractional differential equations have been examined during last decades. Among them, singular equations, are more notable. In this article, we introduce the bi-singularity concept and consider a bi-singular   fractional-order differential equation. Also, the main result is demonstrated through an example. 





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{\small
\noindent{\bf Mehdi Shabibi }

\noindent Assistant Professor of Mathematics

\noindent Department of Mathematics, Meharn Branch, Islamic Azad University

\noindent Mehran, Iran

\noindent E-mail: $mehdi\_math1983@yahoo.com$}\\



\end{document}
