\def\papertitle{ Some existence results to positive solutions for p-Laplacian boundary value problems of fractional differential equations}
\def\paperauthorA{F. Chabane}
%\def\paperauthorB{$^1$F. Chabane, $^2$M. Benbachire}
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\title{\papertitle}
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			\begin{minipage}{11cm}
				 Laboratoire de Math\'{e}matiques et Sciences appliqu\'{e}es, Universit\'{e} de Ghardaia, Algeria\\
				E-mail address: chabanefarid201644@gmail.com
			\end{minipage}
			}}
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%			{ $^1$ Laboratoire de Math\'{e}matiques et Sciences appliqu\'{e}es, Universit\'{e} de Ghardaia, Algeria\\
%				$^2$ E-mail address: chabanefarid201644@gmail.com\\Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria\\
%				E-mail address : mbenbachir2001@gmail.com}
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%\textbf{* F. Chabane$^{1}$ and M. Benbachir$^{2}$ }\newline
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%				$^1$ Laboratoire de Math\'{e}matiques et Sciences appliqu\'{e}es, Universit\'{e} de Ghardaia, Algeria\\
%				E-mail address: chabanefarid201644@gmail.com
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%				$^2$ Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria\\
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\begin{abstract}
In this manuscript, we deal with a study of the existence and the multiplicity of $\rho_1 $-concave positive solutions for a boundary value of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. An application of a functional analysis tools, more specifically, we using some fixed point theorems and under some additional assumptions, some of important results  have been proven and we obtain the existence of at least one solution.

\end{abstract}


% Introduction section 

%\section{section 1}
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%\section{conclusions}

%\begin{center}
%{\fontsize{14}{20}\textbf{A study of existence to positive solutions for p-Laplacian boundary value problems of fractional differential equations }}
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%\begin{center}
%\textbf{* F. Chabane$^{1}$ and M. Benbachir$^{2}$ }\newline
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%				$^1$ Laboratoire de Math\'{e}matiques et Sciences appliqu\'{e}es, Universit\'{e} de Ghardaia, Algeria\\
%				E-mail address: chabanefarid201644@gmail.com
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%			\quad
%			\begin{minipage}{4.5cm}
%				$^2$ Faculty of Sciences, Saad Dahlab University, Blida 1, Algeria\\
%				E-mail address : mbenbachir2001@gmail.com
%			\end{minipage}			
%		}}
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%\vspace{1.5cm}  
%\textbf{Abstract:}  \textit{In this manuscript, we deal with a study of the existence and the multiplicity of $\rho_1 $-concave positive solutions for a boundary value of two-sided fractional differential equations involving generalized-Caputo fractional derivatives. An application of a functional analysis tools, more specifically, we using some fixed point theorems and under some additional assumptions, some of important results  have been proven and we obtain the existence of at least one solution.
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\vspace{.5cm}  \textbf{Key Words:} Fractional differential equations; Caputo-Katugampola (CK) fractional type;
boundary value problem; fixed point theorems; Positive solutions.
\section{Introduction}
Lately, many researches on fractional differential problems have been dealt with by many researchers. Especial, fractional p-Laplacian has been  used in modeling different problems, for exampale  in science, engineering, biology, \cite{AAK,KSM,OKB,IPG,SGS,ea} etc.

In this words, we consider the following fractional boundary value problem (FBVP):  
\begin{equation} \label{pb_main}
\left\{ 
\begin{array}{l}
^{\rho_{2};C}\!\mathcal{D}^{\sigma }_{T^{-}}\left(\phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u\right)\right) (t)+ c f(u(t))=0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  a<t<T, \\ [7pt]
u(a)+  ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)=0, \; \delta_{\rho_{1}}^{2}u(a)=0, \; \delta_{\rho_{1}}^{1}u(T)=\mu \delta_{\rho_{1}}^{1}u(\eta) + \lambda,  \\ [7pt]
^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}}  u(T)= - \delta_{\rho_{2}}^{1}\left[\phi_{p}\left( ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}}u\right)\right](a) = \delta_{\rho_{2}}^{2}\left[\phi_{p}\left( ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}}u\right)\right](T)= 0, 
\end{array}
\right.
\end{equation}
\noindent where 
\begin{itemize}
	\item $^{\rho_{1};C}\!\mathcal{D}^{\sigma }_{a^{+}},^{\rho_{2};C}\!\mathcal{D}^{\alpha }_{T^{-}}$ are  the left and  right-sided Caputo-Katugampola fractional derivatives with $\rho_{1}, \rho_{2} \in \mathbb{R} - \{ 1 \}$ and  
	$2< \sigma,\alpha \leqslant 3$, 
	\item $ \phi_{p} $ is  the p-Laplacian operator, i.e., $ \phi_{p}(s)=|s|^{p-2} s, p>1$, $\delta_{\rho}^k=\left(t^{1-\rho}\dfrac{\mathrm{d}}{\mathrm{d}t}\right)^k$,
	\item  $f$ is continuous and positive. 
	\item    $\eta \in(a,T)$, $0\leq\mu <1$, $c\geq 0$  and $\lambda \geq 0$.  
\end{itemize}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%REFERENCES BIBLIOGRAPHIQUES                                %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 In \cite{ BMHS}, Chuanzhi Bai  used the Guo-Krasnoselskii 
fixed point theorem and the Banach contraction mapping principle to prove  the existence and uniqueness of positive solutions for the FBVP: 
\begin{equation}
\left\{\begin{array}{l}
\left(\phi_{p}\left(D_{0^{+}}^{\alpha} u\right)\right)^{\prime}(t)+f(t, u(t))=0, \quad \quad 0<t<1, \\
u(0)=D_{0^{+}}^{\alpha} u(0)=0, \quad { }^{c}D_{0^{+}}^{\beta} u(0)={ }^{c}D_{0^{+}}^{\beta} u(1)=0,
\end{array}\right.
\end{equation}
\noindent where $0<\beta \leq 1,2<\alpha<2+\beta, D_{0^{+}}^{\alpha}$ and ${ }^{c} D_{0^{+}}^{\beta}$ are the Riemann-Liouville and Caputo fractional derivatives of orders $\alpha, \beta$, respectively, $p>1,$ and $f:[a, b] \times \mathbb{R} \rightarrow \mathbb{R}$
is a continuous function.
 
 Chai \cite{BZZY}, obtained  some results for the existence of at least one nonnegative solution and two positive solutions by using a fixed point theorem on a cone for the nonlinear FBVP  
\begin{equation}
\left\{\begin{array}{l}
D_{0+}^{\beta}\left(\phi_{p}\left(D_{0+}^{\alpha} u\right)\right)(t)+f(t, u(t))=0, \quad 0<t<1, \\
u(0)=0, u(1)+\sigma D_{0+}^{\gamma} u(1)=0, \quad D_{0+}^{\alpha} u(0)=0,
\end{array}\right.
\end{equation}
where $1<\alpha \leq 2$, $0 <\beta, \gamma \leq 1$, $0 \leq \alpha-\gamma-1$, and $\sigma > 0$.
 
Using the fixed point index theory, Su et al. \cite{HLC2} studied the existence of positive solution for a nonlinear four-point singular FBVP 
 \begin{equation}
\left\{\begin{array}{ll}
\left(\phi_{p}\left(u^{\prime}\right)\right)^{\prime}(t) +a(t) f(u(t))=0, & 0<t<1, \\
\alpha \phi_{p}(u(0))-\beta \phi_{p}\left(u^{\prime}(\xi)\right)=0, & \gamma \phi_{p}(u(1))+\delta \phi_{p}\left(u^{\prime}(\eta)\right)=0,
\end{array}\right.
\end{equation}
where 
$\alpha, \gamma>0, \beta, \delta \geq 0, \xi, \eta \in(0,1)$, and $\xi<\eta$. 
 
 Su \cite{ LG} applied  the fixed-point index theory to study the existence of positive solutions for the nonlinear third-order two-Point singular boundary value problem 
 \begin{equation}
\left\{\begin{array}{l}
\left(\phi_{p}\left(u^{(n-1)}\right)\right)^{\prime}(t) + a(t) f(u(t))=0, \quad 0<t<1, \\
u(0)=u^{\prime}(0)=\cdots=u^{(n-3)}(0)=u^{(n-1)}(0)=0, \quad u(1)=\sum_{i=1}^{m-2} \alpha_{i} u\left(\eta_{i}\right),
\end{array}\right.
\end{equation}
where 
$0<\eta_{1}<\eta_{2}<\cdots<\eta_{m-2}<1, \alpha_{i}>0$ with $\sum_{i=1}^{m-2} \alpha_{i} \eta_{i}^{n-2}<1$.
 
Using the coincidence degree theory, Tang et al. \cite{WJ2} gave a new result on the existence   of positive solutions to  the  FBVP
\begin{equation}
\left\{\begin{array}{l}
{ }^{c}D_{0^{+}}^{\alpha} \left( \phi_{p}{ }^{c}D_{0^{+}}^{\beta} u\right)(t)=f\left(t, u(t), { }^{c}D_{0^{+}}^{\beta} u(t)\right), \\
u(0)=0, \quad { }^{c}D_{0^{+}}^{\beta} u(0)={ }^{c}D_{0^{+}}^{\beta} u(1),
\end{array}\right.
\end{equation}
where $0<\alpha, \beta \leq 1,1<\alpha+\beta$ $\leq 2$.
 
 Torres \cite{NK1}, studied the existence and multiplicity  for a mixed-order three-point boundary value  problem of fractional differential equation evolving Caputo's differential operator and the boundary conditions with integer order derivatives
 \begin{equation}\label{ooo1}
\left\{\begin{array}{l}
\left(\varphi_{p}\left({ }^{c}D_{0^{+}}^{\alpha} u\right)\right)^{\prime}(t) + a(t) f(t, u(t))=0, \quad 0<t<1, \\
{ }^{c}D_{0^{+}}^{\alpha} u(0)=u(0)=u^{\prime \prime}(0)=0, \quad u^{\prime}(1)=\gamma u^{\prime}(\eta),
\end{array}\right.
 \end{equation}
where $\eta, \gamma \in(0,1), \alpha \in(2,3]$.
 
Base on the coincidence degree theory, Chen et al. \cite{HZS}  gave new results about the problem
 \begin{equation}
\left\{\begin{array}{l}
{ }^{c}D_{0^{+}}^{\beta} \phi_{p}\left({ }^{c}D_{0^{+}}^{\alpha} x\right)(t)=f\left(t, x(t), { }^{c}D_{0^{+}}^{\alpha} x(t)\right), t \in[0,1], \\
{ }^{c}D_{0^{+}}^{\alpha} x(0)={ }^{c}D_{0^{+}}^{\alpha} x(1)=0,
\end{array}\right.
 \end{equation}
where $0<\alpha, \beta \leq 1,1<\alpha+\beta \leq 2$.

 It can be seen that our work presented in this words has some following features which are different from those \cite{HZS, NK1, WJ2, LG, HLC2, BZZY, BMHS}, in addition to this, the appropriate manipulations of the controls $\rho_1$, $\rho_2$, $\alpha$, $\sigma$, $c$, $\lambda$, $\mu$ and $\eta$, allow us to see the impact and the extent of our study.
%  For example one can see that, when we put $^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t) = -v(t)$, the fractional boundary value problem \eqref{pb_main} reduce to this FGBVP
% \begin{equation} \label{pb_main2}
%\left\{ 
%\begin{array}{l}
%^{\rho_{2};C}\!\mathcal{D}^{\sigma }_{T^{-}}\left(\phi_{p}\left(v\right)\right) (t)= \tilde{f}(t), \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  a<t<T, \\ [7pt]
%v(T)= - \delta_{\rho_{2}}^{1}\left[\phi_{p}\left( v\right)\right](a) = \delta_{\rho_{2}}^{2}\left[\phi_{p}\left( v\right)\right](T)= 0, 
%\end{array}
%\right.
%\end{equation}
%where the function $ \tilde{f}(t)$ can be extracted from the first equation and under some boundary value conditions in \eqref{pb_main}.
%
%We can also notice that, the fractional boundary value problem \eqref{pb_main} can be formulated as follows 
%\begin{equation} \label{pb_main212}
%\left\{ 
%\begin{array}{l}
%\left(\phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u\right)\right) (t)+ c \hat{f}(t)=0, \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  a<t<T, \\ [7pt]
%u(a)+  ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)=0, \; \delta_{\rho_{1}}^{2}u(a)=0, \; \delta_{\rho_{1}}^{1}u(T)=\mu \delta_{\rho_{1}}^{1}u(\eta) + \lambda,  
%\end{array}
%\right.
%\end{equation} 
%where $ \hat{f}(t)$ is  twice continuously differentiable functions on $[a,T]$, with respect to $\delta_{\rho_2}$ and has a continous fractional direvative on $[a,T]$, with respect to order $\sigma$.
Add to this there is no known research paper that  delves deeper than us into extracting some of the properties of the Greens functions which used to study the existence of solutions, special when the non-integer order of the fractional derivatives is huge.

In this words, we obtain some sufficient conditions ensuring the existence of at least one positive solutions for the BVP \eqref{pb_main}. 
The rest of the paper is organized as follows. Section 2 presents some
basic definitions, lemmas, and preliminary results. In Section 3, we present some important lemmas. In Section 4, we derive some conditions on the parameter $\lambda$ to obtain the existence of at least one positive solution.  Finally, we give some illustrative examples in Section 5.


\section{Preliminaries and background materials}
%In this section ,we will give and recall it to the reader  some definitions,fixed point theorem, lemmas, properties and notation which we use later in the proofs of our main results \\
 
 \subsection*{Notations}
 In addition to the notations introduced with the problem \eqref{pb_main}, let $J=[a,T] \subset (0,\infty)$, and $\rho > 0$.
 \begin{enumerate}
 	\item $C(J)$ denotes the Banach space of continuous functions $h$ on $J$ endowed with the norm $$\Vert h\Vert_{C} = \max _{x\in J}| h(x) |.$$
 	\item  $AC(J)$ and $C^n(J)$ denote the spaces of absolutely continuous and $n$ times continuously differentiable functions on $J$ respectively.
 	\item $L^{p}(a, T)$ denotes the space of Lebesgue integrable functions on $(a, T)$.
 	\item $C^{n}_{\rho}(J)$ is the Banach space of $n$ continuously differentiable functions on $J$, with respect to $\delta_{\rho}$:
 	$$C^{n}_{\rho}(J)= \left\lbrace h \in C(J): \, \delta_{\rho}^{k} h \in C(J) ,k=0,1,...,n\right\rbrace, $$
 	endowed with the norm $$\Vert h\Vert_{C^{n}_{\rho}} = \sum_{k=0}^{n} \Vert \delta_{\rho}^{k}h\Vert_{C}.$$
 	\item $[\alpha]$ is the largest integer less than or equal to $\alpha$. Throughout the words, we use $n=[\alpha]+1$.
 	\item $C^{+}(J)=\{y \in C(J), \; y(t) \geq 0 \; \; \forall t \in J\}.$
 \end{enumerate}
 
\subsection{Fractional calculus}

We present here basic definitions and lemmas from fractional calculus theory (See \cite{AAK,KSM,OKB,IPG,SGS,BAI2} and references therein). 


\begin{definition}[Function space]
For $ c \in \mathbb{R} $, consider the Banach space 
$$\mathcal{X}_{c}^{p}(a,T)=\left\lbrace h : J\rightarrow \mathbb{R}: \; \Vert h\Vert_{\mathcal{X}_{c}^{p}} :=\left(\int_{a}^{T} \mid t^{c} \, h(t)\mid ^{p}\frac{\mathrm{d}t}{t}\right)^{\frac{1}{p}} < +\infty\right\rbrace.$$
%For $p=\infty$ 
%$$ \Vert h \Vert_{\mathcal{M}_{c}^{p}} =ess~\sup_{a\leq t\leq T}\left[\mid t^{c}h(t)\mid \right].$$ 
\end{definition}



Now, we recall the Katugampola (K) and Caputo-Katugampola (CK) fractional integrals and derivatives \cite{BAI2}.
\begin{definition}
	\label{D2.1}
For a  function $u\in \mathcal{X}_{c}^{p}(a,T)$, the Katugampola left-sided ($^\rho\mathcal{I}^{\alpha}_{a^+} $)   and right-sided ($^\rho\mathcal{I}^{\alpha}_{T^-}$) fractional integrals of order $\alpha >0$  are defined by \[^\rho\mathcal{I}^{\alpha}_{a^+}u(t)=\frac{\rho ^{1-\alpha}} {\Gamma(\alpha)}\int_{a}^{t}(t^{\rho}-s^{\rho})^{\alpha-1}s^{\rho-1} u(s) ds, \quad t > a,\]
\[^\rho\mathcal{I}^{\alpha}_{T^-}u(t)=\frac{\rho ^{1-\alpha}} {\Gamma(\alpha)}\int_{t}^{T}(s^{\rho} - t^{\rho})^{\alpha-1}s^{\rho-1}u(s) ds,\quad t< T.\]

\end{definition}
\begin{remark}\label{rema1}
For a  function $u\in \mathcal{X}_{c}^{p}(a,T)$, the Katugampola left-sided ($^\rho\mathcal{I}^{\alpha}_{a^+} $)   fractional integrals of order $\alpha >0$  can be geven by the generalized convolution
of u and h \[^\rho\mathcal{I}^{\alpha}_{a^+}u(t)= \left(h\star_g u\right)(t)=\int_{a}^{t}h(s) u(g^{-1}\left(g(t)+g(a)-g(s)\right)g^{\prime}(s) ds, \quad t > a,\]
where $h(t)=\frac{1} {\Gamma(\alpha)}(g(t) - g(a))^{\alpha-1}$ and $g(t)=\frac{t^\rho}{\rho}$. For more detailes see \cite{fahd}. This generalized convolution of two functions is commutative.
\end{remark}

Next, we present some properties for left-sided integrals. But, the same properties are also true for the right-sided ones. 

\begin{lemma} \label{le2.5}
Let $c \in \mathbb{R} $, $\alpha, \beta, \rho > 0$, and $1\leq p \leq \infty $. Then, for all  $u \in \mathcal{X}_c^{p}  (a, T )$, we have the following.
\begin{itemize}
	 \item $^{\rho}\mathcal{I}_{a^+}^{\alpha} : \mathcal{X}_c ^{p}(a,T) \rightarrow \mathcal{X}_c ^{p}(a,T).$ 
\item $^{\rho}\mathcal{I}_{a^{+}}^{\alpha}$ is  linear. 
 \item  $ ^{\rho}\mathcal{I}_{a^+}^{\alpha} \circ ^{\rho}\mathcal{I}_{a^+}^{\beta} = ~  ^{\rho}\mathcal{I}_{a^+}^{\alpha+\beta}.$
\end{itemize}
\end{lemma}

\begin{definition}\label{d2.11}  For  a function  $u \in C_{\delta}^{n}([a, T])$ (or $\in AC_{\delta}^{n}([a, T])$), the  Caputo-Katugampola fractional derivatives  are defined by:
$$ ^{\rho_{1};C}\!\mathcal{D}_{a^+}^{\alpha} u (t) = \!^{\rho_{1}}\mathcal{I}^{n-\alpha}_{a^{+}} \delta_{\rho_{1}}^{n} u(t). $$
$$ ^{\rho_{1};C}\!\mathcal{D}_{T^-}^{\alpha} u (t) = (-1)^{n} ~\!^{\rho_{1}}\mathcal{I}^{n-\alpha}_{T^{-}} \delta_{\rho_{1}}^{n} u(t). $$
\end{definition}


 \begin{lemma}\label{le2.6}
 Let $ \beta >  \alpha > 0$, $ h \in \mathcal{M}_{c}^{p} (a,T) $, $u \in AC_{\delta}^{n}([a, T])$ or $  C_{\delta}^{n}([a, T])$.  Then we have
 \begin{align}
 ^{\rho;C}\!\mathcal{D}_{a^+}^{\alpha}  \!^{\rho}\mathcal{I}_{a^+}^{\beta}h(t)  = \!^{\rho}\mathcal{I}_{a^+}^{\beta -\alpha} h(t),
 \end{align}
\begin{align}
 ^{\rho}\mathcal{I}_{a^+}^{\alpha} \left( ^{\rho;C}\!\mathcal{D}_{a^+}^{\alpha} u\right)(t)&=u(t) - \sum_{k=0}^{n-1} C_k\left(\frac{t^{\rho} -a^{\rho}}{\rho} \right)^{k},\label{eq2.6}    \\
   ^{\rho}\mathcal{I}_{T^-}^{\alpha}\left( ^{\rho;C}\!\mathcal{D}_{T^-}^{\alpha} u\right)(t)&=u(t) - \sum_{k=0}^{n-1} D_k \left(\frac{T^{\rho}-t^{\rho} }{\rho} \right)^{k}, \label{eq2.7} 
\end{align}
\noindent for some real constants $C_k$ and $D_k$. 
\end{lemma}

\begin{lemma}\label{le2.8}
 If  $ ^{\rho;C}\!\mathcal{D}_{a^+}^{\alpha}u \in C(J)$, then $u \in C^{n-1}_{\rho}(J)$.
\end{lemma}
\begin{lemma}\label{le2.82}
 If  $u \in C^1(J)$, then 
 $$\delta^{1}_\rho\left(^{\rho}\mathcal{I}_{a^{+}}^{\alpha} u\right)(t) = \frac{1}{\Gamma(\alpha)}\int_a ^t \left(\frac{s^\rho - a^\rho}{\rho}\right)^{\alpha-1} \frac{s^{\rho -1}}{\left(t^\rho + a^\rho -s^\rho\right)^{\frac{\rho-1}{\rho}}} u^\prime (\left(t^\rho + a^\rho -s^\rho\right)^{\frac{1}{\rho}})\mathrm{d}s \in C(J)$$.
\end{lemma}

\subsection{Fixed point theorems}
 Let $\mathbb{E}$ be a real Banach function space, endowed with the infinity norm.
 
\begin{definition}\label{D2.9} 
 A nonempty closed convex
set $K \subset \mathbb{E}$ is called cone if the following properties are satisfied.
\begin{enumerate}
\item $\forall u \in K, \, \forall \lambda>0: \; \lambda u \in K.$ 
\item $\forall u \in K: -u \in K \implies u=0$. 
\end{enumerate}
\end{definition}
\begin{definition}\label{D2.10}
A continuous operator is called completely continuous if it maps bounded sets into precompact sets.
\end{definition}

Let  $K \subset \mathbb{E}$ be a cone, $r>0$, $\mathcal{P}_{r}=\{u \in K :\|u\|<r\}$, and $\mathbf{i}$ is the fixed point index function.


\begin{theorem}[ \cite{BAI1},\cite{BAI0}] \label{tho 2.11}
 Let $\mathcal{L}: K \cap\overline{\mathcal{P}}_{r} \rightarrow K$ be a
completely continuous operator such that $\mathcal{L} u \neq u$ for all $u \in \partial \mathcal{P}_{r}$. 
\begin{enumerate}
\item If $\|\mathcal{L} u\|\leqslant\|u\|$ for all $u \in \partial \mathcal{P}_{r}$, then $\mathbf{i}\left(\mathcal{L}, \Omega_{r}, K\right)=1$.  
\item If $\|\mathcal{L} u\|\geqslant\|u\|$ for all $u \in \partial \mathcal{P}_{r}$, then $ \mathbf{i}\left(\mathcal{L}, \Omega_{r}, K\right)=0$. 
\end{enumerate}


\end{theorem}
\begin{theorem}[Guo-Krasnoselskii \cite{AAK}] \label{the2.12}
Assume that $\mathcal{P}_1 $  and $\mathcal{P}_2$ are open subsets of $\mathbb{E}$ with $0 \in \mathcal{P}_1$ and $\overline{\mathcal{P}_1} \subset\mathcal{P}_2 $. Let
$\mathcal{L}: K \cap (\overline{\mathcal{P}_2}\setminus\mathcal{P}_1) \rightarrow K$ be completely continuous operator. Consider
\\
$(\text{D}_1)$ $\Vert \mathcal{L} u \Vert \leq \Vert u\Vert, \forall u \in  K \cap\partial\mathcal{P}_1 ~~\text{and}~ \Vert \mathcal{L} u\Vert \geq  \Vert u\Vert, \forall u \in  K \cap\partial\mathcal{P}_2 $,\\
$(\text{D}_2)$ $\Vert \mathcal{L} u \Vert \leq \Vert u\Vert, \forall u \in  K \cap\partial\mathcal{P}_2 ~~\text{and}~ \Vert \mathcal{L} u\Vert \geq  \Vert u\Vert, \forall u \in  K \cap\partial\mathcal{P}_1 $.

If $(\text{D}_1)$ or $(\text{D}_2)$ holds, then $\mathcal{L}$ has a fixed point in $K \cap (\overline{\mathcal{P}_2}\setminus\mathcal{P}_1)$.
\end{theorem}


\section{Major results}

We present some important lemmas which play a key role to prove the
major results.

\subsection{Lemmas}

 Consider the linear generalized boundary-value problem associated to \eqref{pb_main}
 \begin{equation}\label{pb_sub1}
 \left\{ 
  \begin{array}{ll} 
\!^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t) + y(t)=0, \quad a<t<T, \\ [5pt]
 u(a)+  \!^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)=0,\delta^{2}_{\rho_{1}}  u(a)=0 ,\delta^{1}_{\rho_{1}}  u(T)-\mu \delta^{1}_{\rho_{1}}  u(\eta)=\lambda.
\end{array} 
 \right.
 \end{equation}
\begin{lemma}\label{le2.12} For $y \in C(J)$, the integral solution of \eqref{pb_sub1} is given by 
\begin{equation}\label{0.01}
u(t)=\int_{a}^{T} G(t, s) y(s) \mathrm{d}s + \left( \frac{t^{\rho_{1}} - a^{\rho_{1}}}{\rho_1 (1-\mu)  } \right)\mu \int_{a}^{T} H(\eta, s) y(s) \mathrm{d} s + \left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}(1-\mu) }\right)\lambda  +    y(a). \nonumber
\end{equation}
where 
\begin{equation} \label{2.5}
G(t, s)= \left\{\begin{array}{ll}{ \frac{1}{\Gamma(\alpha -1)}\left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right) \left(\frac{T^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1} - \frac{1}{\Gamma(\alpha)}\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-1}s^{\rho_{1} -1}  ,} & {a\leq s \leq t\leq T}, \\ {\frac{1}{\Gamma(\alpha -1)}\left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right) \left(\frac{T^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1},} & {a\leq t \leq s\leq T,}\end{array}\right.
\end{equation}
and
\begin{equation}\label{2.6}
H(\eta, s)= \left\{\begin{array}{cc}{
	\frac{1}{\Gamma( \alpha -1)} \left(\frac{T^{\rho_{1}} - s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1}   -  \frac{1}{\Gamma( \alpha -1)} \left(\frac{\eta^{\rho_{1}} - s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1} ,} & {a\leq s \leq \eta\leq T,} \\ 
\frac{1}{\Gamma( \alpha -1)} { \left(\frac{T^{\rho_{1}} - s^{\rho_{1}}}{\rho}\right)^{\alpha-2}s^{\rho_{1}-1},} & {a\leq\eta \leq s\leq T.}\end{array}\right.
\end{equation}
\end{lemma}


\begin{proof}
	By applying (\ref{eq2.6}) and from the  boundary conditions of \eqref{pb_sub1} we get the desired result. The converse follows by direct computation. The proof is completed. \\
\end{proof}

 Now, consider the generalized boundary-value problem associated to \eqref{pb_main}
 \begin{equation} \label{pb_sub2}
 \left\{ 
\begin{array}{l}
^{\rho_{2};C}\!\mathcal{D}^{\sigma }_{T^{-}}\left(\phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t)\right)\right)=g(t), \quad a<t<T,  \\ [5pt]
u(a)+ ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)=0, \delta_{\rho_{1}}^{2}u(a)=0, \quad \delta_{\rho_{1}}^{1}  u(T)-\mu \delta_{\rho_{1}}^{1}  u(\eta)=\lambda, \\ [5pt]
^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^+}  u(T)=0, -\delta_{\rho_{2}}^{1} \left[\phi_{p}\left(  ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u\right)\right] (a)=0, \delta_{\rho_{2}}^{2}\left[\phi_{p}\left( ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u\right)\right](T) = 0. 
\end{array}
\right.
 \end{equation}
 
 \begin{lemma}
Let $g(t)\in C^{+}(J)$ and $y(t) = \phi_{\bar{p}}\left( \int_{a}^{T} K(t, \tau)g(\tau)\mathrm{d}\tau\right)$, the problem \eqref{pb_sub2} is equivalent to the problem \eqref{pb_sub1}
where
\begin{equation}\label{2.11}
K(t, s)=\left\{\begin{array}{cc}{    \frac{1}{\Gamma(\sigma-1)}  \left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right)\left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma-2}s^{\rho_{2}-1} - \frac{1}{\Gamma(\sigma)}\left(\frac{s^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -1}s^{\rho_{2}-1}         ,} & {t \leq s,} \\ 
{\frac{1}{\Gamma(\sigma-1)}  \left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right)\left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma- 2}s^{\rho_{2}-1},} & {s\leq t,}\end{array}\right.,
\end{equation}
 \end{lemma}
 \begin{proof}
 From Lemma \ref{le2.6}, the equation \eqref{pb_sub2} is equivalent to the equation 
$$
 \begin{array}{ll}
\phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t)\right)& = - D_0  - D_{1}\left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right) - D_{2}\left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right)^{2} + ^{\rho_{2}}\!\mathcal{I}^{\sigma}_{T^-} g(t) , \\
 \end{array}
 $$
for some constants $D_0 ,D_1 , D_2 \in \mathbb{R}$. Using the second boundary condition, we get 
 \begin{align*}
 \phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t)\right)=&  ^{\rho_{2}}\!\mathcal{I}^{\sigma}_{T^-} g(t) \;  -\left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right) \times  \frac{1}{\Gamma(\sigma -1)}\int_{a}^{T} \left(\frac{\tau^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma-2}\tau^{\rho_{2}-1} g(\tau) \mathrm{d}\tau    \\
      &=- \int^{T}_{a} K(t,\tau) q(\tau) \mathrm{d}\tau.
 \end{align*}
 Consequently, $$ ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t)= - \phi_{\bar{p}}\left(\int^{T}_{a} K(t,\tau) g(\tau) \mathrm{d}\tau\right),$$
 \noindent where $\bar{p} = \frac{p}{p-1}$.  Thus, the problem \eqref{pb_sub2} can be written as 
\begin{equation} \label{pb_main1}
%\tag{$P^{\star}$}
\left\{ 
\begin{array}{l}
   ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t) + \phi_{\bar{p}}\left(\int^{T}_{a} K(t,\tau) q(\tau) \mathrm{d}\tau\right)=0 ,\quad t \in(a,T),\\ 
 u(a)+\!  ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)=0, \delta_{\rho_{1}}^{2}u(a)=0, \quad \delta_{\rho_{1}}^{1}  u(T)=\mu \delta_{\rho_{1}}^{1} u(\eta)+\lambda,
\end{array}
\right.
 \end{equation}
which, according to Lemma \ref{le2.12}, has a unique solution of the form \eqref{pb_sub2_sol}.
\end{proof}

\begin{lemma}\label{le2.13}
For $g(t)\in C^{+}(J)$,  the BVP \eqref{pb_sub2} has a unique solution 
\begin{align} \label{pb_sub2_sol} 
u(t) =&\int_{a}^{T} G(t, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d}s + \left( \frac{t^{\rho_{1}} - a^{\rho_{1}}}{ \rho_{1} - \mu \rho_{1}} \right)\mu \int_{a}^{T} H(\eta, s)  \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d} s \nonumber\\  
&\qquad + \left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1} - \mu \rho_{1}}\right) \lambda  +    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d}\tau\right),
\end{align}
\noindent where  $G(t,s)$, $H(t,s)$ and $K(t, s)$ are defined in the previous  Lemmas \ref{le2.12}.
\end{lemma}


\begin{lemma}\label{le2.14}
The functions  $G$, $H$ and $K$ (Equ. \eqref{2.5}, \eqref{2.6},  and \eqref{2.11}) satisfy the following.
\begin{enumerate}[(i)]
\item $G(t,s)$ , $H(t,s)$ and $K(t,s)$ are continuous on $[a,T]\times[a,T]$. 
\item $\forall (t,s) \in [a,T]\times[a,T]$: 
$$G(t,s) \leqslant \left(\frac{T^{\rho_{1}} - a^{\rho_{1}} }{\rho_{1}}\right)^{\alpha -1}\frac{T^{\rho_{1}-1}}{\Gamma(\alpha -1)},  \; 
\int^{T}_{a} G(t,s) \mathrm{d}s = \left(\frac{t^{\rho_1} - a^{\rho_1}}{\Gamma(\alpha)\rho_1}\right)\left(\frac{T^{\rho_1} - a^{\rho_1}}{\rho_1}\right)^{\alpha-1} - \frac{1}{\Gamma(\alpha +1)}\left(\frac{t^{\rho_1} - a^{\rho_1}}{\rho_1}\right)^{\alpha},$$


$$ H(n,s) \leqslant \left(\frac{T^{\rho_{1}} - a^{\rho_{1}} }{\rho_{1}}\right)^{\alpha -2}\frac{T^{\rho_{1}-1}}{\Gamma(\alpha -1)}, \; \int_{a}^{T}  H(n,s) \mathrm{d}s = \frac{1}{\Gamma(\alpha)}\left(\left(\frac{T^{\rho_{1}} - a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-1} -\left( \frac{\eta^{\rho_{1}} - a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -1}\right),$$
$$K(t,s) \leqslant \left(\frac{T^{\rho_{2}} - a^{\rho_{2}} }{\rho_{2}}\right)^{\sigma -1}\frac{T^{\rho_{2}-1}}{\Gamma(\sigma -1)}, \; \int_{a}^{T}  K(s,\tau) \mathrm{d}\tau = \frac{T^{\rho_{2}}-s^{\rho_{2}}}{\rho_{2}\Gamma (\sigma)} \left(\left(\frac{T^{\rho_{2}} - a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -1} -\frac{1}{\sigma}\left(\frac{T^{\rho_{2}} - s^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -1}\right).$$
\item $\forall (t,s) \in [a,T]^2: \quad G(t,s)\geqslant 0, H(t,s)\geqslant 0, K(t,s)\geqslant 0$;
\item The function $t \to G(t,s)$ is increasing and $t \to K(t,s)$ is decreasing $\forall s\in [a,T]$. In addition, we have 
$$ \forall (t,s) \in (a,T)^2: \; \left(\frac{t^{\rho_{1}} - a^{\rho_{1}}}{T^{\rho_{1}} - a^{\rho_{1}}}\right)^{\alpha-1} G(T, s)\leqslant G(t, s) \; \text{ and } \; \left(\frac{T^{\rho_{2}} - t^{\rho_{2}}}{T^{\rho_{2}} - a^{\rho_{2}}}\right)^{\sigma-1} K(a, s)\leqslant K(t, s);$$
\item $\forall (t,s) \in (a,T)^2: \quad  \frac{t^{\rho_{1}-1} \rho_{1}}{T^{\rho_{1}} -a^{\rho_{1}}}\left[1 - \left(\frac{t}{T}\right)^{\rho_{1}(\alpha -2)}\right] G(T, s)\leqslant G^{\prime}_t(t, s) \leqslant\frac{\alpha -1}{\alpha -2}\frac{t^{\rho_{1}-1} \rho_{1}}{T^{\rho_{1}} -a^{\rho_{1}}} \, G(T, s)$ and $  \mid K^{\prime}_t(t, s) \mid\leqslant\frac{t^{\rho_{2}-1} \rho_{2}}{T^{\rho_{2}} -a^{\rho_{2}}} \, K(a, s)$.
\end{enumerate}
\end{lemma}

\begin{proof}
Using the definitions of $G$, $H$ and $ K$, $(i)$ and $(ii)$ are obtained straightforwardly. 
\begin{itemize}
	\item Property $(iii)$. We only consider the case $ s \leq t $ as the other case is straightforward.
\end{itemize}
 When  $s \leq t $, we have 
\begin{align*}
G(t,s) &\geqslant\frac{1}{\Gamma(\alpha -1)}\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right) \left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}a^{\rho_{1}-1} - \frac{1}{\Gamma(\alpha)}\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-1}a^{\rho_{1} -1}\nonumber,  \\
&\geqslant\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -1}a^{\rho_{1} -1}\left[\frac{1}{\Gamma(\alpha -1)} - \frac{1}{\Gamma(\alpha )}\right], \\
&\geqslant 0, \qquad \qquad \Big(\text{because } \Gamma(\alpha -1) \leqslant \Gamma(\alpha ) \; \text{ for } 2 < \alpha \leq 3 \Big). 
\end{align*}
Similarly, we can easily prove that $H(t,s)\geqslant 0$ and $K(t,s)\geqslant 0, \forall (t,s) \in [a,T]^2$.

 \begin{itemize}
 	\item Property $(iv)$. Through direct accounts  we can  check that $G(t,s)$ is nondecreasing w.r.t.  $t\in [a,T]$.
 \end{itemize}
% From (\ref{2.5}) and (\ref{2.6}) we have
\begin{equation}\label{2.14}
\frac{\partial G}{\partial t} (t, s)= \left\{
\begin{array}{ll}{\frac{t^{\rho_1-1}}{\Gamma(\alpha -1)}\left(\frac{T^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1} - \frac{t^{\rho_1-1}}{\Gamma(\alpha-1)}\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1} -1},} & {a \leqslant s \leqslant t \leqslant T}, \\ 
{\frac{t^{\rho_1-1}}{\Gamma(\alpha -1)}\left(\frac{T^{\rho_{1}}-s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}s^{\rho_{1}-1},} & {a \leqslant t \leqslant s \leqslant T.}\end{array}\right.
\end{equation}
Thus, $G(t, s)$ is increasing with respect to $t \in[a,T]$ 
and therefore, $G(t, s) \leqslant G(T, s),$ for $a \leqslant t, s \leqslant T .$ Furthermore, for $ t \leqslant s $, we have  
 \begin{align*}
 \frac{\partial K(t,s) }{\partial  t} &= -\frac{t^{\rho_2-1}}{\Gamma(\sigma-1)} \left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma-2}s^{\rho_{2}-1} + \frac{(\sigma -1)t^{\rho_{2}-1}}{\Gamma(\sigma)}\left(\frac{s^{\rho_2} -t^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -2}s^{\rho_{2}-1},     \\
& =\frac{t^{\rho_{2}-1}}{\Gamma(\sigma-1)}s^{\rho_{2} -1}  \left[ \left(\frac{s^{\rho_{2}} -t^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -2} -\left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -2}\right], \\
&\leqslant   \frac{t^{\rho_{2}-1}}{\Gamma(\sigma-1)}s^{\rho_{2} -1}  \left[ \left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -2} -\left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma -2}\right]=  0.
 \end{align*}
For $a\leqslant s \leqslant t \leqslant T$, we have
  \begin{align*}
 \frac{\partial K(t,s) }{\partial  t} &=\frac{-t^{\rho_{2}-1}}{\Gamma(\sigma-1)}   \left(\frac{s^{\rho_{2}} -a^{\rho_{2}}}{\rho_{2}}\right)^{\sigma-2}s^{\rho_{2}-1} \leqslant 0.
   \end{align*}
  Thus, $K(t,s)$ is nonincreasing with respect to $t$ . Consequently $ K(t,s) \leqslant K(a,s) \quad \forall t,s \in[a, T]. $ On the other hand,  when $t \geqslant s $ 
\begin{align*}
\frac{G(t, s)}{G(T, s)}&=\frac{ (\alpha-1) (t^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2}	- (t^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} }
{ (\alpha-1) (T^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2} 
- (T^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} },  \\
& = 
\frac{ (\alpha-1) (t^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2}	- (t^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} 
	\left(\frac{t^{\rho_{1}}-s^{\rho_{1}}  }{ t^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}}
{ (\alpha-1) (T^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2} 
	- (T^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} }.
\end{align*}
As $\left(\frac{t^{\rho} - s^{\rho}}{t^{\rho}-a^{\rho}}\right)^{\beta}\leq \left(\frac{T^{\rho} - s^{\rho}}{T^{\rho}-a^{\rho}}\right)^{\beta}$ for $\beta>0$, we obtain
\begin{align*}
\frac{G(t, s)}{G(T, s)} & \geqslant 
\frac{ (\alpha-1) (t^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2}	- (t^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} 
	\left(\frac{T^{\rho_{1}}-s^{\rho_{1}}  }{ T^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}}
{ (\alpha-1) (T^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2} 
	- (T^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} } , \\
& \geqslant \frac{(t^{\rho_{1}}-a^{\rho_{1}})^{\alpha -1}}{\left(T^{\rho_{1}} -a^{\rho_{1}}\right)^{\alpha-1} }
\frac{ 
	(\alpha-1) \left(T^{\rho_{1}} -a^{\rho_{1}}\right)^{\alpha-1} (t^{\rho_{1}} -a^{\rho_{1}})^{2 - \alpha} (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2}	
	- 
	\left(	T^{\rho_{1}}-s^{\rho_{1}} 	\right)^{\alpha -1}
}
{ (\alpha-1) (T^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2} 
	- (T^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} }, \\
& \geqslant \frac{(t^{\rho_{1}}-a^{\rho_{1}})^{\alpha -1}}{\left(T^{\rho_{1}} -a^{\rho_{1}}\right)^{\alpha-1} }
\frac{ 
	\left(\frac{T^{\rho_{1}} -a^{\rho_{1}}}{t^{\rho_{1}} -a^{\rho_{1}}}\right)^{\alpha-2}  
	(\alpha-1) \left(T^{\rho_{1}} -a^{\rho_{1}}\right)  
	(T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2}	
	- 
	\left(	T^{\rho_{1}}-s^{\rho_{1}} 	\right)^{\alpha -1}
}
{ (\alpha-1) (T^{\rho_{1}} -a^{\rho_{1}}) (T^{\rho_{1}} -s^{\rho_{1}})^{\alpha-2} 
	- (T^{\rho_{1}}-s^{\rho_{1}})^{\alpha -1} }, \\
& \geqslant  \frac{(t^{\rho_{1}}-a^{\rho_{1}})^{\alpha -1}}{\left(T^{\rho_{1}} -a^{\rho_{1}}\right)^{\alpha-1} }.
\end{align*}
For $t \leqslant s$, we have 
$$ \frac{G(t, s)}{\left(t^{\rho_{1}} -a^{\rho_{1}} \right)^{\alpha-1}}  =
 \frac{\rho_{1}^{\alpha-1} \, s^{\rho_{1}-1}}{ \Gamma(\alpha -1)}
 %\left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)
  \left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} \, 
\frac{1}{\left(t^{\rho_{1}} -a^{\rho_{1}} \right)^{\alpha-2}},
$$
which is a nonincreasing function as $\alpha \geq 0$. Consequently, 
$ \frac{G(t, s)}{\left(t^{\rho_{1}} -a^{\rho_{1}} \right)^{\alpha-1}}  \geq \frac{G(T, s)}{\left(T^{\rho_{1}} -a^{\rho_{1}} \right)^{\alpha-1}} $, which implies $G(t, s) \geqslant \left(\frac{t^{\rho_{1}}-a^{\rho_{1}} }{T^{\rho_{1}}-a^{\rho_{1}} }\right)^{\alpha -1} G(T, s)$.

Using similar techniques,  one can prove that $K(t, s) \geqslant \left( \frac{T^{\rho_{2}} -t^{\rho_{2}}}{T^{\rho_{2}} - a^{\rho_{2}}}\right)^{\sigma -1} K(a, s),$ for $a \leqslant s, t < T$. Therefore ($iv$) of Lemma \ref{le2.14} holds.
\begin{itemize}
	\item Property $(v)$. Again, we can consider two cases. Nevertheless, we prove the results for the case $s\leq t$ only. The simpler case $a \leq t\leq s< T$ can be treated similar arguments.    
\end{itemize}
When $s\leq t$ we have 
\begin{align*}
 \frac{G_t^{\prime}(t, s)}{G(T, s)} \, 
 \frac{(T^{\rho_1} - a^{\rho_1}) }{t^{\rho_1-1} \rho_1 (\alpha-1)} =& 
 \frac{
 	\left[
 	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} - \left(t^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2}
 	\right]
 }
 {
 	(\alpha-1)
 	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} - 
 	\frac{\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-1}}{(T^{\rho_1} - a^{\rho_1})}
 }. 
  \end{align*}
Consequently,
\begin{align*}
\frac{G_t^{\prime}(t, s)}{G(T, s)} \, 
\frac{(T^{\rho_1} - a^{\rho_1}) }{t^{\rho_1-1} \rho_1 (\alpha-1)} \leq &
\frac{
	\left[
	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} \right]
}
{
	(\alpha-1)
	\left(T^{\rho_1}-s^{\rho_1}\right)^{\alpha-2} - 
	\frac{\left(T^{\rho_1}-s^{\rho_1}\right)^{\alpha-1}}{(T^{\rho_1} - a^{\rho_1})}
}, \\ 
\leq &
\frac{1}
{
	(\alpha-1) - 
	\frac{\left(T^{\rho_1}-s^{\rho_{1}}\right)}{(T^{\rho_1} - a^{\rho_1})}
}, \\ 
\leq & \frac{1}{(\alpha-2)}. 
\end{align*}
Using similar techniques, we get 
\begin{align*}
\frac{\mid K_t^{\prime}(t, s)\mid}{K(a, s)} \, 
\frac{(T^{\rho_2} - a^{\rho_2}) }{t^{\rho_2-1} \rho_2 } \leq & 1. 
\end{align*}
On the other hand, 
\begin{align*}
\frac{G_t^{\prime}(t, s)}{G(T, s)} \, 
\frac{(T^{\rho_1} - a^{\rho_1}) }{t^{\rho_1-1} \rho_1 } =& 
\frac{(\alpha-1)
	\left[
	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} - \left(t^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2}
	\right]
}
{
	(\alpha-1)
	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} - 
	\frac{\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-1}}{(T^{\rho_1} - a^{\rho_1})}
}, \\
%\geq& 
%\frac{(\alpha-1)
%	\left[
%	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} - \left(t^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2}
%	\right]
%}
%{
%	(\alpha-1)
%	\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} 
%}, \\
\geq& 1 - \frac{ \left(t^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2}}
{\left(T^{\rho_{1}}-s^{\rho_{1}}\right)^{\alpha-2} }, \\
 \geq &
1 -\left(\frac{t}{T}\right)^{\rho_{1} (\alpha -2)} \left(\frac{1 -\left(\frac{s}{t}\right)^{\rho_{1}}}{1 -\left(\frac{s}{T}\right)^{\rho_{1}}}\right)^{\alpha -2}, \\
 \geq &
1 -\left(\frac{t}{T}\right)^{\rho_{1} (\alpha -2)} . 
 \end{align*}
Thus, the proof is completed.
 \end{proof}

Now, consider  the Banach space $\mathbb{E} =C_{\rho_{1}}^{3}(J)$. We can define norm

$$\|u\|=
\begin{array}{ll}
 $$\sum\limits_{k=0}^3 \max _{a \leq t \leq T}|\delta_{\rho_{1}}^{k} u(t)| ,
 \end{array}$$
and the cone  
$$ E =\left\lbrace u\in \mathbb{E}: u \text{ is nonnegative and increasing }\right\rbrace.$$

\begin{lemma} \label{normu}
	Let $u$ be the unique solution of the BVP \eqref{pb_sub2} associated to a given $g(t)\in C^{+}(J)$. Then $u\in E$, and the following inequality hold for $t \in [\bar{\theta}, \theta] \subset (a,T)$. 	
	
	
\begin{equation}
\min_{t\in [\bar{\theta} ,\theta]}u(t) \geqslant \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha -1} \; \mathrm{M} \; \Vert u\Vert,
\label{minu_theta} 
\end{equation}	
	where   
	$$
	\mathrm{M} = 4 \min
	\left\lbrace 1,
	\frac{\alpha -2}{\alpha -1}\left(\frac{T^{\rho_{1}} - a^{\rho_{1}}}{\rho_{1}}\right), 
	\min\left\lbrace\; \Gamma(\alpha -1)  \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{2-\alpha}, \frac{1}{(\bar{p}-1)\varsigma}\right\rbrace\,\times \right. \\ \left. \left(\frac{\bar{\theta}^{\rho_1}- a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1}Z(\theta)\int^{T}_{a} G(T,s)\mathrm{d}s,  
	\right\rbrace,
	$$	
	 $ Z(t) =\phi_{\bar{p}}\left(\left(\frac{T^{\rho_{2}} -t^{\rho_{2}}}{T^{\rho_{2}} -a^{\rho_{2}}}\right)^{\sigma-1} \right)$ and $\varsigma=\frac{\rho_2}{(T^{\rho_2} - a^{\rho_2})\Gamma(\alpha -2)}\max_{a\leqslant t\leqslant T}\left\lbrace\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \; (t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_2 -\rho_1}{\rho_1}}\mathrm{d}s\right\rbrace $.
\end{lemma}
\begin{proof}
From Lemma \eqref{le2.13}, we have
\begin{align}
u(t)& = \int_{a}^{T} G(t, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d}s + \left( \frac{t^{\rho_{1}} - a^{\rho_{1}}}{ \rho_{1} - \mu \rho_{1}} \right)\mu \int_{a}^{T} H(\eta, s)  \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d} s \nonumber\\  &\qquad + \left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1} - \mu \rho_{1}}\right) \lambda  +    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d}\tau\right).
\nonumber 
\end{align}
\hspace{0.25cm} (1) The functions $G, H,$ and $K$ are non-negative (Lemma \eqref{le2.14}-($iii$)). Thus, $u$ is also non-negative. Furthermore, as $G$ is increasing w.r.t. $t$ (Lemma \eqref{le2.14}-($iv$), so it is the function $u$.  

(2) As $u$ is non-negative and increasing, we have 
	\begin{multline*}
	\max_{a\leqslant t \leqslant T}|u(t)| = u(T) = \int_{a}^{T} G(T, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d}s \\ 
	+ \left( \frac{T^{\rho_{1}} - a^{\rho_{1}}}{ \rho_{1} - \mu \rho_{1}} \right)\mu \int_{a}^{T} H(\eta, s)  \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d} s \\  
	+ \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1} - \mu \rho_{1}}\right) \lambda  +     \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d}\tau\right).
	\end{multline*}
	For $t\in [\bar{\theta} ,\theta] $, using ($iv$) of Lemma \ref{le2.14} and the fact that $\left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{} <1$, we get 
	\begin{multline*}
	u(t) \geqslant \int_{a}^{T}\left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha -1}  G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d}s  
	\\  + \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha-2} \times \left( \frac{t^{\rho_{1}} - a^{\rho_{1}}}{ \rho_{1} - \mu \rho_{1}} \right)\mu \int_{a}^{T} H(\eta, s)  \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d} s 
	\\   + \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha-2} \times \left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1} - \mu \rho_{1}}\right) \lambda  
	\; + \; \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha-1} \times    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d}\tau\right).
	\end{multline*}
	Consequently,
	\begin{equation}\label{2.21}
	u(t)  \geqslant \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha -1}\max_{a \leqslant t \leqslant T}|u(t)|
	\end{equation}.
%\comment{Why do we add another term in the inequality? We can avoid adding a term and modifying the expression of M, right?-You are right.}

(3) We have 
	\begin{multline}
	\delta_{\rho_{1}}^{1}u(t) = t^{1-\rho_1}\int_{a}^{T} G^{\prime}_{t}(t, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s
	\\ +\frac{\mu }{\left(1-\mu \right)} \int_{a}^{T} H(\eta, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s  +\frac{\lambda }{(1-\mu )}. 
	\end{multline}
	
	From Lemma \ref{le2.14} (($iii$) and ($v$)), we can deduce that $\delta_{\rho_{1}}^{1}u(t) \geq 0$ and
	\begin{align}
	\delta_{\rho_{1}}^{1}u(t)\leqslant  & \int_{a}^{T}\frac{\alpha -1}{\alpha - 2}\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s \nonumber\\ 
	&+\frac{\mu }{\left(1-\mu \right)} \int_{a}^{T} H(\eta, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s +\frac{\lambda }{(1-\mu )}, \nonumber\\
	\leqslant &\frac{\alpha -1}{\alpha - 2}\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}} \times\left[ \int_{a}^{T} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s  \right.\nonumber\\ 
	& + \left. \mu\left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\mu\rho_{1} }\right) \int_{a}^{T} H(\eta, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s + \lambda \left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\mu\rho_{1} }\right)\right],    \nonumber\\	
	\leqslant&  \frac{\alpha -1}{\alpha - 2}\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}} \times\left[ \int_{a}^{T} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s+ \lambda \left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\mu\rho_{1} }\right)+ \right. \nonumber\\ 
	&\left.\mu\left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\mu\rho_{1} }\right) \int_{a}^{T} H(\eta, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)d\tau\right) \mathrm{d} s  +  \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)d\tau\right)\right], \nonumber\\
	\leqslant&  \frac{\alpha -1}{\alpha - 2}\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}} \times u(T).\label{2.22}
	\end{align}.
%	\begin{align*}
%	\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{1}u(t)| &\leqslant\frac{\alpha -1}{\alpha - 2}\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}}\left(\frac{\bar{\theta}^{\rho_1} - a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1}\max_{a\leqslant t\leqslant T}| u(t)|. 
%	\end{align*}
%\comment{Why do we add another term in the inequality?-We had previously thought that u(T) is not equal to $\max_{a\leqslant t\leqslant T}| u(t)|$, that's why we had to find an other estimation.}

	
(4) A straightforward calculus gives 
	\begin{equation}\label{bacg}
		 \delta_{\rho_{1}}^{2}u(t) = -\frac{1}{\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{t^{\rho_{1}} -s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \; \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d} \tau\right) \mathrm{d} s.
		 \end{equation}
	Then, we get:	
	\begin{align*}
	|\delta_{\rho_{1}}^{2}u(t)|&\leqslant \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right)\times \frac{1}{\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{t^{\rho_{1}} -s^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3} s^{\rho_{1}-1}\mathrm{d} s, \\
	& \leqslant \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right)\times \frac{1}{\Gamma(\alpha -1)}\left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}.
	\end{align*}
	Thus,
	\begin{align*}
	& \max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)|\leqslant \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right) \; \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}. 
	\end{align*}
	By multiplying both sides of the  previous inequality by $\phi_{\bar{p}}\left(\left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right) $, we get
	\begin{align*}
	& \phi_{\bar{p}}\left(\left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right)\;\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)|\leqslant \phi_{\bar{p}}\left(  \int_{a}^{T} \left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1}K(a, \tau)g(\tau)\mathrm{d} \tau\right) \; \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2},\\
	\end{align*}
	using  Lemma \eqref{le2.14}-$(iv)$, we get:
	\begin{align}
	\phi_{\bar{p}}\left(\left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right)\;\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)|\leqslant \phi_{\bar{p}}\left(  \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d} \tau\right) \; \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}. \label{-2}
	\end{align}
	We multiplying both sides by $G(t,s)$ and integrate over $[a,T]$ w.r.t. $s$, we get 
	\begin{align*}
	\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)| \; &\int^{T}_{a} G(t,s)\phi_{\bar{p}}\left(\left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right)\mathrm{d}s \\ &\leqslant \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2} \; \int^{T}_{a} G(t,s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d} \tau\right)\mathrm{d}s,\\
&\leqslant \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2} \;\left[	\int_{a}^{T} G(t, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d}s + \left(\frac{t^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1} - \mu \rho_{1}}\right) \lambda  \right. \\
 &\left. \left( \frac{t^{\rho_{1}} - a^{\rho_{1}}}{ \rho_{1} - \mu \rho_{1}} \right)\mu \int_{a}^{T} H(\eta, s) \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)g(\tau)\mathrm{d}\tau\right) \mathrm{d} s   +     \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d}\tau\right)\right],\\
&=:\frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2} \;u(t),\\
&\leqslant \frac{1}{\Gamma(\alpha -1)} \left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2} \;\max_{a\leqslant t\leqslant T}| u(t)|
\nonumber  
	\end{align*}
Furthermore, for $t\in [\bar{\theta} ,\theta] $  
\begin{align*}
\int^{T}_{a} G(t,s)\phi_{\bar{p}}\left(\left(\frac{T^{\rho_2}-s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right)\mathrm{d}s \geqslant& 
\left(\frac{t^{\rho_1}- a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1} Z(\theta) \int^{T}_{a} G(T,s)\mathrm{d}s, 
\end{align*}
	\begin{multline}\label{2.23}
	\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)| \; \int^{T}_{a} G(t,s)\phi_{\bar{p}}\left(\left(\frac{T^{\rho_2} - s^{\rho_2}}{T^{\rho_2} - a^{\rho_2}}\right)^{\sigma -1} \right)\mathrm{d}s \geqslant \\ \left(\frac{t^{\rho_1}- a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1} Z(\theta) \int^{T}_{a} G(T,s)\mathrm{d}s\;\max_{a\leqslant t\leqslant T}| \delta_{\rho_{1}}^{2}u(t)|, 
	\end{multline}. 
%	\begin{multline*}
%	\max_{a\leqslant t\leqslant T}|\delta_{\rho_{1}}^{2}u(t)|\leqslant \\ \left(\left(\frac{t^{\rho_1}- a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1} Z(\theta) \int^{T}_{a} G(T,s)\mathrm{d}s\right)^{-1}\;\frac{1}{\Gamma(\alpha -1)}\left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2} \, \max_{a\leqslant t\leqslant T} |u(t)|. 
%	\end{multline*}
	
	
(5) From the  equation in  \eqref{bacg} and Remark \ref{rema1} , one can see that 
	\begin{multline}\label{27}
	\delta_{\rho_{1}}^{2}u(t) = -\frac{1}{\Gamma(\alpha -1)}\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}s^{\rho_{1}-1} \; \phi_{\bar{p}}\left( \int_{a}^{T} K\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\right) \mathrm{d} s. 
	\end{multline}
	
	Then, from \eqref{27} and Lemma \ref{le2.82} we get 
	\begin{equation}
	\delta_{\rho_{1}}^{3}u(t) = \delta_{\rho_{1}}^{1}\left(\delta_{\rho_{1}}^{2}u\right)(t)= -\frac{t^{1-\rho_1}}{\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1}  \phi_{\bar{p}}\left( \int_{a}^{T}K\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)^{\prime}g(\tau)\mathrm{d} \tau\right) \mathrm{d} s\nonumber,
	\end{equation}
%	and 
%	\begin{equation}
%	^{\rho_{1};C}\!\mathcal{D}^{\beta }_{a^{+}}u(t)=\!^{\rho_{1};C}\!\mathcal{I}^{3-\beta }_{a^{+}}\delta_{\rho_{1}}^{3}u(t)
%	\end{equation}
	where 
	\begin{multline}
	 \phi_{\bar{p}}\left( \int_{a}^{T}K\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\right)^{\prime}= \int_{a}^{T} K_t ^{\prime}\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\;\times\\  \frac{(\bar{p}-1) t^{\rho_{1} -1}}{(t^{\rho_1} + a^{\rho_1}- s^{\rho_1} )^{\frac{\rho_1 -1}{\rho_1}}}\left\vert \int_{a}^{T} K\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\right\vert^{p-2}.
	 \end{multline}
	Thus,
	
	\begin{align*}
	\mid\delta_{\rho_{1}}^{3}u(t)\mid &= \frac{-1}{\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \;\int_{a}^{T} K_t ^{\prime}\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\;\times \\ &\qquad\qquad\qquad  \frac{(\bar{p}-1) }{(t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_1-1}{\rho_1}}}\left\vert \int_{a}^{T} K\left((t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{1}{\rho_1}}, \tau\right)g(\tau)\mathrm{d} \tau\right\vert^{p-2}\mathrm{d}s\nonumber,\\
	&\leqslant 
	 \frac{\rho_2}{(T^{\rho_2} - a^{\rho_2})\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \; (t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_2 -1}{\rho_1}}\int_{a}^{T} K\left(a, \tau\right)g(\tau)\mathrm{d} \tau\;\times \\ \qquad\qquad & \frac{(\bar{p}-1)}{(t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_1 -1}{\rho_1}}}\left\vert \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right\vert^{p-2}\mathrm{d} s\nonumber\\
	 &= (\bar{p}-1)\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right)\frac{\rho_2}{(T^{\rho_2} - a^{\rho_2})\Gamma(\alpha -2)}\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \; (t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_2 -\rho_1}{\rho_1}}\mathrm{d}s\;\\
	 &\leqslant(\bar{p}-1)\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right)\frac{\rho_2}{(T^{\rho_2} - a^{\rho_2})\Gamma(\alpha -2)}\max_{a\leqslant t\leqslant T}\left\lbrace\int_{a}^{t}\left(\frac{s^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -3}s^{\rho_{1}-1} \; (t^{\rho_1} + a^{\rho_1}- s^{\rho_1})^{\frac{\rho_2 -\rho_1}{\rho_1}}\mathrm{d}s\right\rbrace,
	\end{align*}
	Then,
	\begin{equation}\label{2.24}
	\max_{a\leqslant t\leqslant T}|\delta_{\rho_{1}}^{3}u(t)|\leqslant (\bar{p}-1)\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau)\mathrm{d} \tau\right)\times\varsigma.
	\end{equation}
	As in (2).  
%	that
%	\begin{equation}
%	\max_{a\leqslant t\leqslant T}|\delta_{\rho_{1}}^{3}u(t)|\leqslant \left(\left(\frac{t^{\rho_1}- a^{\rho_1}}{T^{\rho_1} - a^{\rho_1}}\right)^{\alpha-1}\min_{s\in[\bar{\theta},\theta]}Z(s)\int^{T}_{a} G(T,s)\mathrm{d}s\right)^{-1}\;\times \max_{a\leqslant t\leqslant T} |u(t)|.
%	\end{equation}
	
	(6) From the inequalities \eqref{2.21}, \eqref{2.22}, \eqref{2.23} and \eqref{2.24} one  can see that, equation\eqref{minu_theta} is a direct consequence of the previous results. \\
\end{proof}

Then, for  selected $[\bar{\theta}, \theta] \subset (a,T)$, we define the cone $$ \mathcal{P} = \left\lbrace u\in E  :
\min_{t\in [\bar{\theta}, \theta] }u(t) \geqslant \left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha -1} \mathrm{M} \Vert u\Vert\right\rbrace, 
$$
and the  integral operator $\mathcal{N}_\lambda :\mathcal{P} \rightarrow \mathbb{E} $ defined for $t \in [\bar{\theta}, \theta]$ by  
 \begin{multline}
	\mathcal{F}_\lambda(u)(t) = \int_{a}^{T} G(t, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right)d\tau\right) \mathrm{d} s \\
	+\mu\left(\frac{ t^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\int_{a}^{T} H(\eta, s) \; \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right) d\tau\right) \mathrm{d} s \\
	+\left(\frac{ t^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\lambda  +   \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cf\left( u(\tau)\right)d\tau\right). \label{2.29}
\end{multline}
We have $ \mathcal{F}_{\lambda}\left(\mathcal{P}\right)  \subset \mathcal{P} $  and fixed points of $\mathcal{F}_\lambda$ are solutions of \eqref{pb_main}.  To use some fixed point Theorems, we need to show that $\mathcal{N}_\lambda$ is completely continuous.
\begin{lemma}\label{le2.18} 
\cite{BZZY} Let $h, s>0$. For any $x,y \in [0,c]$, the following propositions hold.  
\begin{enumerate}[1]
\item If $\; s>1$, then $ |x^s - y^s|\leqslant s h^{s -1} |x - y| $.
\item If $\; 0< s \leqslant 1$, then $ |x^s - y^s|\leqslant |x - y|^s$.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{le2.19} 
The integral operator $\mathcal{F}_\lambda :\mathcal{C}\rightarrow \mathcal{C} $  is  continuous and compact.
\end{lemma}
\begin{proof}

The continuity of $\mathcal{F}_\lambda$ is a consequence of the continuity and positiveness  of  $ G, K, c$ and $f$. 

To prove that $\mathcal{F}_\lambda$ is compact, let us consider a bounded subset $\Omega \subset \mathcal{P} $. Then, there exists $L > 0$ such that for any $ u  \in \Omega $ we have 
$$|f(u(t))|\leqslant L. $$
For any $ u  \in \Omega $, as  $\mathcal{F}_\lambda(u)$ is positive , continuous and $G$ is increasing w.r.t. $t$, The maximum is at the value $ t=T$ i.e.,  $$\max_{a\leqslant t\leqslant T}\mid \mathcal{F}_\lambda(u(t))\mid=\mathcal{F}_\lambda(u(T)).$$ 
Consequently, from the above inequality, we get 
\begin{align}  
\max_{a\leqslant t\leqslant T}\mid \mathcal{F}_\lambda(u(t))\mid 
%=& \int_{a}^{T} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right)d\tau\right) \mathrm{d} s \\ 
%&+\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\int_{a}^{T} H(\eta, s) \; \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right) d\tau\right) \mathrm{d} s \\ 
%&+\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1}}\right)\lambda  +    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cf\left( u(\tau)\right)d\tau\right), \\
%	\end{multline*}
%Since  $g$ positive and $\phi_{\bar{p}}$ is increasing, we get 
%\begin{equation*}
\leq &\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cL  d\tau\right)\times\left[\int_{a}^{T} G(T, s) \mathrm{d} s + \mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right) \int_{a}^{T} H(\eta, s) \mathrm{d} s +1\right] \nonumber\\  
&+\lambda \left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)    \quad \quad \doteq \quad \quad  \bar{L} . \label{2.33}
\end{align}
%Thus,	
%\begin{align}	
%\max_{a\leqslant t\leqslant T}\mid \mathcal{N}_\lambda(u(t))\mid	\leqslant&\frac{A}{B} \left[\int_{a}^{T} G(T, s)\mathrm{d} s + \mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right) \int_{a}^{T} H(\eta, s)\mathrm{d} s  \right] \left(\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau) L\mathrm{d}\tau\right)\right) \nonumber\\
%	&+\lambda\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)+ A\int_{a}^{T} K(a, \tau)g(\tau) L\mathrm{d}\tau \nonumber\\
%	\leq& \frac{A}{B}\left[ \frac{1}{\Gamma(\alpha)}\left(\frac{T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha}\left(\frac{\alpha -1}{\alpha } + \frac{\mu}{1-\mu} \right)-  \frac{1}{\Gamma(\alpha)}\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\left(\frac{ \eta^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1} }\right)^{\alpha -1}  \right] \nonumber\\
%	& \times \left(\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau) L\mathrm{d}\tau\right)\right) 
%	+ A\int_{a}^{T} K(a, \tau)g(\tau) L\mathrm{d}\tau +\lambda\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\nonumber \\
%	\leqslant& \left[ \frac{1}{\Gamma(\alpha)}\left(\frac{T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha}\left(\frac{\alpha -1}{\alpha } + \frac{\mu}{1-\mu} \right)  \right] \times \frac{A\phi_{\bar{p}}\left(L\right)}{B}\left(\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)g(\tau) \mathrm{d}\tau\right)\right) \nonumber\\
%	&  + A\times L\int_{a}^{T} K(a, \tau)g(\tau) \mathrm{d}\tau +\lambda\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right) \quad \quad \doteq \quad \quad  \bar{L} . \label{2.33}
%	\end{align}
Then, as in Lemma \eqref{normu}, we obtain  
%\begin{align}
%\max_{a\leqslant t\leqslant T}\mid \delta_{\rho_{1}}^{1}\left(\mathcal{N}_\lambda(u(t))\right)\mid  &\leqslant \frac{\alpha -1}{\alpha -2}\left(\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}}\right)\left(\frac{\bar{\theta}^{\rho_{1}}-a^{\rho_{1}}}{T^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}\ell\label{2.34} , \\
%\max_{a\leqslant t\leqslant T}\mid \delta_{\rho_{1}}^{2}\left(\mathcal{N}_\lambda(u(t))\right)\mid  &\leqslant \frac{1}{\Gamma(\alpha -1)}\left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}\left(\left(\frac{\bar{\theta}^{\rho_{1}}-a^{\rho_{1}}}{T^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}z(\theta)\int^{T}_{a} G(T,s)\mathrm{d}s\right)^{-1}\ell\label{2.35} ,\\
%\max_{a\leqslant t\leqslant T}\mid \delta_{\rho_{1}}^{2}\left(\mathcal{N}_\lambda(u(t))\right)\mid  &\leqslant \left(\left(\frac{\bar{\theta}^{\rho_{1}}-a^{\rho_{1}}}{T^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}z(\theta)\int^{T}_{a} G(T,s)\mathrm{d}s\right)^{-1}\ell\label{2.36}.
%\end{align}
%So, from(\ref{2.33}), (\ref{2.34}), (\ref{2.35}) and (\ref{2.36}); we obtain 
$$\Vert \mathcal{F}_\lambda u \Vert \leqslant \mathbf{N} \; \bar{L},$$
\noindent where
$
 \mathbf{N}  = 4\max \left\lbrace 1, \frac{\alpha -1}{\alpha -2}\left(\frac{\rho_{1}}{T^{\rho_{1}}-a^{\rho_{1}}}\right), \max\left\lbrace \frac{1}{\Gamma(\alpha -1)}\left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha -2}, 1\right\rbrace\times \left(\left(\frac{\bar{\theta}^{\rho_{1}}-a^{\rho_{1}}}{T^{\rho_{1}}-a^{\rho_{1}}}\right)^{\alpha -1}Z(\theta)\int^{T}_{a} G(T,s)\mathrm{d}s\right)^{-1} \right\rbrace .
$

  Hence, $\mathcal{F}_\lambda(\Omega)$ is uniformly bounded. Furthermore, by using Lemmas \eqref{le2.13}, \eqref{le2.18}, \eqref{le2.14} and Lebesgue dominated convergence theorem, we deduce the equicontinuouty of $\mathcal{F}_\lambda(\Omega)$. Therefore, $\mathcal{F}_\lambda $ is completely continuous by Arzela-Ascoli Theorem.
\end{proof}

In order to avoid repetition in the remaining, we use the following notations.
\begin{equation*} 
\quad f_0 :=\lim_{r \to 0^+}  \frac{f(r)}{\phi_{p}\left(r\right)}, ~ f_\infty = \lim_{r \to \infty}  \frac{f(r)}{\phi_{p}\left(r\right)}.
\end{equation*}
\begin{equation*}
\gamma :=\left(\frac{\bar{\theta}^{\rho_{1}}- a^{\rho_{1}}}{T^{\rho_{1}}- a^{\rho_{1}}}\right)^{\alpha -1} \mathrm{M},
\end{equation*}
\begin{equation*}
\Lambda_1 := \left[\left(\int_{a}^{T} G(T, s) \mathrm{d} s+\mu\left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\rho_{1}\mu} \right)\int_{a}^{T} H(\eta, s)  \mathrm{d} s +1 \right)\times \phi_{\bar{p}}\left(\int_{a}^{T} K(a, \tau)cd\tau\right) \right]  ^{-1},
\end{equation*}
\begin{equation*}
\Lambda_2 :=\left[\frac{\alpha -1}{\alpha-2}\!\frac{\rho_{1}}{T^{\rho_{1}} - a^{\rho_{1}}}\left(\int_{a}^{T} G(T, s) \mathrm{d} s+\frac{\mu }{\left(1-\mu \right)} \int_{a}^{T} H(\eta, s)  \mathrm{d} s \right)\times \phi_{\bar{p}}\left(\int_{a}^{T} K(a, \tau)cd\tau\right) \right]^{-1}, 
\end{equation*}
\begin{equation*}
\Lambda_3 := \left[\frac{ 1}{\Gamma(\alpha-1)}\left(\frac{T^{\rho_{1}}-a^{\rho_{1}}}{\rho_{1}}\right)^{\alpha-2}\times    \phi_{\bar{p}}\left(\int_{a}^{T} K(a, \tau)cd\tau\right)\right]^{-1}, \quad 
\Lambda_4 :=  \left[  ((\bar{p}-1)\varsigma \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cd\tau\right)\right]^{-1},
\end{equation*}
\begin{equation*}
\Lambda_5 := \left[\gamma \left(\frac{\bar{\theta}^{\rho_1}-a^{\rho_1}}{T^{\rho_1}-a^{\rho_1}}\right)^{\alpha-1}\min_{a \leq s\leq \theta}(Z(s))\left(\int_{a}^{T} G(T, s)  +\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)H(\eta, s)\mathrm{d} s\right)\times \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)c d\tau\right)\right]^{-1}, 
\end{equation*}


\section{Study the existence of solutions in a cone}

In this section, we set some hypotheses to  derive an interval for $\lambda$ which ensures the existence of   positive solutions of the BVP \eqref{pb_main}.\\

\textbf{(H1)} There exist \; $ R_{1} >0$ and  $m_1\in (0,\min \left\lbrace \frac{\Lambda_{1}}{8}, \frac{\Lambda_{2}}{8} ,\frac{\Lambda_{3}}{4} ,\frac{\Lambda_{4}}{4}  \right\rbrace)$  such that
$$ \quad f(r) \leqslant \phi_{p}\left(m_1 R_1\right), \quad  \forall r \in [0,R_1],$$
\textbf{(H2)} There exist \; $ R_{2} >0$ and $m_2 \in (\Lambda_5,\infty)$ such that
  $$\quad f(r)\geqslant \phi_{p}\left(m_2 R_2\right), \quad \forall r\in [\gamma R_2,R_2],$$
  
   \textbf{(H3)} $$ \quad f_0  = \omega\prime \in [ 0,   k^{p-1} ),\quad k=\frac{\min \left\lbrace \frac{\Lambda_{1}}{8} ,\frac{\Lambda_{2}}{8} ,\frac{\Lambda_{3}}{4}, \frac{\Lambda_{4}}{4} \right\rbrace}{4}~~~~~~~~~~~~~~~~~~~~$$\label{2.37254}
   \textbf{(H4)}$$\quad f_\infty =   \xi \in (\left(\frac{2\Lambda_5}{\gamma}\right)^{p-1},\infty).$$\label{2.37125}
   
   
\begin{theorem}\label{the 2.19} Assume that the conditions \textbf{(H1)}and \textbf{(H2)} hold, and \; $0 < R_{1} < R_{2}$. Then, the BVP \eqref{pb_main} has at  least one  positive solution  for $\lambda>0$ small enough.
\end{theorem}

\begin{proof}
Let $\mathcal{P}_1 = \left\lbrace u \in E : \Vert u \Vert \leq R_1 \right\rbrace $ and $\lambda $ satisfying 
\begin{equation}\label{2.38}
0 < \lambda\leqslant \frac{(1-\mu)R_1 }{8} \; \min \left(1,\frac{\rho_{1}}{T^{\rho_{1}} -a^{\rho_{1}}}\right).
\end{equation}

%As  $\mathcal{N}_\lambda(u)$ is positive and $G$ is increasing w.r.t. $t$, we have 
Let $u \in  E\cap\partial\mathcal{P}_1 $, i.e., $ \Vert u \Vert =R_1$. From (\textbf{H1}) and \eqref{2.38}, we have 

\begin{multline*}
\max_{a \leqslant t\leqslant T}\mid \mathcal{F}_\lambda(u(t))\mid  =  \mathcal{F}_\lambda(u(T)) =\int_{a}^{T} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right)d\tau\right) \mathrm{d} s
\\ +\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\int_{a}^{T} H(\eta, s) \; \phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right) d\tau\right) \mathrm{d} s 
\\ +\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\lambda  +    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cf\left( u(\tau)\right)d\tau\right).
\end{multline*}
Then, 
\begin{multline*}
\max_{a \leqslant t\leqslant T}\mid \mathcal{F}_\lambda(u(t))\mid 	\leqslant  \frac{\Lambda_1 R_1}{8}\left[\left(\int_{a}^{T} G(T, s) \mathrm{d} s
+\mu\left(\frac{T^{\rho_{1}} -a^{\rho_{1}} }{\rho_{1}-\rho_{1}} \right)\int_{a}^{T} H(\eta, s)  \mathrm{d} s +1\right) \times \phi_{\bar{p}}\left(\int_{a}^{T} K(a, \tau)cd\tau\right)\right] 
 + \frac{R_1}{8}.
\end{multline*}
Consequently, 
$$ 
\max_{a \leqslant t\leqslant T}\mid \mathcal{F}_\lambda(u(t))\mid	\leqslant \frac{R_1}{8} + \frac{R_1}{8}  =  \frac{\Vert u \Vert}{4}.$$
Similarly, we obtain 
\begin{align*} 
\sum\limits_{k=1}^3 \max_{a \leqslant t\leqslant T}\mid \delta^{k}_{\rho_{1}}\mathcal{F}_\lambda(u(t))\mid  \leqslant &\frac{3\Vert u \Vert}{4}.
\end{align*}
Therefore, we conclude that 
\begin{equation}
\Vert \mathcal{F}_\lambda u\Vert \leqslant \Vert u \Vert,\quad \forall u  \in E\cap\partial\mathcal{P}_1\label{2.42*}.
\end{equation}

On the other hand, we set $\mathcal{P}_2 = \left\lbrace u\in E: \Vert u \Vert \leqslant R_2 \right\rbrace$. Then, for any $ u \in E\cap\partial\mathcal{P}_2$ by Lemma \eqref{normu} one has 
$$R_2 \geqslant\min_{\tau\in [\bar{\theta} ,\theta]}u(\tau) \geqslant \gamma R_2.$$ 
Using hypothesis $ \textbf{H2} $, we get 
\begin{align*}
\mathcal{F}_\lambda(u(T))&~~~~ \geqslant \left(\frac{\bar{\theta}^{\rho_1}-a^{\rho_1}}{T^{\rho_1}-a^{\rho_1}}\right)^{\alpha-1}\left[\int_{a}^{T} G(T, s)\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right)d\tau\right) \mathrm{d} s+\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1}}\right)\int_{a}^{T} H(\eta, s)\right. \nonumber\\ &\left. 
	\phi_{\bar{p}}\left( \int_{a}^{T} K(s, \tau)cf\left( u(\tau)\right) d\tau\right) \mathrm{d} s +\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)\lambda  +    \phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)cf\left( u(\tau)\right)d\tau \right)\right]\\
%	\geqslant & \left(\frac{\bar{\theta}^{\rho_1}-a^{\rho_1}}{T^{\rho_1}-a^{\rho_1}}\right)^{\alpha-1}m_2 R_2 Z(\theta)\left(\int_{a}^{T} G(T, s)  +\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)H(\eta, s)\mathrm{d} s\right)\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)c\phi_{p}\left( u(\tau)\right) d\tau\right)  \\ 
	\geqslant &\left(\frac{\bar{\theta}^{\rho_1}-a^{\rho_1}}{T^{\rho_1}-a^{\rho_1}}\right)^{\alpha-1}m_2 R_2 \gamma \; Z(\theta)  \left(\int_{a}^{T}G(T, s)  +\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)H(\eta, s)\mathrm{d} s\right)\times\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)c d\tau\right)\\
	\geqslant &\left(\frac{\bar{\theta}^{\rho_1}-a^{\rho_1}}{T^{\rho_1}-a^{\rho_1}}\right)^{\alpha-1}\Lambda_5 R_2 \gamma \; Z(\theta)  \left(\int_{a}^{T}G(T, s)  +\mu\left(\frac{ T^{\rho_{1}} -a^{\rho_{1}}}{\rho_{1}-\mu\rho_{1} }\right)H(\eta, s)\mathrm{d} s\right)\times\phi_{\bar{p}}\left( \int_{a}^{T} K(a, \tau)c d\tau\right)\\
	 &:=R_2 = \Vert u\Vert.
\end{align*} 
Which implies that 
\begin{equation}
\Vert \mathcal{F}_\lambda u\Vert \geqslant \Vert u  \Vert ,\quad\text{for any}\quad  u \in E\cap\partial\mathcal{P}_2.\label{2.421}
\end{equation}

 Therfore, from (\ref{2.42*}), (\ref{2.421}), $R_1 <R_2$ and by applying the first part of Theorem \eqref{the2.12}, we deduce that, the operator $\mathcal{F}_\lambda$ has at least one fixed point $u \in E\cap \overline{\mathcal{P}_2} \backslash\mathcal{P}_1$, which is  positive
solution of BVP \eqref{pb_main}. \\
\end{proof}

\begin{theorem}\label{the 2.19*}
Assume that all conditions \textbf{(H1)}, \textbf{(H2)} hold. Then, the BVP \eqref{pb_main} has no  positive solution for $\lambda$ large enough. 
\end{theorem}
\begin{proof}
Fome Technical Contradiction, we get the desired result
\end{proof}
\begin{remark}\label{remark4.3}
\end{remark} If $ f_0 = 0,~~f_\infty = \infty.$ hold, then the condotions \textbf{(H1)} and \textbf{(H2)} hold respectively.
Moreover, if the function $f$ is nondecreasing, the following theorem holds.
\begin{theorem}\label{2.20}
Assume  that the hypotheses of Theorem \eqref{the 2.19} hold and that $f$ is nondecreasing. Then, there exists $\lambda^*>0$ such that the BVP \eqref{pb_main} has at least one  positive
solution for $\lambda \in(0, \lambda^*)$ and has no  positive solution for $\lambda \in( \lambda^*, \infty)$. %\comment{We can have  $\lambda^* = \infty$? $\Theta$ is bounded. Why? From  Theorem \eqref{the 2.19} we have $\Theta$ .}
\end{theorem}
\begin{proof} 
%\comment{I changed the proof. Please check again.}
	
Let $ \Theta= \left\lbrace \lambda, ~~\text{the BVP \eqref{pb_main} has at least one  positive
solution} \right\rbrace  \subset\mathbb{R}_{+}^{*}$ and $\lambda^{*} =\sup \Theta $. It follows from Theorem \eqref{the 2.19} that   $\Theta \neq \emptyset$ and thus  $\lambda^{*}$ exists. We denote by $u_{0}$ the solution of  BVP
\eqref{pb_main} associated to $\lambda_{0}$ and 
%
%Now, let From the definition of $\lambda^{*}$, we know that for any $\lambda\in (0,\lambda^{*})$, there is a $\lambda_0 >\lambda$ such that BVP  \eqref{pb_main} associated to $\lambda_0$
%%\begin{equation*}
%%\begin{cases}
%% &^{\rho_{2};C}\!\mathcal{D}^{\sigma }_{T^{-}}\left(\phi_{p}\left(^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(t)\right)\right)+ g(t) f(u(t))=0, \quad a<t<T \\
%%&u(a)-B_{0}\left(  ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{a^{+}} u(a)\right)=0, \delta_{\rho_{1}}^{2}u(a)=0,\delta_{\rho_{1}}^{1}u(T)=\mu \delta_{\rho_{1}}^{1}u(\eta) + \lambda_0 \\
%%&\!^{\rho_{1};C}\!\mathcal{D}_{a^+}  u(T)=0,-\delta_{\rho_{2}}^{1}\left[\phi_{p}\left( ^{\rho_{1};C}\!\mathcal{D}_{a^+} u\right)\right](a) = 0,\delta_{\rho_{2}}^{2}\left[\phi_{p}\left( ^{\rho_{1};C}\!\mathcal{D}^{\alpha }_{0^{+}} u\right)\right](T)= 0
%%
%%\end{cases}
%%\end{equation*}
%has a $\rho_{1}$-concave positive solution $u_{0}$ \comment{why? }. 
%
%Now, we need to prove that for any $\lambda \in (0, \lambda_0)$, BVP
%\eqref{pb_main} has a $\rho_{1}$-concave positive solution. In fact, let 
\[  
\mathcal{K}(u_{0}) =\left\lbrace u\in E :u(t)<u_{0}(t), \forall t\in [a,T]\right\rbrace
 .\]
Let $\lambda \in (0,\lambda_0)$ and $u \in \mathcal{K}(u_{0})$. It follows from the definition of $	\mathcal{N}_\lambda$ \eqref{2.29} and the monotonicity of $f$ that for any $t$
\begin{equation*}
	\mathcal{F}_\lambda (u(t)) \leq 	\mathcal{F}_\lambda (u_0(t)) = u_{0}(t).
\end{equation*}
Thus, $\mathcal{F}_\lambda (\mathcal{K}(u_0))\subseteq \mathcal{K}(u_0)$. By Shaulder's fixed point theorem we know that, 
there exists a fixed point $u \in \mathcal{K}(u_0)$, which is a positive solution of  \eqref{pb_main}. The proof is completed.
\end{proof}

\begin{theorem}\label{theo2.21}
Suppose that  $f$  satisfies \textbf{(H3)} and \textbf{(H4)}. Then, the BVP \eqref{pb_main} has at least one  positive solution forfor $\lambda>0$ small enough.
\end{theorem}
\begin{proof}
Firstly, from the definition of $f_0$, for all $\epsilon>0$  there exists an adequate small positive number $\bar{\delta}(\epsilon)$ such that  

\begin{equation*}
f(r) \leqslant (\epsilon + \omega\prime) r^{p-1}\leqslant (\epsilon + \omega\prime)\bar{\delta}(\epsilon)^{p-1}, \quad \forall r\in [0, \bar{\delta}(\epsilon)].
\end{equation*} 
Then, for $\epsilon = k^{p-1}-\omega\prime $   we have  
\begin{equation}
f(r) \leqslant  k^{p-1} \bar{\delta}(\epsilon)^{p-1}\leqslant \left(2k\bar{\delta}(\epsilon)\right)^{p-1}.
\end{equation} 
It's enough to take $ R_1 =\bar{\delta}(\epsilon) $ and $m_1= 2k \in (0 ,\min \left\lbrace \frac{\Lambda_{1}}{8} ,\frac{\Lambda_{2}}{8} ,\frac{\Lambda_{3}}{4}, \frac{\Lambda_{4}}{4}  \right\rbrace)$,
 i.e.,  the conditions \textbf{(H1)} holds.\\
 Next, since $f_\infty = \xi \in (\left(\frac{2\Lambda_5}{\gamma}\right)^{p-1} ,\infty)$, then for every $\epsilon>0$, there exists an adequate big positive number $R_2 \neq R_{1}$ 
 such that 
 \begin{equation*}
 f(r) \geqslant (\xi -\epsilon)r^{p-1}  \geqslant(\xi -\epsilon) \left(\gamma R_2\right)^{p-1},\quad  \forall r\geqslant\gamma R_2.
 \end{equation*}
 Hence, for $\epsilon =\xi - \left(\frac{2\Lambda_5}{\gamma}\right)^{p-1} $ we get 
 \begin{equation}
 f(r) \geqslant\left(\frac{2\Lambda_5}{\gamma}\right)^{p-1} \left(\gamma R_2\right)^{p-1}  =\left(2\Lambda_{5} R_2\right)^{p-1}
 .\end{equation}
 Let $m_2= 2\Lambda_5  >\Lambda_5$, thus condition \textbf{(H2)} holds, by applying Theorem \ref{the 2.19}, we complete the proof.
\end{proof}




\section{An applications}

In this section, we give an example to illustrate the usefulness of our main results.

\begin{example}
Let us consider the following fractional BVP

\begin{equation} \label{5.1}
\left\{ 
\begin{array}{l}
 ^{C-H}\!\mathcal{D}^{\frac{5}{2} }_{e^{2^-}}\left(\phi_{p}\left(^{C-H}\!\mathcal{D}^{\frac{5}{2}}_{e^{1^+}} u\right)\right)(t)+  \frac{5\sqrt{\pi}}{2^{\frac{7}{2}}}u^{\frac{3}{2}}(t)=0,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad  e^1 <t< e^2 
, \\ [7pt]
u(e^1)+ \!^{C-H}\!\mathcal{D}^{\frac{5}{2}}_{e^{1^+}} u(e^1)=0,~\delta_{0^+}^{2}u(e^1)=0,~\delta_{0^+}^{1}u(e^2)=\frac{1}{2} \delta_{0^+}^{1}u(e^{\frac{8}{5}}) + \lambda
,  \\ [7pt]
^{C-H}\!\mathcal{D}^{\frac{5}{2}}_{e^{1^+}}  u(e^2)=0,~-\delta_{0^+}^{1}\left(\phi_{p}\left( ^{C-H}\!\mathcal{D}^{\frac{5}{2}}_{e^{1^+}}u\right)\right)(e^1) = 0,~\delta_{0^+}^{2}\left(\phi_{p}\left( ^{C-H}\!\mathcal{D}^{\frac{5}{2}}_{e^{1^+}}u\right)\right)(e^2)= 0
. 
\end{array}
\right.
\end{equation}
\noindent Here
\begin{equation*}
\begin{array}{ccc}
\rho_1=\rho_2=0 ,~&\alpha =\sigma=\frac{5}{2},~ &\mu =\frac{1}{2}\\
\eta= e^{\frac{8}{5}} ,~&p=\frac{3}{2},~ &\bar{p}=3 \\
\theta=e^{\frac{7}{4}},~ &\bar{\theta}=e^{\frac{3}{2}},c=\frac{5\sqrt{\pi}}{2^{\frac{7}{2}}},
\end{array}
\end{equation*}

$^{C-H}\!\mathcal{D}^{\frac{5}{2} }_{e^{1^+}}$ and $^{C-H}\!\mathcal{D}^{\frac{5}{2} }_{e^{2^-}}$ are  the left and  right-sided Caputo-Hadamard fractional derivatives.
 
We can easily show that $f(u(t))= u^{\frac{3}{2}}(t)$ satisfy:
$$ f_0 = \lim_{u\to 0^+} \frac{f(u)}{\phi_{\frac{3}{2}}(u)} = \lim_{u\to 0^+} u(t)=0,\quad f_\infty = \lim_{u\to \infty} \frac{f(u)}{\phi_{\frac{3}{2}}(u)} = \lim_{u\to 0^+} u(t)=\infty. $$
Then obviously,
\begin{align*}
\left(\int_{e^1}^{e^2} K(e^1,\tau )c\mathrm{d}\tau \right)^{2}=1,~
~\min\left\lbrace \frac{\Lambda_1}{8} ,~\frac{\Lambda_2}{8},~\frac{\Lambda_3}{4},~\frac{\Lambda_4}{4}\right\rbrace \simeq 0.014589936 ,~~\Lambda_5 \simeq 15.13154296.
\end{align*}
So, all conditions of Theorem \ref{the 2.19} hold, then we can choose $R_2>R_1$ and for $\lambda$ satisfies $0<\lambda\leqslant \frac{1}{32}R_1$, such that 
$$ \mathcal{P}_1 = \left\lbrace u\in E :\Vert u\Vert <R_1\right\rbrace ,\mathcal{P}_2 = \left\lbrace u\in E:\Vert u\Vert <R_2\right\rbrace
.$$ 
Then, we can show that, the BVP (\ref{5.1}) has at least one  positive solution $u\in E\cap(\overline{\mathcal{P}_2}\setminus\mathcal{P}_1 )$ for $\lambda$ small enough.
\end{example}
\section{Conclusion}
In this words we have discussed the existence and the uniqueness of solutions for a
class of nonlinear fractional differential equations with a boundary value, by using
the properties of the Green functions associated to \eqref{pb_main}, the Guo-Krasnosel'skii and Banach fixed point theorems. 


\begin{thebibliography}{}                                                                                                %
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%Exemple de format des BIBLIOGRAPHIQUES                            %
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%\textsc{G.D. ANDERSON, M.K. VAMANAMURTHY and M. VUORINEN}, \emph{"Generalized convexity and inequalities"}; \textbf{ Journal of Mathematical Analysis and Applications}, Volume 335, No 2, pp. 1294-1308, (2007). 

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