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\fancyhead[CE]{A.  MALEKPOUR,  M. SHABIBI } 
\fancyhead[CO]{INVESTIGATING OF A COMMON SOLUTION    ...}



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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 
Investigation of a common solution for a multi-singular fractional system by using control functions method\\} 
\vspace{2mm}

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

{\bf A. Malekpour}\vspace*{-2mm}\\
\vspace{2mm} {\small Department of Mathematics, South Tehran Branch, Islamic Azad University, Tehran, Iran} \vspace{2mm}

{\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}


\end{center}

\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In the proposed article, the existence of a  solution is investigated for a pointwise multi-singular fractional differential equation with integral boundary conditions. In the following, the existence of a common solution will be considered for two singular fractional equations.  Likewise, some examples are represented to demonstrate our main results. 
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 34A08; 37C25; 46F30

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

\section{Introduction}



Besides the fact that fractional differential calculation had been dated back to the last three centuries, it is of high significance among the recent researches and academians (see, for instance,  \cite{5ab}- \cite{d}), that sometimes are singular at some points (see  \cite{m6}- \cite{ms10}).\\
 In \cite{22}, the author investigated the fractional equation
$\mathcal{D}^{\sigma} \nu(t) + y(t,\nu(t))=0$ with initial conditions $\nu(0)=\nu''(0)=0$ and $\nu(1)= \tau \int_{0}^{1} \nu(s)ds$, where $0<t<1$, $2<\sigma<3$, $0<\tau<2$, $\mathcal{D}^{\sigma}$ is the Caputo fractional derivative and 
$y:[0,1] \times[0,\infty) \to [0,\infty)$ is a continuous functions.\\
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, $y$ may have singularity at $\xi=1$ and $\int_{0}^{1} d \gamma(s)<1$ (\cite{5}). \\  In 2015, 
the fractional problem
$\mathcal{D}^{\rho}y(t)=\psi(t,y(t), \mathcal{D}^{\sigma}y(t))$ with boundary conditions $y(0)+y'(0)=g(x)$, $\int_{0}^{1} y(t) dt=m_0$ and
$y''(0)=y^{(3)}(0)=\dots=y^{(n_{\rho}-1)}(0)=0$ was studied where, $0<t<1$, $m_0$ 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 continuous with $\psi(t,u,v)$ that may be singular at $t=0$ (\cite{a}).\\
In 2018, the existence of a solution for  the following three steps crisis problem was investigated: 
\begin{eqnarray}
\mathcal{D}^{\eta} z(t)+ \psi(t , z(t), z'(t), \mathcal{D}^{\sigma}z(t), \int_{0}^{t} \Omega(\xi) z(\xi) d\xi , \omega(x(t)))=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 non-negative real numbers $\iota_0$ and $\iota_1 \in [0,\infty)$ and all $x_1, x_2 \in C^1[0,1]$,
$\mathcal{D}^{\eta}$is the $\eta$-order Caputo fractional derivative, 
$\psi(t,z_1(t),..., z_5(t))=\psi_1(t,z_1(t),..., z_5(t))$ for all $t\in[0,\lambda)$,
$\psi(t,z_1(t),..., z_5(t))=\psi_2(t,z_1(t),..., z_5(t))$ for all $t\in[\lambda ,\mu]$ and
$\psi(t,z_1(t),..., z_5(t))=\psi_3(t,z_1(t),..., z_5(t))$ for all $t\in(\mu,1]$, $\psi_1(t,.,.,.,.,.)$ and $\psi_3(t,.,.,.,.,.)$ are continuous on $[0,\lambda)$ and $(\mu,1]$ and $\psi_2(t,.,.,.,.,.)$ is multi-singular (\cite{kh}).
\\
In 2019, the existence and uniqueness of solutions were discussed for the following class of boundary value problems of nonlinear fractional differential equations depending with non-separated type integral boundary conditions were discussed
$$ 
\mathcal{D}^{q} z(t)= \Psi(t , z(t),  \mathcal{D}^{r}z(t)) $$
with the  conditions $z(0) - \iota_1 z(\tau) = \kappa_1 \int_0^{\tau} U(s, z(s))ds$ and  \\$z'(0) - \iota_2 z'(\tau) = \kappa_2 \int_0^{\tau} V(s, z(s))ds,$ where $t \in [0,\tau]$, $t>0$, $1 < q \leq 2$, $0 < r \leq 1$, $\mathcal{D}^{q}$ is the $q$-th order of the Caputo fractional derivative, $\Psi \in C([0,\tau] \times \mathbb{R} \times \mathbb{R},  \mathbb{R} ), U,V : [0,\tau] \times \mathbb{R} \to \mathbb{R}$ are given continuous functions and $\iota_1, \iota_2, \kappa_1, \kappa_2 \in \mathbb{R}$ with $\iota_1 \neq 1$ and $\iota_2 \neq 1$ (\cite{nn}).

In 2020, the existence of solutions were examined for the following nonlinear differential pointwise defined system: 
\begin{eqnarray}
\left\{ 
\begin{array}{ll}
\mathcal{D}^{\alpha_1} \nu_1(t) = h_1(t, \nu_1(t),  \nu'_1(t), \mathcal{D}^{\beta_1}\nu_1(t),  I^{p_1}\nu_1(t), \\
..., \nu_m(t),  \nu'_m(t), \mathcal{D}^{\beta_m}\nu_m(t),  I^{p_m}\nu_m(t)),  \\
.\\
.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,~~~~~~~ t \in [0,1] \\
.\\
\mathcal{D}^{\alpha_m} \nu_m(t) = h_m(t, \nu_1(t),  \nu'_1(t), \mathcal{D}^{\beta_1}\nu_1(t),  I^{p_1}\nu_1(t), \\
..., \nu_m(t),  \nu'_m(t), \mathcal{D}^{\beta_m}\nu_m(t),  I^{p_m}\nu_m(t)),
\end{array} \right. \nonumber
\end{eqnarray}
with boundary value conditions $\nu^{(j)}_{k}(0)= 0$ for $2 \leq j \leq n_k -1$ and $k= 1,\dots,m$,  $$\nu_k(\theta_{k})=\sum_{i=1}^{n_0} \lambda_{i,k} \mathcal{D}^{\mu_{i,k} }\nu_k(\gamma_{i,k})$$ and $\nu'_{k}(0)= \nu_{k}(\eta_{k})$ for all $k= 1,2, ...,m$, where $ \lambda_{i,k} \geq 0$, $\beta_k, \gamma_{i,k}, \mu_{i,k}, \theta_{k}, \eta_{k} \in (0,1)$, $p_k >0$, $m, n_0 \in \mathbb{N}$, $k= 1,2, ...,m$, $i= 1,2, ...,n_0$, 
$\mathcal{D}^{\alpha_k}$ is the Caputo fractional derivative of order $\alpha_{k} \geq 2$,  $n _k= [\alpha_k] + 1$, 
and  $h_k :[0,1] \times X ^{4m} \to \mathbb{R}$, is singular at some points $[0,1]$,  where  $X =C^1[0,1]$  (\cite{mshkh}).







Regarding the main ideas of the papers, we  investigate the non-controlled multi-singular fractional differential pointwisly defined equation
\begin{eqnarray}
\label{prb1}
\mathcal{D}^\sigma w(t) +\mathcal{U}(t,w(t), w'(t), \mathcal{D}^{\beta}w(t), \phi(w(t))) = 0
\end{eqnarray}
with boundary conditions 
$w(0)=0$ for $ \sigma \in [2,3)$ and $w(0)=w''(0)=w^{(n_0)}(0)=0$ where $n_{0}=[\sigma]-1$ for $\sigma \in [3,\infty)$ and also $w(\eta)+\int_0^1 w(s)ds=0$ where $\sigma \geq 2$, $\eta, \beta \in (0,1)$, $\phi: X \rightarrow X$ is a mapping such that for all $w_1,w_2 \in X$, $\| \phi(w_1) - \phi(w_2)\| \leq a_0 \|w_1-w_2\| + a_1 \|w_1'- w_2' \|$ for some $a_0, a_1 \in [0,\infty)$,
$\mathcal{D}^{\sigma}$ is the Caputo fractional derivative of order $\sigma$ and
$\mathcal{U}:[0,1] \times \mathbb{R}^{4} \to \mathbb{R}$ is a function such that
$\mathcal{U}(t,.,.,.,.)$ is singular at some points $t\in [0,1]$. In fact,  $\mathcal{U}$ is stated to be multi-sigular when  it is singular at more than one point $t$ (see Example 2.1 and 2.2).
Likewise, $\mathcal{D}^{\alpha}w(t)+\mathcal{U}(t)=0$ 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's obvious that every 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 $s \geq 0$ for a function $ y:(s,\infty)\rightarrow \mathbb{R} $ is defined by
	$\mathcal{I}^{r}_{s^+}y(t)=\frac{1}{\Gamma(r)} \int_{s}^{t} (t-s)^{r-1} y(s)ds$
	provided that the right-hand side s pointwise defined on $(s,\infty)$. we denote $\mathcal{I}^{r}y(t)$ for $\mathcal{I}^{r}_{0^+}y(t)$. Also, The Caputo fractional derivative of order $r>0$ of a function $ y:(0,\infty)\to \mathbb{R}$ is defined by $^{c}\mathcal{D}^{r}y(t)=\frac{1}{\Gamma(n-r)}\displaystyle\int_{0}^{t}\!\frac{y^{n}(s)}{(t-s)^{r+1-n}}ds$, 
where $n=[r]+1$ (\cite{4}). \\ Let $\Psi$ be the family of nondecreasing functions $\psi :[0,\infty) \to [0,\infty)$ such that $\sum_{n=1}^{\infty} \psi^{n}(t)<\infty$ for all $t> 0$ (\cite{9}).
One can check that $\psi(t)<t$ for all $t>0$ (\cite{9}). Let $\mathcal{T}:X \to X$ and $\alpha :X \times X \to [0,\infty)$ be two maps. Then $\mathcal{T}$ is called an $\alpha$-admissible map whenever $\alpha(x,y) \geq 1$ implies $\alpha(\mathcal{T}x,\mathcal{T}y) \geq 1$ (\cite{10}). Let $(X,d)$ be a complete metric space, $\psi \in \Psi$ and $\alpha :X \times X \to [0,\infty)$ a map. A self-map $\mathcal{T}:X \to X$ is called an $\alpha$-$\psi$-contraction whenever $\alpha(x,y) d(\mathcal{T}x,\mathcal{T}y) \leq \psi(d(x,y))$ for all $x,y \in X$ (\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})Let $X$ is a Banach space and $\mathcal{C} \subseteq X$ is closed and convex. Suppose that $\Xi$ be a relatively open subset of $\mathcal{C}$ with $0 \in \Xi$ and let $\mathcal{T}: \Xi \rightarrow \mathcal{C}$ be a continuous and compact mapping. Then either 
\\i) the mapping $\mathcal{T}$ has a fixed point in $\bar{\Xi}$, or
\\ii) there exists $w_0 \in \partial \Xi$ and $\gamma \in (0,1)$ with $w_0= \gamma \mathcal{T}w_0.$ 
\end{lemma}
\begin{lemma}\label{l1.2}(\cite{22a}) Let $(X,d)$ be a complete metric space, $\psi \in \Psi$, $\alpha :X \times X \to [0,\infty)$ is a map and $\mathcal{S},\mathcal{T}:X \to X$ are mappings satisfying the following conditions
\\
i) for $x,y \in X$, $\alpha(x,y) \geq 1$ implies $\alpha(\mathcal{S}x,\mathcal{T}y) \geq 1$ or $\alpha(\mathcal{T}x,\mathcal{S}y) \geq 1$,
\\
ii) there exists  $x_0 \in X$ such that $\alpha(x_0 ,\mathcal{S}x_0) \geq 1$,
\\
iii) $\mathcal{S}$ and $\mathcal{T}$ are continuous
\\
iv) for all $x,y \in X$, $\alpha(x,y) d(\mathcal{S}x,\mathcal{T}y) \leq \psi(d(x, y))$ and $\alpha(y,x) d(\mathcal{S}x,\mathcal{T}y) \leq \psi(d(x, y))$.
\\
Then $\mathcal{T}$ and $\mathcal{S}$ have a common fixed point.
\end{lemma}

\section{Main Results}
\begin{lemma}\label{l2.1} Let $\sigma \geq2$, $ \eta \in (0,1)$ and $\mathcal{U} \in L^{1}[0,1]$. Then $w(t)= \int^1_0 \kappa(t,s) \mathcal{U}(s) ds$ is a solution for the pointwise defined problem $\mathcal{D}^\sigma w(t) +\mathcal{U}(t) = 0$ with boundary value conditions $w(0)=0$ for $ \sigma \in [2,3)$ and $w(0)=w''(0)=w^{(n_0)}(0)=0$ where $n_{0}=[\sigma]-1$ for $\sigma \in [3,\infty)$ and also $w(\eta)+\int_0^1 w(s)ds=0$ for all $ \sigma \in [2, \infty)$, where
$$\kappa(t,s)= \left\{ \begin{array}{ll}
\frac{-(t-s)^{\sigma-1} }{\Gamma(\sigma)} + \frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2t(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )} \hfill~~~~ 0\leq s \leq t \leq 1, \ s\leq \eta
\\\\
\frac{-(t-s)^{\sigma-1} }{\Gamma(\sigma)} + \frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} \hfill~~~~ 0\leq \eta \leq s \leq t \leq 1
\\\\
\frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} \hfill~~~~ 0\leq t \leq s \leq 1, \ \eta \leq s
\\\\
\frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2t(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )} \hfill~~~~ 0\leq t \leq s \leq \eta \leq 1.
\end{array} \right. $$
\end{lemma}
\begin{proof} Let for all $t \in E \subset [0,1]$ the equation $\mathcal{D}^\sigma w(t) +\mathcal{U}(t) = 0$ is held, where $m(E^c)=0$ and $m$ is the Lebesgue measure on $\mathbb{R}$. Also let $\mathcal{U}_0 \in L^1[0,1] \cap C[0,1]$ be a function such that $\mathcal{U}_0=\mathcal{U}$ on $E$. Note that if this problem has a solution then $\mathcal{U}_0$ exists, because if $w_0 \in C[0,1]$ is a solution for the pointwise defined problem, it is enough to consider $\mathcal{U}_0(t)=-\mathcal{D}^{\sigma}w_0(t)$ for all $t\in [0,1]$, so we have $\mathcal{U}_0 \in L^1[0,1] \cap C[0,1]$ and $\mathcal{U}_0=\mathcal{U}|_{E}$. Hence if $t\in E$, we have
\begin{eqnarray}
 & & \mathcal{I}^{\sigma}(\mathcal{U}(t))= \frac{1}{\Gamma(\sigma)}\int_0^t (t-s)^{\sigma-1}\mathcal{U}(s)ds \nonumber \\
& &= \frac{1}{\Gamma(\sigma)}(\int_{[0,t]\cap E} (t-s)^{\sigma-1}\mathcal{U}(s)ds +\int_{[0,t]\cap E^c} (t-s)^{\sigma -1}\mathcal{U}(s)ds) \nonumber \\
& &=\frac{1}{\Gamma(\sigma)}\int_{[0,t]\cap E} (t-s)^{\sigma-1}\mathcal{U}_0(s)ds \nonumber \\
& &= \frac{1}{\Gamma(\sigma)}(\int_{[0,t]\cap E} (t-s)^{\sigma-1}\mathcal{U}_0(s)ds +\int_{[0,t]\cap E^c} (t-s)^{\sigma-1}\mathcal{U}_0(s)ds) \nonumber \\
& &=\frac{1}{\Gamma(\sigma)}\int_0^t (t-s)^{\sigma-1}\mathcal{U}_0(s)ds= \mathcal{I}^{\sigma}(\mathcal{U}_0(t)). \nonumber
\end{eqnarray}
If $t\in E^c | \{0\}$, then there exists $\{ t_n\} \subset E$ such that $t_n \to t ^-$ as $n\to \infty$, so
\begin{eqnarray}
& &\mathcal{I}^{\sigma}(\mathcal{U}(t)) = \frac{1}{\Gamma(\sigma)}\int_0^t (t-s)^{\sigma-1}\mathcal{U}(s)ds \nonumber \\
& &= \lim_{n\to \infty} \frac{1}{\Gamma(\sigma)} \int_0^{t_n} (t_n -s)^{\sigma-1}\mathcal{U}(s)ds = \lim_{n\to \infty} \mathcal{I}^{\sigma}(\mathcal{U}(t_n)) \nonumber \\
& &= \lim_{n\to \infty} \mathcal{I}^{\sigma}(\mathcal{U}_0(t_n)) =\lim_{n\to \infty} \frac{1}{\Gamma(\sigma)} \int_0^{t_n} (t_n-s)^{\sigma-1}\mathcal{U}_0(s)ds  \nonumber \\
& &= \frac{1}{\Gamma(\sigma)}\int_0^t (t-s)^{\sigma-1}\mathcal{U}_0(s)ds =\mathcal{I}^{\sigma}(\mathcal{U}_0(t))\nonumber
\end{eqnarray}
and in the case $t=0 \in E^c $, we have $\mathcal{I}^{\sigma}(\mathcal{U}(t))=\mathcal{I}^{\sigma}(\mathcal{U}_0(t))=0$. So for all $t\in [0,1]$, $\mathcal{I}^{\sigma}(\mathcal{U}(t))=\mathcal{I}^{\sigma}(\mathcal{U}_0(t))$. Therefore  if $\mathcal{D}^\sigma w(t) +\mathcal{U}(t) = 0$ for all $t \in E$, then $\mathcal{I}^{\sigma}(\mathcal{D}^\sigma w(t))= \mathcal{I}^{\sigma}(-\mathcal{U}(t))$ for all $t\in [0,1]$, consequently $\mathcal{I}^{\sigma}(D^\sigma w(t))= \mathcal{I}^{\sigma}(-\mathcal{U}_0(t))$ on $ [0,1]$. 

Thus, regarding Lemma (\ref{l1.22}) and the boundary conditions, we obtain
$$ w(t)= - \frac{1}{\Gamma (\sigma)} \int^t_0 (t-s)^{\sigma - 1} \mathcal{U}(s) ds + \iota_{1} t.$$
Putting $t=\eta$, we have
$$ w(\eta)= - \frac{1}{\Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds + \iota_{1} \eta.$$
On the other hand,
\begin{eqnarray}
\int_0^1 w(s)ds&=&\int_0^1 w(t)dt= - \frac{1}{\Gamma (\sigma)} \int^1_0 \int^t_0 (t-s)^{\sigma - 1} \mathcal{U}(s) ds dt + \frac{\iota_{1}}{2} \nonumber\\
&=& - \frac{1}{\Gamma (\sigma)} \int^1_0 \int^1_s (t-s)^{\sigma - 1} dt \mathcal{U}(s) ds + \frac{\iota_{1}}{2} \nonumber\\
&=& - \frac{1}{\Gamma (\sigma)} \int^1_0 (\frac{1}{\sigma}(t-s)^{\sigma}|_s^1)\mathcal{U}(s)ds + \frac{\iota_{1}}{2} \nonumber\\
&=& - \frac{1}{\Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s)ds + \frac{\iota_{1}}{2}. \nonumber
\end{eqnarray}
By hypothesis $w(\eta)=-\int^1_0w(s) ds$, so we have
$$- \frac{1}{\Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds +\iota_1 \eta = \frac{1}{\Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s)ds - \frac{\iota_{1}}{2},$$
hence,
$$\iota_1(\eta+\frac{1}{2})=\frac{1}{\Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s)ds + \frac{1}{\Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds. $$
Therefore,
$$\iota_1=\frac{2}{(2\eta+1)}(\frac{1}{\Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s)ds + \frac{1}{\Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds).$$
So we obtain the following equations
\begin{eqnarray}
& &w(t) =- \frac{1}{\Gamma (\sigma)} \int^{t}_0 (t-s)^{\sigma - 1} \mathcal{U}(s) ds \nonumber\\
& &+ \frac{2t}{2\eta+1}(\frac{1}{\Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s) ds + \frac{1}{\Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds)\nonumber\\
& &= - \frac{1}{\Gamma (\sigma)} \int^{t}_0 (t-s)^{\sigma - 1} \mathcal{U}(s) ds + \frac{2t}{(2\eta+1) \Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s) ds \nonumber\\
& &+ \frac{2t}{(2\eta+1) \Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds. \nonumber
\end{eqnarray}
If $\eta \geq t$, then 
\begin{eqnarray}
w(t)= &-& \frac{1}{\Gamma (\sigma)} \int^{t}_0 (t-s)^{\sigma - 1} \mathcal{U}(s) ds \nonumber\\
&+& \frac{2t}{(2\eta+1) \Gamma (\sigma+1)} (\int^t_0 + \int^{\eta}_t + \int^1_{\eta}) (1-s)^{\sigma}\mathcal{U}(s) ds \nonumber\\
&+& \frac{2t}{(2\eta+1) \Gamma (\sigma)} (\int^t_0 +\int^{\eta}_t) (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds. \nonumber
\end{eqnarray}

If $\eta \leq t$ then 
\begin{eqnarray}
w(t)= &-& \frac{1}{\Gamma (\sigma)} (\int^{\eta}_0 + \int_{\eta}^t) (t-s)^{\sigma - 1} \mathcal{U}(s) ds \nonumber\\
&+& \frac{2t}{(2\eta+1) \Gamma (\sigma+1)} (\int^{\eta}_0 + \int_{\eta}^t + \int^1_{t}) (1-s)^{\sigma}\mathcal{U}(s) ds \nonumber\\
&+& \frac{2t}{(2\eta+1) \Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s) ds. \nonumber
\end{eqnarray}
So  $w(t)= \int_0^1 \kappa(t,s) \mathcal{U}(s) ds$ can be written, where 

$$\kappa(t,s)= \left\{ \begin{array}{ll}
\frac{-(t-s)^{\sigma-1} }{\Gamma(\sigma)} + \frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2t(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )} \hfill~~~~ 0\leq s \leq t \leq 1, \ s\leq \eta
\\\\
\frac{-(t-s)^{\sigma-1} }{\Gamma(\sigma)} + \frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} \hfill~~~~ 0\leq \eta \leq s \leq t \leq 1
\\\\
\frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} \hfill~~~~ 0\leq t \leq s \leq 1, \ \eta \leq s
\\\\
\frac{2t(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2t(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )} \hfill~~~~ 0\leq t \leq s \leq \eta \leq 1.
\end{array} \right. $$
\end{proof}
\begin{lemma}
Let $\kappa(t,s)$ be given in Lemma (\ref{l2.1}). Then for all $t,s\in [0,1]$, $\kappa(t,s)$ has the following properties
\\
i) $|\kappa(t,s)|\leq A_{\sigma, \eta} t(1-t)^{\sigma-1},$
\\
ii) $|\frac{\partial \kappa(t,s)}{\partial t}| \leq A_{\sigma, \eta} (1-t)^{\alpha-1}$, \\
where $A_{\sigma, \eta} = \frac{2(1+\sigma)}{(2\eta+1) \Gamma(\sigma +1 )}$.
\end{lemma}
\begin{proof}
i) For all $t,s\in [0,1]$ we have
\begin{eqnarray}
|\kappa(t,s)| &\leq & \frac{2t(1- s)^{\sigma}}{(2\eta+1) \Gamma(\sigma+1)} + \frac{2t(\eta- s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma)} \nonumber\\
&=& \frac{2t(1- s)^{\sigma} +2t \sigma (\eta-s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma+1)}
\leq \frac{2t(1- s)^{\sigma} +2t \sigma (1-s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma+1)} \nonumber\\
&=& \frac{2t(1- s)^{\sigma -1} (1-s+\sigma)}{(2\eta+1) \Gamma(\sigma+1)}
\leq \frac{2t(1- t)^{\sigma-1} (1+\sigma)}{(2\eta+1) \Gamma(\sigma+1)} \nonumber\\
&= & A_{\sigma, \eta} t(1-t)^{\sigma-1}.\nonumber
\end{eqnarray}
\\
ii) By derivation from the $\kappa(t,s)$ with respect to $t$, it is deduced that
$$\frac{\partial \kappa}{\partial t}(t,s)=\frac{-(\sigma-1)(t-s)^{\sigma-2} }{\Gamma(\sigma)} + \frac{2(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )}$$
for $0\leq s < t < 1$ and $s \leq \eta$,
$$\frac{\partial \kappa}{\partial t}(t,s)=\frac{-(\sigma-1)(t-s)^{\sigma-2} }{\Gamma(\sigma)} + \frac{2(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)}$$
for $0\leq \eta \leq s < t < 1$,
$$\frac{\partial \kappa}{\partial t}(t,s)=\frac{-(\sigma-1)(t-s)^{\sigma-2} }{\Gamma(\sigma)} + \frac{2(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)}$$
for $0\leq \eta \leq s < t < 1$,
$$\frac{\partial \kappa}{\partial t}(t,s)=\frac{2(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)}$$
for $0< t < s \leq 1$ and $ \eta \leq s$,
and finally 
$$\frac{\partial \kappa}{\partial t}(t,s)=\frac{2(1-s)^{\sigma} }{(2\eta+1)\Gamma(\sigma +1)} + \frac{2(\eta-s)^{\sigma-1} }{(2\eta+1)\Gamma(\sigma )}$$
for $0< t < s \leq \eta \leq 1$,
hence
\begin{eqnarray}
|\frac{\partial \kappa(t,s)}{\partial t}| &\leq & \frac{2(1- s)^{\sigma}}{(2\eta+1) \Gamma(\sigma+1)} + \frac{2(\eta- s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma)} \nonumber\\
&=& \frac{2(1- s)^{\sigma} +2 \sigma (\eta-s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma+1)}
\leq \frac{2(1- s)^{\sigma} +2\sigma (1-s)^{\sigma-1}}{(2\eta+1) \Gamma(\sigma+1)} \nonumber\\
&=& \frac{2(1- s)^{\sigma -1} (1-s+\sigma)}{(2\eta+1) \Gamma(\sigma+1)}
\leq \frac{2(1- t)^{\sigma-1} (1+\sigma)}{(2\eta+1) \Gamma(\sigma+1)} \nonumber\\
&= & A_{\sigma, \eta} (1-t)^{\sigma-1},\nonumber
\end{eqnarray}
for all $t,s \in [0,1]$ that $t \neq s, t\neq 0$ and $t\neq1$. In the case $t= s, t=0$ or $t=1$, the same result is obtained.
\end{proof}
\\Now, let $\mathcal{F}:X \rightarrow X$ be defined as
\begin{eqnarray}
& & \mathcal{F}w(t) = \int_0^1 \kappa(t,s) \mathcal{U}(s,w(s), w'(s), D^{\beta}w(s), \phi(w(s))) ds \nonumber\\
& &=- \frac{1}{\Gamma (\sigma)} \int^{t}_0 (t-s)^{\sigma - 1} \mathcal{U}(s,w(s), w'(s), D^{\beta}w(s), \phi(w(s))) ds \nonumber\\
& &+  \frac{2t}{(2\eta+1) \Gamma (\sigma+1)} \int^1_0 (1-s)^{\sigma}\mathcal{U}(s,w(s), w'(s), D^{\beta}w(s), \phi(w(s))) ds \nonumber\\
& & + \frac{2t}{(2\eta+1) \Gamma (\sigma)} \int^{\eta}_0 (\eta-s)^{\sigma - 1} \mathcal{U}(s,w(s), w'(s), D^{\beta}w(s), \phi(w(s))) ds, \nonumber
\end{eqnarray}
where $0<\beta<1$ and $\phi: X \rightarrow X$ is a mapping such that 
$$\| \phi(w_1)- \phi(w_2) \| \leq a_0 \| w_1- w_2 \| + a_1 \| w_1'- w_2' \|,$$
for all $w_1,w_2 \in X$ and some $a_0, a_1 \in [0, \infty)$. By taking $l_0=a_0 + a_1$, it can be seen  that $\| \phi(w_1)- \phi(w_2) \| \leq l_0 \| w_1- w_2 \|_{*},$ for all $w_1,w_2 \in X$. According to the defintion of Caputo derivative, for all $t \in [0,1]$ and $w_1,w_2 \in X$ it follows
$$| \mathcal{D}^{\beta}w_1(t) - \mathcal{D}^{\beta}w_2(t) | \leq \frac{1}{\Gamma(1-\beta)} \int_0^t (t-s)^{- \beta} |w_1'(s) - w_2'(s)| ds \leq \frac{\|w_1' - w_2'\|}{\Gamma(2-\beta)} t^{1-\beta},$$
so
$$\| \mathcal{D}^{\beta}w_1 - \mathcal{D}^{\beta}w_2 \| \leq \frac{\|w_1' - w_1'\|}{\Gamma(2-\beta)} \leq \frac{\|w_1 - w_2|_{*}}{\Gamma(2-\beta)}.$$
Now, we consider $\mathcal{F} : X \rightarrow X$, to prove that the pointwise problem (\ref{prb1}) has a solution in $X$. For this,  by lemma (\ref {l2.1}), we indicate that $\mathcal{F}$ 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, investigate the existence of a sloution  for the singular fractional differential problem. 
\begin{theorem}\label{t2.3}
Let $\mathcal{U}:[0,1]\times(C[0,1])^{4}\to \mathbb{R}$ be a singular function   at some points $t \in [0,1]$ such that $\mathcal{U}(t , \mathcal{O}, \mathcal{O}, \mathcal{O}, \mathcal{O}) \in L^{1}[0,1]$ where $\mathcal{O}$ is the zero function on $[0,1]$, i.e for all $s \in [0,1]$, $\mathcal{O}(s)=0$. Assume that there exists a nondecreaing mapping $\Lambda : X^{4}\to \mathbb{R}^{^+}:= [0,\infty)$  such that 
$ \frac{\Lambda(z,z,z,z)}{z} \to q_0< \infty$ as $z \to 0^{+}$ and $ \frac{\Lambda(z,z,z,z)}{z} \to 0$ as $z \to \infty$. If  the  inequality
\begin{eqnarray}
& &|\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{U}(t, z_1, z_2, z_3, z_4)| \nonumber\\ & & \leq b(t) \Lambda(w_1 - z_1, w_2 - z_2, w_3 - z_3, w_4 - z_4), \nonumber
\end{eqnarray}
be established 
for almost all $t \in [0,1]$, all $(w_1, w_2, w_3, w_4), (z_1, z_2, z_3, z_4) \in X^{4}$ and some $b \in L^{1}[0,1]$,
then  the poinwise defined problem (\ref{prb1})
has a solution.
\end{theorem}
\begin{proof}
Let $\epsilon$ be arbitary. Regardig to the properties $\lim_{z \to 0^{+}} \frac{\Lambda(z,z,z,z)}{z}=q_0< \infty$,  there exists  $0<\delta(\epsilon) \leq \epsilon$ such that for all $z \in (0, \delta(\epsilon)]$, 
$ \frac{\Lambda(z,z,z,z)}{z} <q_0 +\epsilon$, and so
$\Lambda(z,z,z,z)<(q_0 +\epsilon)z$. Hence taking  $z= \delta(\epsilon):= \delta$, we have
\begin{eqnarray}
\Lambda(\delta,\delta,\delta,\delta)<(q_0 +\epsilon)\delta< (q_0 +\epsilon)\epsilon.
\end{eqnarray}
Now, let $\{w_n\}_{n \geq 1}$ be a sequence such that $w_n \to w$ in $X$ as $n \to \infty$. So $\|w_n - w\|_{*} \to 0$ as $n \to \infty$. Therefore,  there exists $m \in \mathbb{N}$ such that $n \geq m$ implies 
$$\|w_n - w\|_{*} = max \{ \|w_n - w\|, \|w'_n - w' \|\}< \frac{\delta}{l_1},$$
where $l_1:= max \{1, \frac{1}{\Gamma(2-\beta)}, a_0+a_1\}$. So it is concluded that $\|w_n - w\|< \frac{\delta}{l_1}$ and  $\|w'_n - w'\|< \frac{\delta}{l_1},$ for all $n \geq m$. Hence for  all $t \in [0,1]$ and  $n \geq m$, we have
\begin{eqnarray}
& & |\mathcal{F}w_n(t)-\mathcal{F}w(t)| \nonumber\\
&\leq & \int_0^1 |\kappa(t,s)| \bigg| \mathcal{U}(s, w_n(s), w'_n(s), \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) \nonumber\\
& & - \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s)))\bigg| ds \nonumber\\
&\leq & \int_0^1 A_{\sigma, \eta} t(1-t)^{\sigma-1} \bigg|\mathcal{U}(s, w_n(s), w'_n(s), \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) \nonumber\\
& & - \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) \bigg| ds \nonumber\\
&\leq & \int_0^1 A_{\sigma, \eta} t(1-t)^{\sigma-1} b(s) \Lambda((w_n - x)(s), (w'_n - w')(s),  \nonumber\\
& & (\mathcal{D}^{\beta} w_n - \mathcal{D}^{\beta}w)(s), \phi (w_n(s)) - \phi(w(s))) ds \nonumber\\
&\leq & A_{\sigma, \eta} t(1-t)^{\sigma-1} \int_0^1 b(s) \Lambda(\|w_n - w\|, \|w'_n - w'\|, \frac{\|w'_n - w'\|}{\Gamma(2-\beta)}, \nonumber\\
& & a_0\|w_n - w\| + a_1 \|w'_n - w'\|) ds \nonumber\\
&\leq & A_{\sigma, \eta} t(1-t)^{\sigma-1} \Lambda(\frac{\delta}{l_1}, \frac{\delta}{l_1}, \frac{\delta}{l_1 \Gamma(2-\beta)}, (a_0 + a_1) \frac{\delta}{l_1}) \int_0^1 b(s) ds \nonumber\\
&\leq & m_1 A_{\sigma, \eta} t(1-t)^{\sigma-1} \Lambda(l_1 \frac{\delta}{l_1}, l_1 \frac{\delta}{l_1}, l_1 \frac{\delta}{l_1 }, l_1 \frac{\delta}{l_1}) \nonumber\\
&= & m_1 A_{\sigma, \eta} t(1-t)^{\sigma-1} \Lambda(\delta, \delta, \delta, \delta) 
\leq m_1 A_{\sigma, \eta} t(1-t)^{\sigma-1} (q_0+\epsilon)\epsilon, \nonumber
\end{eqnarray}
where $m_1=\int_0^1 b(s)ds$. So $\|\mathcal{F}w_n-\mathcal{F}_w\| \leq m_1 A_{\sigma, \eta} (q_0+\epsilon)\epsilon$, for all $n \geq m$. In a similar mannner for all $t \in [0,1]$ and  $n \geq m$, it is resulted that
\begin{eqnarray}
& & |\mathcal{F}'w_n(t)-\mathcal{F}'w(t)| \nonumber\\
&\leq & \int_0^1 |\frac{\partial \kappa(t,s)}{\partial t} | \bigg|\mathcal{U}(s, w_n(s), w'_n(s),  \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) \nonumber\\
& &- \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) \bigg| ds \nonumber\\
&\leq & m_1 A_{\sigma, \eta} (1-t)^{\sigma-1} (q_0+\epsilon)\epsilon. \nonumber
\end{eqnarray}
Hence $\|\mathcal{F}'w_n-\mathcal{F}'w\| \leq m_1 A_{\sigma, \eta} (q_0+\epsilon)\epsilon$, for all $n \geq m$. Using the above inequalities as well as $*-$norm definition, we conclude that  
$$\|\mathcal{F}w_n-\mathcal{F}w\|_{*}= max \{\|\mathcal{F}w_n- \mathcal{F}w\|, \|F'w_n-\mathcal{F}'w\| \} \leq m_1 A_{\sigma, \eta} (q_0+\epsilon)\epsilon$$ for all $n \geq m$, and since $\epsilon>0$ is arbitary, it is deduced that 
$\mathcal{F}w_n \to \mathcal{F}w$ in $X$ as $w_n \to w$ in $X$, so $\mathcal{F}$ is a continuous mapping on $X$. Now, put $m_2= \int_0^1 |\mathcal{U}(s,\mathcal{O},\mathcal{O},\mathcal{O},\mathcal{O})|ds$. Since $\lim_{z \to \infty} \frac{\Lambda(z,z,z,z)}{z}=0$, therefore
$$\lim_{z \to \infty} \frac{m_2 +m_1 \Lambda(z,z,z,z)}{z}=0.$$
So for $\epsilon>0$, there exists $r(\epsilon)>0$ such that $z\geq r(\epsilon)$ implies that 
$$ \frac{m_2 +m_1 \Lambda(z,z,z,z)}{z}< \epsilon.$$
Thus, for all $z\geq r(\epsilon)$, we have $m_2 +m_1 \Lambda(z,z,z,z)< \epsilon z$. Choose an $\epsilon_0>0$  such that $0<\epsilon_0<\frac{1}{A_{\sigma, \eta} l_1}$ and let $r_0:=r(\epsilon_0)$, then, for all $z\geq r_0$ the following inequality is held: $$m_2 +m_1 \Lambda(z,z,z,z)< \epsilon_0 z,$$
By putting  $z= r_0 l_1$, in the above inequality, we have
$$m_2 +m_1 \Lambda(r_0 l_1, r_0 l_1, r_0 l_1, r_0 l_1)< \epsilon_0 r_0 l_1< \frac{r_0}{A_{\sigma, \eta}}.$$
Now, let $\Xi= \{ w \in X : \|w\|_{*}<r_0\}$,  $\lambda \in (0,1)$ and $w_0 \in \partial \Xi$ be such that $w_0 = \lambda \mathcal{F}w_0$, then for all $t \in [0,1]$, we have
\begin{eqnarray}
& & |w_0(t)| = |\lambda \mathcal{F}w_0(t)|
\leq \int_0^1 |\kappa(t,s)| \bigg|\mathcal{U}(s, w_0(s), w'_0(s), \mathcal{D}^{\beta}w_0(s), \phi(w_0(s))) \bigg| ds \nonumber\\
& &\leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \bigg( \int_0^1 \bigg|\mathcal{U}(s, w_0(s), w'_0(s), \mathcal{D}^{\beta}w_0(s), \phi(w_0(s)))  \nonumber\\
& &- \mathcal{U}(s, \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s)) \bigg| ds + \int_0^1 | \mathcal{U}(s, \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s))| ds \bigg) \nonumber\\
& &\leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \bigg( \int_0^1 b(s) \Lambda(x_0 (s), w'_0(s), \mathcal{D}^{\beta} w_0(s), \phi (w_0(s))) ds + m_2\bigg) \nonumber\\
& & \leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \bigg( \Lambda(\|w_0\|, \|w'_0\|, \|\mathcal{D}^{\beta} w_0\|, \|\phi (w_0(s))\|) \int_0^1 b(s) ds + m_2 \bigg) \nonumber\\
& &\leq  A_{\sigma, \eta} t(1-t)^{\sigma-1} \bigg( \Lambda( l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}) m_1 + m_2 \bigg), \nonumber
\end{eqnarray}
conseqently 
\begin{eqnarray}
\|w_0\| = \lambda \| \mathcal{F}w_0 \| & \leq & A_{\sigma, \eta}  \bigg( \Lambda( l_1 r_0, l_1 r_0, l_1 r_0, l_1 r_0) m_1 + m_2 \bigg) \nonumber\\
&<& A_{\sigma, \eta} \frac{r_0}{A_{\sigma, \eta}} = r_0. \nonumber
\end{eqnarray}
Likewise, for all $t \in [0,1]$, it is infered that 
\begin{eqnarray}
& &|w'_0(t)| = |\lambda \mathcal{F}'w_0(t)|\nonumber\\
& &\leq \int_0^1 | \frac{\partial \kappa(t,s)}{\partial t}| \bigg| \mathcal{U}(s, w_0(s), w'_0(s), \mathcal{D}^{\beta}w_0(s), \phi(w_0(s))) \bigg| ds \nonumber\\
& & \leq A_{\sigma, \eta} (1-t)^{\sigma-1} \bigg( \int_0^1 \bigg|\mathcal{U}(s, w_0(s), w'_0(s), \mathcal{D}^{\beta}w_0(s), \phi(w_0(s))) \nonumber\\
& &- \mathcal{U}(s, \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s)) \bigg| ds + \int_0^1 | \mathcal{U}(s, \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s), \mathcal{O}(s))| ds \bigg) \nonumber\\
& &\leq  A_{\sigma, \eta} (1-t)^{\sigma-1}  \bigg( \int_0^1 b(s) \Lambda(w_0 (s), w'_0(s), \mathcal{D}^{\beta} w_0(s), \phi (w_0(s))) ds + m_2 \bigg) \nonumber\\
&\leq & A_{\sigma, \eta} (1-t)^{\alpha-1}  \bigg( \Lambda(\|w_0\|, \|w'_0\|, \|\mathcal{D}^{\beta} w_0\|, \|\phi (w_0(s))\|) \int_0^1 b(s) ds + m_2 \bigg) \nonumber\\
&\leq & A_{\sigma, \eta} (1-t)^{\sigma-1}  \bigg( \Lambda( l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}, l_1 \|w_0\|_{*}) m_1 + m_2 \bigg), \nonumber
\end{eqnarray}
so 
\begin{eqnarray}
\|w'_0\| = \lambda \| \mathcal{F}'w_0 \| & \leq & A_{\sigma, \eta}  \bigg( \Lambda( l_1 r_0, l_1 r_0, l_1 r_0, l_1 r_0) m_1 + m_2 \bigg) \nonumber\\
&<& A_{\sigma, \eta} \frac{r_0}{A_{\sigma, \eta}} = r_0. \nonumber
\end{eqnarray}
Hence, $r_0 = \|w_0\|_{*} = \max \{ \|w_0\|, \|w'_0\| \}<r_0$ which is a contradiction. Therefore, regarding to  theorem (\ref{l1.1}), $\mathcal{F}:X \to X$ has a fixed point in $X$, so  the pointwise defined fractional differential equation (\ref{prb1}) has a solution.
\end{proof}
\\The final result is illustrated by the folllowing example. 
\begin{example}
Let $\sigma_1, ..., \sigma_n \in (0,1)$ such that $\Sigma_{i=1}^{n}\sigma_i<1$, $\delta_1, ..., \delta_n \in [0,1]$, $$d(t)=\frac{1}{(t-\delta_1)^{\sigma_1} (t-\delta_2)^{\sigma_2} ...(t-\delta_n)^{\sigma_n}},$$
$$ c(t) = \left\{ \begin{array}{ll}
0 \ \ \ \ \ \ \ \ \ t \in [0,1] \cap Q
\\
\\
1 \ \ \ \ \ \ \ \ \ t \in (0,1) \cap Q^{c}. 
\ \ \ \end{array}\right.$$
$b(t)=\frac{1}{c(t)}$ and
$$\mathcal{U}(t,w_1, w_2, w_3, w_4)= b(t)(\Sigma_{i=1}^{4} \frac{|w_i|}{1+|w_i|}) +d(t).$$
Consider the pointwise defined equation
\begin{eqnarray}
\label{ex1} \mathcal{D}^{\sqrt{11}}w(t)+\mathcal{U}(t, w(t), w'(t), \mathcal{D}^{\frac{2}{3}}w(t), \int_0^t w(s) ds)=0
\end{eqnarray}
with boundary condition 
$w(0)=w''(0)=0$ and $w(\eta)+\int_0^1 w(s)ds=0$, in which $ \eta \in (0,1)$ is fixed. Then,   for all $(w_1, w_2, w_3, w_4), (z_1, z_2, z_3, z_4) \in X^{4}$ and almost $t \in [0,1]$ we have 
\begin{eqnarray}
& & \bigg|\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{U}(t, z_1, z_2, z_3, z_4) \bigg| \nonumber\\
& &= b(t) \bigg|\Sigma_{i=1}^{4} (\frac{|w_i|}{1+|w_i|}- \frac{|z_i|}{1+|z_i|}) \bigg|  \leq b(t) \Sigma_{i=1}^{4} \frac{|w_i - z_i|}{1+|w_i - z_i|} \nonumber\\
& & = b(t) \Lambda(w_1-z_1, w_2-z_2, w_3-z_3, w_4-z_4), \nonumber
\end{eqnarray}
where
$$\Lambda(z_1, z_2, z_3, z_4)= \Sigma_{i=1}^{4} \frac{|z_i|}{1+|z_i|}.$$
Simply speaking,
$\lim_{z \to 0^{+}} \frac{\Lambda(z,z,z,z)}{z}=4< \infty$, $\lim_{z \to \infty} \frac{\Lambda(z,z,z,z)}{z}=0$  and $b(t) \in L^1[0,1]$. Note that if $\phi(w(t))= \int^t_0 w(s) ds$, then
$$|\phi(w(t))- \phi(z(t))| \leq \int^t_0 |w(s) - z(s)|ds \leq \|w-z\| t,$$
for all $t \in [0,1]$, so $\|\phi(w)- \phi(z)\| \leq \|w-z\| $. Therefore all the conditions of Theorem  (\ref{t2.3})   are held, so by therem (\ref{t2.3}), the pointwisedefined equation (\ref{ex1}) has a solution.
\end{example}

Now, we want to consider two pointwise defined differential equaions 
\begin{eqnarray}
\label{e1} \mathcal{D}^\sigma w(t) +\mathcal{U}(t, w(t), w'(t), \mathcal{D}^{\beta}w(t), \phi(w(t))) = 0
\end{eqnarray}
and 
\begin{eqnarray}
\label{e2} \mathcal{D}^\sigma z(t) +\mathcal{V}(t, z(t), z'(t), \mathcal{D}^{\gamma}z(t), \phi(z(t))) = 0,
\end{eqnarray}
when $\sigma \geq 2$, $\gamma, \beta \in (0,1)$, $\phi : X \to X$ is a mappings such that for all $w_1, w_2 \in X$, 
$\|\phi(w_1)- \phi(w_2)\| \leq a_0\|w_1-w_2\| + a_1\|w_1'-w_2'\|,$ 
for some $a_0, a_1, \in [0,\infty)$ and $\mathcal{U}, \mathcal{V} :[0,1] \times X^{4} \to \mathbb{R}$ are two fuctions that are singular at some set with measure zero, under boundary conditions 
$w(0)=z(0)=0$ for $ \sigma \in [2,3)$ and
$$w(0)=w''(0)=w^{(n_0)}(0)= z(0)=z''(0)=z^{(n_0)}(0)=0$$
where $n_{0}=[\sigma] +1$
for $\sigma \in [3,\infty)$ and also $w(\eta)+\int_0^1 w(s)ds=z(\eta)+\int_0^1 z(s)ds=0.$
We will show that under some conditions, these two equations have the same solution. 

For this, we define $\mathcal{F}, \mathcal{S}: X \to X$ as
$$\mathcal{F}w(t)= \int_0^1 \kappa(t,s) \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta} w(s), \phi( w(s))) ds$$
and 
$$\mathcal{S}z(t)= \int_0^1 \kappa(t,s) \mathcal{V}(s, z(s), z'(s), \mathcal{D}^{\gamma} z(s), \phi( z(s))) ds$$
where $\kappa(t,s)$ is the Green function that defined by lemma (\ref{l2.1}). We will prove that $\mathcal{F}$ and $\mathcal{S}$ has a common fixed point, so two equations (\ref{e1}) and (\ref{e2}) have a same solution.
\begin{theorem}\label{t3.3}
Let $\mathcal{U}, \mathcal{V}: [0,1] \times X^{4} \to \mathbb{R}$ are continuous on  $E \subset X$ with $m(E^{c})=0$ and there exist $b, \theta \in L^{1}[0,1]$, nondecreasing mapping \\$\Lambda: X^{4} \to \mathbb{R}$ such that 
$$\lim_{\|z_i\| \to 0} \frac{|\mathcal{V}(t, z_1, z_2, z_3, z_4)|}{\|z_i\|} \leq \theta(t)$$
 and $|\mathcal{U}(t, w_1, w_2, w_3, xw_4)| \leq b(t) \Lambda (w_1, w_2, w_3, w_4)$ for all $(w_1, w_2, w_3, w_4) \in X^{4}$, $1 \leq i \leq 4$ and almost all $t \in [0,1]$. Also let
$$\lim_{z \to 0^{+}} \frac{\Lambda (z, z, z, z)}{z}= q_0,$$
$m_1:= \int_0^1 b(s) ds < \frac{1}{A_{\sigma, \eta}}$ and $m_2:= \int_0^1 \theta(s) ds < \frac{1}{l_2 A_{\sigma, \eta}}$, where \\ $l_1= \max \{1, \frac{1}{\Gamma(2-\beta)}, a_0+a_1\}$, $l_2= \max \{1, \frac{1}{\Gamma(2-\gamma)}, a_0+a_1\}$ and $q_0 \in [0, \frac{1}{l_1})$. If for all 
$(w_1, w_2, w_3, w_4), (z_1, z_2, z_3, z_4) \in X^{4}$ that \\ $(w_1, w_2, w_3, w_4) \neq (z_1, z_2, z_3, z_4)$, almost all $t \in [0,1]$ and all $1 \leq i \leq 4$
$$\lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{V}(t, z_1, z_2, z_3, z_4)}{max \| w_i - z_i \|} = 0, $$
then the pointwise defined equations (\ref{e1}) and (\ref{e2}) have a common solution.
\end{theorem}
\begin{proof}
Since 
$$\lim_{z \to 0^{+}} \frac{\Lambda (z, z, z, z)}{z}= q_0,$$
so for each $\epsilon>0$, there exists $0<\delta(\epsilon) \leq \epsilon$ such that $z \in (0, \delta(\epsilon)]$ implies that
$$\frac{\Lambda (z, z, z, z)}{z}< q_0+ \epsilon,$$
therefore
$$\Lambda (z, z, z, z)< (q_0+ \epsilon) z.$$
Let $\epsilon_1>0$ be such that $q_0+ \epsilon_1<\frac{1}{l_1}$, then for all $z \in (0, \delta_1:= \delta(\epsilon_1)]$ it is concluded that 
$$\Lambda (z, z, z, z)< (q_0+ \epsilon_1) z,$$
consequently 
$$\Lambda (l_1 z, l_1 z, l_1 z, l_1 z)< (q_0+ \epsilon_1) l_1 z <z,$$
for all $z \in (0, \frac{\delta_1}{l_1}]$.
On the other hand for all $x \in X$ and $t \in [0,1]$, we have
\begin{eqnarray}
& &|\mathcal{F}w(t)| \leq  \int_0^1 |\kappa(t,s)| \bigg|\mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) \bigg| ds \nonumber\\
& &\leq  \int_0^1 A_{\sigma, \eta} t(1-t)^{\sigma-1} b(s) \Lambda ( w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) ds \nonumber\\
& & \leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \int_0^1 b(s) \Lambda ( \|w\|, \|w'\|, \|\mathcal{D}^{\beta}w\|, \|\phi(w)\|) ds \nonumber\\
& &\leq  A_{\sigma, \eta} t(1-t)^{\sigma-1} \Lambda ( \|w\|, \|w'\|, \frac{\|w'\|}{\Gamma(2-\beta)}, a_0\|w\|+ a_1\|w'\|) \int_0^1 b(s) ds \nonumber\\
& & \leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \Lambda ( l_1 \|w\|_{*}, l_1 \|w\|_{*}, l_1 \|w\|_{*}, l_1 \|w\|_{*}) m_1. \nonumber
\end{eqnarray}
So, if $\|w\|_{*} \in (0, \frac{\delta_1}{l_1}]$, then
$$|\mathcal{F}w(t)| \leq A_{\sigma, \eta} t(1-t)^{\sigma-1} \|w\|_{*} m_1 \leq \|w\|_{*} t(1-t)^{\sigma-1}$$
thus, it is resulted that $\|\mathcal{F}w\| \leq \|w\|_{*}$. Also we have 
\begin{eqnarray}
 & & |\mathcal{F}'w(t)|\leq  \int_0^1 |\frac{\partial \kappa(t,s)}{\partial t} | |\mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s)))| ds \nonumber\\
& &\leq  \int_0^1 A_{\sigma, \eta} (1-t)^{\sigma-1} b(s) \Lambda ( w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) ds \nonumber\\
& &\leq  A_{\sigma, \eta} (1-t)^{\sigma-1} \int_0^1 b(s) \Lambda ( \|w\|, \|w'\|, \|\mathcal{D}^{\beta}w\|, \|\phi(w)\|) ds \nonumber\\
& & \leq A_{\sigma, \eta} (1-t)^{\sigma-1} \Lambda ( \|w\|, \|w'\|, \frac{\|w'\|}{\Gamma(2-\beta)}, a_0\|w\|+ a_1\|w'\|) \int_0^1 b(s) ds \nonumber\\
& &\leq  A_{\sigma, \eta} (1-t)^{\sigma-1} \Lambda ( l_1 \|w\|_{*}, l_1 \|w\|_{*}, l_1 \|w\|_{*}, l_1 \|w\|_{*}) m_1. \nonumber
\end{eqnarray}
Therefore, if $\|w\|_{*} \in (0, \frac{\delta_1}{l_1}]$, then
$$|\mathcal{F}'w(t)| \leq A_{\sigma, \eta} (1-t)^{\sigma-1} \|w\|_{*} m_1 \leq \|w\|_{*} (1-t)^{\sigma-1},$$
so, we conclude that $\|\mathcal{F}'w\| \leq \|w\|_{*}$.
Hence if $\|w\|_{*} \in (0, \frac{\delta_1}{l_1}]$ then
\begin{eqnarray}
\label{rb1} \|\mathcal{F}w\|_{*} = \max \{ \|\mathcal{F}w\|, \|\mathcal{F}'w\| \} \leq \|w\|_{*}.
\end{eqnarray}
By the assumptions,  for all $1 \leq i \leq 4$ and almost all $t \in [0,1]$,
$$\lim_{\|z_i\| \to 0} \frac{|\mathcal{V}(t, z_1, z_2, z_3, z_4)|}{\|z_i\|} \leq \theta(t),$$
so, for each $\epsilon>0$ there exists $\delta(\epsilon)>0$, such that $\|z_i\| \in (0, \delta(\epsilon)]$ implies 
$$|\mathcal{V}(t, z_1, z_2, z_3, z_4)| \leq (\theta(t) + \epsilon) \|z_i\|.$$
 Thus, for  $\epsilon>0$, there exists $\delta(\epsilon)>0$ such that  $l_2 \|z\| \in (0, \delta(\epsilon)]$, it follows
\begin{eqnarray}
|\mathcal{V}(t, z, z', \mathcal{D}^{\gamma}z, \phi(z))| &\leq& (\theta(t) + \epsilon) \max \{ \|z\|, \|z'\|, \|\mathcal{D}^{\gamma}z\|, \|\phi(z)|\| \} \nonumber\\
&\leq & (\theta(t) + \epsilon) l_2 \|z\|_{*}. \nonumber
\end{eqnarray}
 Since $m_2 < \frac{1}{l_2 A_{\sigma, \eta}}$,  there exists $\epsilon_2>0$ such that $m_2 + \epsilon_2 < \frac{1}{l_2 A_{\sigma, \eta}}$. Put $\delta_2 := \delta(\epsilon_2)$, so if $\|z\| \in (0, \frac{\delta_2}{l_2}]$, then  we have
$$|\mathcal{V}(t, z, z', \mathcal{D}^{\gamma}z, \phi(z))| \leq (\theta(t) + \epsilon_2) l_2 \|z\|_{*}.$$
Thus, for $z \in X$ in which $\|z\| \in (0, \frac{\delta_2}{l_2}]$, we conclude that
\begin{eqnarray}
|\mathcal{S}z(t)| & \leq & \int_0^1 |\kappa(t,s)| |\mathcal{V}(s, z(s), z'(s), \mathcal{D}^{\gamma}z(s), \phi(z(s)))| ds \nonumber\\
& \leq & \int_0^1 A_{\sigma, \eta} t(1-t)^{\alpha -1} (\theta(s) + \epsilon_2) l_2 \|z\|_{*} ds \nonumber\\
& =& t(1-t)^{\sigma -1} A_{\sigma, \eta} (\int_0^1 \theta(s) ds + \epsilon_2) l_2 \|z\|_{*} \nonumber\\
& =& t(1-t)^{\sigma -1} A_{\sigma, \eta} (m_2 + \epsilon_2) l_2 \|z\|_{*} \nonumber\\
&\leq & t(1-t)^{\sigma -1} \|z\|_{*}, \nonumber
\end{eqnarray}
so
$\|\mathcal{S}z\| \leq \|z\|_{*}$. Also for all $t \in [0,1]$ and $z \in X$ in which $\|z\| \in (0, \frac{\delta_2}{l_2}]$, we have 
\begin{eqnarray}
|\mathcal{S}'z(t)| & \leq & \int_0^1 |\frac {\partial \kappa(t,s)}{\partial t}| |\mathcal{V}(s, z(s), z'(s), \mathcal{D}^{\gamma}z(s), \phi(z(s)))| ds \nonumber\\
& \leq & \int_0^1 A_{\sigma, \eta} (1-t)^{\alpha -1} (\theta(s) + \epsilon_2) l_2 \|z\|_{*} ds \nonumber\\
& =& (1-t)^{\sigma -1} A_{\sigma, \eta} (\int_0^1 \theta(s) ds + \epsilon_2) l_2 \|z\|_{*} \nonumber\\
& =& (1-t)^{\sigma -1} A_{\sigma, \eta} (m_2 + \epsilon_2) l_2 \|z\|_{*} \nonumber\\
&\leq & (1-t)^{\sigma -1} \|z\|_{*}, \nonumber
\end{eqnarray}
so
$\|\mathcal{S}'z\| \leq \|z\|_{*}$. Therefore,
\begin{eqnarray} 
\label{rb2} \|\mathcal{S}z\|_{*} = \max\{ \|\mathcal{S}z\|, \|\mathcal{S}'z\| \} \leq \|z\|_{*}.
\end{eqnarray}
Likewise, through the given assumptions for almost all  $t \in [0,1]$, we have
$$\lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{V}(t, z_1, z_2, z_3, z_4)}{max \| w_i - z_i \|} = 0. $$
Put $ \| w_k - z_k \|:= max_{1 \leq j \leq 4} \| w_i - z_i \| $ for some $1 \leq k \leq 4$, then for each $\epsilon>0$ there exists $\delta(\epsilon)>0$ such that $\|w_i\|, \|z_i\| \in (0, \delta]$ implies 
$$|\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{V}(t, z_1, z_2, z_3, z_4)| < \epsilon \|w_k - z_k \|.$$
Let $0 < \epsilon_3< \frac{1}{A_{\sigma, \eta}}$ and $\delta_3 := \delta(\epsilon_3)$,  then if $\|w\|, \|z\| \in (0, \frac{\delta_3}{l_3}]$, we have 
\begin{eqnarray}
& &|\mathcal{U}(t, w, w', \mathcal{D}^{\beta}w, \phi(w)) - \mathcal{V}(t, z, z', \mathcal{D}^{\gamma}z, \phi(z))| \nonumber\\
& &< \epsilon_3 \max \{ \|w-z\|, \|w'-z'\|, \| \mathcal{D}^{\beta}w- \mathcal{D}^{\gamma}z\|, \|\phi(w)- \phi(z)\| \} \nonumber\\  & &\leq \epsilon_3 l_3 \|w-z\|_{*}, \nonumber
\end{eqnarray}

where $l_3 = \max \{ l_1, l_2, |\frac{1}{\Gamma(2-\beta)}- \frac{1}{\Gamma(2-\gamma)}| \} = \max \{ l_1, l_2\}$. So if $\|w\|, \|z\| \in (0, \delta_3]$, then
\begin{eqnarray}
\label{rb3} |\mathcal{U}(t, w, w', \mathcal{D}^{\beta}w, \phi(w)) - \mathcal{V}(t, z, z', \mathcal{D}^{\gamma}z, \phi(z))| \leq \epsilon_3 \|w-z\|_{*}.
\end{eqnarray}
Now, let $\delta_{M} = \min \{\frac{\delta_1}{l_1}, \frac{\delta_2}{l_2}, \delta_3 \}$, define
$\alpha: X^{2} \to [0, \infty)$ as 
$$ \alpha(x,y) = \left\{ \begin{array}{ll}
1 \ \ \ \ \ \ \ \ \ \|w\|_{*}, \|z\|_{*} \in (0, \delta_M]
\\
\\
0 \ \ \ \ \ \ \ \ \ other \ wise
\ \ \ \end{array}\right.$$
and  $\psi: \mathbb{R} \to \mathbb{R}$ as $\psi(t)= \epsilon_{3} A_{\sigma, \eta} t$. So,  $\psi \in \Psi$ is obviuos. If $\alpha(w,z) \geq 1$ then $\|w\|_{*}, \|z\|_{*} \in (0, \delta_{M}]$, so by (\ref{rb2}), $\|\mathcal{S}w\|_{*} \leq \|x\|_{*} \leq \delta_{M}$. Likewise, via (\ref{rb1}), $\|\mathcal{F}y\|_{*} \leq \|y\|_{*} \leq \delta_{M}$, so $\alpha(\mathcal{S}w, \mathcal{F}z) \geq 1$.
If  $w \in X$ be such that $\|w\|_{*} \leq \delta_{M}$, then $\|\mathcal{S}w\|_{*} \leq \delta_{M}$, so it is concluded that there exists $w_0 \in X$ such that $\alpha(w_0, \mathcal{S}w_0) \geq 1$. To check the continuity  $\mathcal{F}$, let $E \subset [0,1]$ be a set which $\mathcal{U}(t, ., ., ., .)$ is not continuous on that, then $m(E)=0$ where $m$ is the Lebesgue measure in $\mathbb{R}$, and let $w_n \to w$ as $n \to \infty$. So for all $t \in [0,1]$ we have 
\begin{eqnarray}
\lim_{w_n \to w} \mathcal{F}w_{n}(t) &=& \lim_{w_n \to w} \int_0^1 \kappa(t,s) \mathcal{U}(s, w_n(s), w'_n(s), \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) ds \nonumber\\
&=& \lim_{n \to \infty} \int_{E^{c}} \kappa(t,s) \mathcal{U}(s, w_n(s), w'_n(s), \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) ds \nonumber\\
&+& \lim_{w_n \to x} \int_E \kappa(t,s) \mathcal{U}(s, w_n(s), w'_n(s), \mathcal{D}^{\beta}w_n(s), \phi(w_n(s))) ds \nonumber\\
&=& \int_{E^{c}} \kappa(t,s) \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) ds \nonumber\\
&=& \int_0^1 \kappa(t,s) \mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) ds \nonumber\\
&=& \mathcal{F}w(t). \nonumber
\end{eqnarray}
Similarly, $\lim_{w_n \to x} \mathcal{F}'w_{n}(t) = \mathcal{F}'w(t)$ is obtained for all $t \in [0,1]$, so it is concluded that  $\mathcal{F}$ is a continuous mapping in $(X, \| . \|_{*})$. On the other hand, for all $t \in [0,1]$ we deduce that
\begin{eqnarray}
& & |\mathcal{F}w(t) - \mathcal{S}z(t)| 
\leq \int_0^1 |\kappa(t,s)| \bigg|\mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s))) \nonumber\\
& &- \mathcal{V}(s, z(s), z'(s), \mathcal{D}^{\beta}z(s), \phi(z(s))) \bigg| ds \nonumber\\
&\leq & A_{\sigma, \eta} t(1-t)^{\sigma -1} \int_0^1 \bigg|\mathcal{U}(s, w(s), w'(s), \mathcal{D}^{\beta}w(s), \phi(w(s)))  \nonumber\\
& &- \mathcal{V}(s, z(s), z'(s), \mathcal{D}^{\beta}z(s), \phi(z(s))) \bigg| ds. \nonumber
\end{eqnarray}
Therefore, when $\|w\|_{*}, \|z\|_{*} \in (0, \delta_M]$, by (\ref{rb3}),  it implies that
\begin{eqnarray}
|\mathcal{F}w(t) - \mathcal{S}z(t)| &\leq & A_{\sigma, \eta} t(1-t)^{\sigma -1} \epsilon_3 \| w - z\|_{*}, \nonumber
\end{eqnarray}
consequently
\begin{eqnarray}
\| \mathcal{F}w - \mathcal{S}z \| &\leq & A_{\sigma, \eta} \epsilon_3 \| x - y\|_{*} = \psi(\| w - z\|_{*}). \nonumber
\end{eqnarray}
In a similar manner,  we have 
\begin{eqnarray}
\| \mathcal{F}'w - \mathcal{S}'z \| &\leq & A_{\sigma, \eta} \epsilon_3 \| w - z\|_{*} = \psi(\| w - z\|_{*}), \nonumber
\end{eqnarray}
hence
$$ \| \mathcal{F}w(t) - \mathcal{S}z \|_{*} = \max \{ \| \mathcal{F}'w - \mathcal{S}'z \| , \| \mathcal{F}w - \mathcal{S}z \|\} \leq \psi(\| w - z\|_{*}).$$
Therefore, regarding Lemma (\ref{l1.2}), both equations (\ref{e1}) and (\ref{e2}) have a common solution.
\end{proof}
\begin{example}
Consider the following pointwise defined equations 
$$\mathcal{D}^{\frac{5}{2}}w(t) + \frac{0.5}{p(t)}(\| w(t)\|^2 + \|w'(t)\|^2 + \| \mathcal{D}^{\frac{1}{2}} w(t)\|^2 + \| \int_0^t w(s) ds\|^2)=0$$
and
$$\mathcal{D}^{\frac{5}{2}}z(t) + \frac{0.3}{\sqrt{t}}(\| z(t)\| + \|z'(t)\| + \| \mathcal{D}^{\frac{1}{3}} z(t)\| + \| \int_0^t z(s) ds\|)=0$$
with boundary conditions 
$w(0)=z(0)=0$ 
and $w(\frac{1}{2})+\int_0^1 w(s)ds=z(\frac{1}{2})+\int_0^1 z(s)ds=0,$
where 
$$ p(t) = \left\{ \begin{array}{ll}
1 \ \ \ \ \ \ \ \ \ t \in [0, 1] | \{\delta_1, ..., \delta_k \}
\\
\\
0 \ \ \ \ \ \ \ \ \ t \in \{\delta_1, ..., \delta_k \}.
\ \ \ \end{array}\right.$$

Put $\Lambda(w_1, w_2, w_3, w_4) = \Sigma_{i=1}^4 \|w_i\|^{2}$, 
$\phi(w(t))= \int_0^t w(s) ds$, $b(t)= \frac{0.5}{p(t)}$, $$\mathcal{U}(t, w_1, w_2, w_3, w_4)= \Lambda(w_1, w_2, w_3, w_4),$$ $\theta(t)=  \frac{0.3}{\sqrt{t}}$ and $$\mathcal{V}(t, z_1, z_2, z_3, z_4)=\theta(t) \Sigma_{i=1}^4 \|z_i\|,$$ then $\| \phi(w) - \phi(z)\| \leq \|w-z\|$, $l_1= \max \{1, \frac{1}{\Gamma(2- \frac{1}{2})} \} = \frac{2}{\sqrt{\pi}}$, \\
$l_2= \max \{1, \frac{1}{\Gamma(2- \frac{1}{3})} \} = \frac{1}{\Gamma( \frac{5}{3})}$,
$q_0 = \lim_{z \to 0^{+}} \frac{\Lambda(z,z,z,z)}{z} = 0 < \frac{1}{l_1},$
$$A_{\sigma, \eta} = \frac{2(1+\frac{5}{2})}{(1+ 1) \Gamma(\frac{5}{2} + 1)}= \frac{28}{15 \sqrt{\pi}},$$
$b, \theta \in L^{1}[0,1]$, $m_1 = \int_0^1 b(s) ds = 0.5 < \frac{1}{A_{\sigma, \eta}}$, 
$m_2 = \int_0^1 \theta(s) ds = 0.6 < \frac{1}{l_2 A_{\sigma, \eta}}$ and for all 
$(w_1, w_2, w_3, w_4), (z_1, z_2, z_3, z_4) \in X^{4}$ that $(w_1, w_2, w_3, w_4) \neq (z_1, z_2, z_3, z_4)$, almost all $t \in [0,1]$ and all $1 \leq i \leq 4$
\begin{eqnarray}
\lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } & & \frac{|\mathcal{U}(t, w_1, w_2, w_3, w_4) - \mathcal{V}(t, z_1, z_2, z_3, z_4)|}{max \| w_i - z_i \|} \nonumber\\
&\leq & |b(t) - \theta(t)| \lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\Sigma_{i=1}^{4} |\|w_i\|^2 - \|z_i\| |}{max \| x_i - z_i \|}\nonumber\\
&\leq & |b(t) - \theta(t)| \lim_{(\|x_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\Sigma_{i=1}^{4} |\|w_i\|^2 - \|w_i\| \|z_i\| |}{max \| w_i - z_i \|}\nonumber\\
& = & |b(t) - \theta(t)| \lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\Sigma_{i=1}^{4} |\|w_i\|( \|w_i\| - \|z_i\| )|}{max \| x_i - y_i \|}\nonumber\\
& \leq & |b(t) - \theta(t)| \lim_{(\|w_i\|, \|z_i\|) \to (0^{+}, 0^{+}) } \frac{\Sigma_{i=1}^{4} \|w_i\| \|w_i - z_i\| }{max \| w_i - z_i \|}  \nonumber\\
&\leq & |b(t) - \theta(t)| \lim_{\|w_i\| \to 0^{+} } \Sigma_{i=1}^{4} \|w_i\| = 0. \nonumber
\end{eqnarray}
Hence, based on Theorem (\ref{t3.3}) there is a common solution for both mentioned equations.
\end{example}
\begin{corollary}
Let $\mathcal{U}: [0,1] \times X^{4} \to \mathbb{R}$ be continuous on set $E \in X$ with $m(E^{c})=0$, there exists $b \in L^{1}[0,1]$ and nondecreasing mapping $\Lambda: X^{4} \to \mathbb{R}$ such that 
$|\mathcal{U}(t, w_1, w_2, w_3, w_4)| \leq b(t) \Lambda (w_1, w_2, w_3, w_4)$ for all $(w_1, w_2, w_3, w_4) \in X^{4}$ and almost all $t \in [0,1]$, also let
$$\lim_{z \to 0^{+}} \frac{\Lambda (z, z, z, z)}{z}= q_0,$$
$m_1:= \int_0^1 b(s) ds < \frac{1}{A_{\sigma, \eta}}$, where $l_1= \max \{1, \frac{1}{\Gamma(2-\beta)}, a_0+a_1\}$ and $q_0 \in [0, \frac{1}{l_1})$.
Then, the pointwise defined equation (\ref{e1}) has a solution.
\end{corollary}
\begin{proof}
In theorem (\ref{t3.3}), let for all $t \in [0,1]$ and $(w_1, w_2, w_3, w_4) \in X^{4}$,
$$\mathcal{V}(t, w_1, w_2, w_3, w_4)= \mathcal{U}(t, w_1, w_2, w_3, w_4).$$
Indicating all conditions of Theorem (\ref{t3.3}) is feasible. Therefore, the pointwise defined equation  (\ref{e1}) has a solution.
\end{proof}



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\bibitem{22a} H. Afshari, M. Sajjadmanesh, {\it Fixed point theorems for $\alpha$-contractive
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\end{thebibliography}

{\small 

\noindent{\bf Abdolhamid Malekpour}

\noindent Ph.D Student of Mathematics

\noindent Department of Mathematics, South Tehran Branch, Islamic Azad University

\noindent Tehran, Iran

\noindent E-mail: $St\_ah\_malekpour@azad.ac.ir$}\\

{\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: $mehdi\_math1983@yahoo.com$}\\



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