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\fancyhead[CE]{A. El Allaoui,  T. Abdeljawad, Y. Allaoui And M. Hannabou} 
\fancyhead[CO]{Investigation of fractional higher order boundary value problems}

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

{\Large \bf 
Investigation of fractional higher order boundary value problems within the Mittag-Leffler Law}


\let\thefootnote\relax\footnote{\scriptsize Received: .... ...; Accepted: .... .... }

{\bf  A. El Allaoui$^*$\let\thefootnote\relax\footnote{$^*$Corresponding Author}}\vspace*{-2mm}\\
\vspace{2mm} {\small  Cadi Ayyad University} \vspace{2mm}

{\bf T. Abdeljawad$^1$}\vspace*{-2mm}\\
\vspace{2mm} {\small  Prince Sultan University} \vspace{2mm}

{\bf Y. Allaoui$^2$}\vspace*{-2mm}\\
\vspace{2mm} {\small  Sultan Moulay Slimane University} \vspace{2mm}

{\bf M. Hannabou$^2$}\vspace*{-2mm}\\
\vspace{2mm} {\small  Sultan Moulay Slimane University} \vspace{2mm}

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\vspace{4mm}


{\footnotesize
\begin{quotation}
{\noindent \bf Abstract.} In this paper, we investigate a fractional differential equation of order $2< \mu <3$ with fractional boundary conditions, employing the Mittag-Leffler Law. We adopt a comprehensive approach to analyze the problem's key aspects. Firstly, we establish the existence of solutions by employing the Leray-Schauder alternative fixed point theorem. Secondly, we examine the uniqueness of the solutions using the Banach principle. Thirdly, we explore how solutions depend continuously on their initial data. Finally, the theoretical findings are supported by a practical example that demonstrates the applicability of the obtained results. This study contributes to the understanding of fractional differential equations with fractional boundary conditions, offering insights into their existence, uniqueness, and sensitivity to initial conditions.
\end{quotation}
\begin{quotation}
\noindent{\bf AMS Subject Classification:} 32F18; 34A12.

\noindent{\bf Keywords and Phrases:} Boundary conditions, Continuous dependence, Mittag-Leffler Law.
\end{quotation}}

\section{Introduction}
The concept of differentiation, a fundamental pillar of calculus, has been extended beyond its integer-order confines to embrace the complexities of non-local and memory-dependent phenomena. This extension, known as fractional calculus, introduces the notion of fractional derivatives. Unlike traditional derivatives, these fractional counterparts allow for the exploration of intricate behaviors and dynamic patterns that are prevalent in various natural and artificial systems.

The applications of fractional derivatives are wide-ranging, permeating diverse fields of science and engineering \cite{b01,b03,b13}. From modeling anomalous diffusion in porous media to describing the viscoelastic properties of materials, fractional derivatives have proven their efficacy in capturing intricate features that elude traditional integer-order derivatives. These applications extend into disciplines like biology, where they find use in explaining neuron firing patterns and drug dispersion in biological tissues, as well as in finance, where they offer insights into complex price dynamics.

Among the various formulations of fractional derivatives, the Atangana-Baleanu fractional derivative in the sense of Caputo (ABC-derivative) emerges as a notable and unique mathematical tool. Its distinct advantage lies in its ability to simultaneously capture both fractal and non-fractal characteristics within a system. Unlike other fractional derivatives, the ABC-derivative excels at modeling processes that exhibit power-law growth rates and self-similar behaviors. This exceptional capability lends itself to the accurate representation of complex real-world systems that exhibit a blend of regularity and irregularity, making the ABC-derivative a potent instrument for dissecting intricate phenomena \cite{ab8,ab9,ab7}.




%%%%% la suite


 In this paper, we focus on a fractional differential equation of order $2<\mu< 3$, incorporating fractional boundary conditions through the utilization of the ABC-fractional derivative:
 \begin{align}\label{1}
\begin{cases}
^{\mathcal{ABC}} \mathcal{D}^{\mu}_{0^+}u(t)=\psi\left(t,u(t),^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)\right),\quad t\in I=[0,1],\quad 2<\mu<3,\quad 0<\nu<1,\\
u(0)=u_0,\\
^{\mathcal{ABC}} \mathcal{D}^{\mu-1}_{0^+}u(1)=u_1,\\
^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+}u(1)=u_2.
\end{cases}
\end{align}
where $u_0,u_1,u_2\in \mathbb{R}$ and $\psi:\,I\times \mathbb{R}^2\longrightarrow \mathbb{R}$ is a real-valued function which is continuous and verified certain conditions.

  This specific form of boundary conditions introduces a novel aspect to the problem, leading to intriguing mathematical challenges and opportunities for analysis.

The primary contribution of this paper lies in its comprehensive investigation of the considered fractional differential equation with ABC-fractional derivative and fractional boundary conditions. While previous studies have explored fractional differential equations and their solutions, our research introduces a unique blend of aspects that enriches the field. Specifically, our work presents the following notable contributions:
\begin{itemize}
\item Novel Boundary Conditions: By incorporating the ABC-fractional derivative and fractional boundary conditions, our study ventures into uncharted territory, offering a new perspective on the behavior of fractional differential equations. Fractional boundary conditions possess a broader scope and can serve as a means to extend and generalize boundary conditions of the Dirichlet or Neumann types. This contribution is particularly valuable as it extends the applicability of fractional calculus to scenarios with non-standard boundary conditions.
\item Triple Analysis Approach: To address different facets of the problem, we adopt a three-fold analytical approach. We establish the existence of solutions using the Leray-Schauder alternative fixed point theorem, ensuring the robustness of our results. Moreover, we delve into the uniqueness of solutions using the Banach principle, providing insights into the distinctiveness of solutions within the problem domain. Additionally, we explore the continuous dependence of solutions on initial data, shedding light on the stability of the system under consideration.
\item Practical Relevance: Theoretical findings are often validated by an illustrative examples. A concrete example that illustrates the application of our results is given. This example validates the theoretical framework.
\end{itemize}
In summary, this paper significantly extends the current understanding of fractional differential equations by introducing fractional boundary conditions through the ABC-fractional derivative. The combination of theoretical rigor, a comprehensive analytical approach, and practical relevance distinguishes our work and paves the way for further exploration of this intriguing area of mathematical research.

%---------------------------------------------------------------------
\section{preliminaries}\label{sec:2}

%This part is devoted to present some basic definitions and lemmas concerning the  fractional calculus which will be used in our results. For more details, see  \cite{b01, b04,b14,b18,AS} and the references therein.\\
%%%%  definition 1 %%%%
This section is dedicated to introducing fundamental definitions and lemmas related to fractional calculus, which will be employed in our findings. For a deeper understanding, refer to \cite{b01,b04,b14,b18,AS} and the sources cited therein.

In this manuscript, we use the notation $\mathcal{C}(I)$ to represent the set of all real-valued continuous functions on interval $I$, equipped with the norm $\Vert u\Vert=\displaystyle\sup_{t\in I}\vert u(t)\vert$, by $\mathcal{AC}(I)$ the set of all real-valued absolutely continuous functions on interval $I$. Additionally, we denote by $$\mathcal{AC}^n(I)=\left\{w:I\longrightarrow \mathbb{R}:w^{(n-1)}\in \mathcal{AC}(I)\right\},\qquad\text{for}\quad n\in\mathbb{N}^{*}.$$.
\begin{definition}\label{Def1}(see \cite{ab1})
Let $\delta,\,\eta\in \mathbb{C}$. The Mittag-Leffler function is given as
\begin{eqnarray*}
\mathcal{E}_{\delta,\eta}(z)=\sum_{n=0}^{\infty}\dfrac{1}{\Gamma(n\delta +\eta)}z^n,\quad \text{for}\quad z\in \mathbb{C}\quad\text{and}\quad Re(\delta)>0.
\end{eqnarray*}
Where $\Gamma$ represents the Euler Gamma function.
\end{definition}
We denote by
\begin{eqnarray*}
\mathcal{E}_{\delta}(z):=\mathcal{E}_{\delta,1}(z)=\sum_{n=0}^{\infty}\dfrac{1}{\Gamma(n\delta +1)}z^n,\quad \text{for}\quad \delta,\,z\in \mathbb{C}\quad\text{and}\quad Re(\delta)>0.
\end{eqnarray*}
%%%%  definition 2 %%%%
\begin{definition}(see \cite{b01,b04})
Let $\mu> 0$,  the Riemann-Liouville fractional integral of order $\mu$ is defined by
$$
\mathcal{I}_{0^+}^{\mu}\varphi(t)=\frac{1}{\Gamma(\mu)}\int_{0}^{t}(t-\theta)^{\mu-1}\varphi (\theta)d\theta,\quad\text{for}\quad a.e.\quad t\in I,
$$
where $\varphi\in L^1(0,1)$.
\end{definition}
In the rest of this paper, $\mathcal{N}$ designates a normalization function that satisfies $\mathcal{N}(0)=\mathcal{N}(1)=1$.
\begin{definition}(see \cite{ab2,ab3,ab4})
\begin{enumerate}
\item The ABC-fractional derivative of order $0< \mu <1$ is given by
$$
{^{\mathcal{ABC}}}\mathcal{D}_{0^+}^{\mu}\varphi(t)=\frac{\mathcal{N}(\mu)}{1-\mu}\int_{0}^{t}\mathcal{E}_{\mu}\left[ -\dfrac{\mu}{1-\mu}(t-\theta)^{\mu}\right]\varphi^{\prime} (\theta)d\theta,\quad\text{for all}\quad  t\in I,
$$
where $\varphi\in \mathcal{AC}(I)$.
\item The associated AB-fractional integral of order $0<\mu <1$ is given by
$$
^{\mathcal{AB}}\mathcal{I}_{0^+}^{\mu}\varphi(t)=\frac{1-\mu}{\mathcal{N}(\mu)}\varphi(t)+\frac{\mu}{\mathcal{N}(\mu)}\mathcal{I}_{0^+}^{\mu}\varphi(t),\quad\text{for}\quad a.e.\quad t\in I,
$$
where $\varphi\in L^1(0,1)$.
\item For $n\in \mathbb{N}^{*}$, the ABC-fractional derivative of order $n< \mu <n+1$ is given by
$$
{^{\mathcal{ABC}}}\mathcal{D}_{0^+}^{\mu}\varphi(t):={^{\mathcal{ABC}}}\mathcal{D}_{0^+}^{\mu-n}\varphi^{(n)}(t),\quad\text{for all}\quad  t\in I,
$$
where $\varphi\in \mathcal{AC}^{n+1}(I)$.
\item For $n\in \mathbb{N}^{*}$, the associated AB-fractional integral of order $n<\mu <n+1$ is given by
$$
^{\mathcal{AB}}\mathcal{I}_{0^+}^{\mu}\varphi(t):=\mathcal{I}_{0^+}^{n}{^{\mathcal{AB}}}\mathcal{I}_{0^+}^{\mu -n}\varphi(t),\quad\text{for}\quad a.e.\quad t\in I,
$$
where $\varphi\in L^1(0,1)$.
\end{enumerate}
\end{definition}

\begin{lemma}\label{lemma1}(see \cite{ab5}) For $n\in \mathbb{N}^{*}$ and $n<\mu <n+1$. We have
$$
\left({^{\mathcal{AB}}}\mathcal{I}_{0^+}^{\mu}{^{\mathcal{ABC}}}\mathcal{D}_{0^+}^{\mu}u\right)(t)=u(t)+\sum_{i=0}^n\sigma_{i}t^i,
$$
$\sigma_{i}\in\mathbb{R},$ $i=0,1,2,...,n,$ and $u\in\mathcal{AC}^{n+1}(I)$.
\end{lemma}
\section{Main results}\label{sec:3}
\subsection{Existence and uniqueness results}{}
\hfill \break
Within this subsection, we initiate by introducing the solution to our presented problem. This solution is formulated in the subsequent lemma:
\begin{lemma}
Let $\Psi\in \mathcal{AC}(I)$ and $\Psi(0)=0$, the following problem
\begin{eqnarray}\label{2}
\begin{cases}
^{\mathcal{ABC}} \mathcal{D}^{\mu}_{0^+}u(t)=\Psi\left(t\right),\quad t\in I=[0,1],\quad 2<\mu<3,\\
u(0)=u_0,\\
^{\mathcal{ABC}} \mathcal{D}^{\mu-1}_{0^+}u(1)=u_1,\\
^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+}u(1)=u_2.
\end{cases}
\end{eqnarray}
has a solution given as follows:
\begin{align}\label{3}
u(t)&=u_0+\dfrac{1}{\omega}\left[u_2-\int_0^1(1-\theta)\Psi(\theta)d\theta-\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(u_1-\int_0^1\Psi(\theta)d\theta\right)\right]t\nonumber\\
&\quad+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^t(t-\theta)\Psi(\theta)d\theta +\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^t(t-\theta)^{\mu-1}\Psi(\theta)d\theta \nonumber\\
&\quad +\dfrac{1}{2\omega}\left(u_1-\int_0^1\Psi(\theta)d\theta\right)t^2.
\end{align}
where $$\omega=\dfrac{\mathcal{N}(\mu-2)}{3-\mu}\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right].$$
\end{lemma}
\begin{proof}
By applying operator $ ^{\mathcal{AB}} \mathcal{I}^{\mu}_{0^+}$ on both sides of the first equation of problem (\ref{2}) (lemma \ref{lemma1}), we obtain:
\begin{align}\label{4}
u(t)&=\sigma_0 +\sigma_1 t+\sigma_2 t^2+ ^{\mathcal{AB}} \mathcal{I}^{\mu}_{0^+} \Psi(t)\nonumber\\
&=\sigma_0 +\sigma_1 t+\sigma_2 t^2+ \mathcal{I}^{2}_{0^+} { ^{\mathcal{AB}}} \mathcal{I}^{\mu -2}_{0^+} \Psi(t)\nonumber\\
&= \sigma_0 +\sigma_1 t+\sigma_2 t^2+ \dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^t(t-\theta)\Psi(\theta)d\theta \nonumber\\
&\quad+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^t(t-\theta)^{\mu-1}\Psi(\theta)d\theta.
\end{align}
where $\sigma_0,\,\sigma_1$ and $\sigma_2$ are real numbers to be determined.

According  to the previous equation, when $t=0$, we obtain
\begin{eqnarray}\label{5}
\sigma_0=u(0)=u_0.
\end{eqnarray}
Note that, for $x\in \mathcal{AC}^{3}(I)$, we have
\begin{align*}
 & ^{\mathcal{ABC}} \mathcal{D}^{\mu-1}_{0^+} x(t)\\
 &= ^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+} x^{\prime}(t)\\
&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\int_{0}^{t}\mathcal{E}_{\mu-2}\left[ -\dfrac{\mu-2}{3-\mu}(t-\theta)^{\mu-2}\right]x^{\prime} (\theta)d\theta\\
&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\int_{0}^{t}
\sum_{n=0}^{\infty}\dfrac{1}{\Gamma(n(\mu-2) +1)}\left( -\dfrac{\mu-2}{3-\mu}(t-\theta)^{\mu-2}\right)^n x^{\prime} (\theta)d\theta\\
&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\sum_{n=0}^{\infty}\dfrac{1}{\Gamma(n(\mu-2) +1)}\left( -\dfrac{\mu-2}{3-\mu}\right)^n\int_{0}^{t}(t-\theta)^{n(\mu-2)} x^{\prime} (\theta)d\theta\\
&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\sum_{n=0}^{\infty}\dfrac{1}{\Gamma(n(\mu-2) +2)}\left( -\dfrac{\mu-2}{3-\mu}\right)^n\int_{0}^{t}(t-\theta)^{n(\mu-2)+1} x^{\prime} (\theta)d\theta\\
&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\sum_{n=0}^{\infty}\left( -\dfrac{\mu-2}{3-\mu}\right)^n\mathcal{I}_{0^+}^{n(\mu-2)+2}x^{\prime}(t),\\
 ^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+} x(t)&=\frac{\mathcal{N}(\mu-2)}{3-\mu}\sum_{n=0}^{\infty}\left( -\dfrac{\mu-2}{3-\mu}\right)^n\mathcal{I}_{0^+}^{n(\mu-2)+2}x(t).
\end{align*}
In particular,
\begin{align*}
& ^{\mathcal{ABC}} \mathcal{D}^{\mu-1}_{0^+}t=0,\quad ^{\mathcal{ABC}} \mathcal{D}^{\mu-1}_{0^+}t^2=2 ^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+}t=2\dfrac{\mathcal{N}(\mu-2)}{3-\mu}\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}t^{\mu -2}\right]t,\\
& ^{\mathcal{ABC}} \mathcal{D}^{\mu-2}_{0^+}t^2=2\dfrac{\mathcal{N}(\mu-2)}{3-\mu}\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}t^{\mu -2}\right]t^2.
\end{align*}
By applying  operator $ ^{\mathcal{AB}} \mathcal{D}^{\mu-1}_{0^+}$ on both sides of equation (\ref{4}) and then substituting $t=1$, we get:
\begin{eqnarray*}
u_1=2\omega \sigma_2+\int_0^1\Psi(\theta)d\theta.
\end{eqnarray*}
Which leads to
\begin{eqnarray}\label{6}
\sigma_2=\dfrac{1}{2\omega}\left(u_1-\int_0^1\Psi(\theta)d\theta\right).
\end{eqnarray}
Now, by applying  operator $ ^{\mathcal{AB}} \mathcal{D}^{\mu-2}_{0^+}$ on both sides of equation (\ref{4}) and then substituting $t=1$, we get:
\begin{align*}
u_2&=\omega \sigma_1+2\sigma_2\dfrac{\mathcal{N}(\mu-2)}{3-\mu}\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]+\int_0^1(1-\theta)\Psi(\theta)d\theta\\
&=\omega \sigma_1+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(u_1-\int_0^1\Psi(\theta)d\theta\right)+\int_0^1(1-\theta)\Psi(\theta)d\theta.
\end{align*}
Which implies that
\begin{eqnarray}\label{7}
\sigma_1=\dfrac{1}{\omega}\left[u_2-\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(u_1-\int_0^1\Psi(\theta)d\theta\right)-\int_0^1(1-\theta)\Psi(\theta)d\theta\right].
\end{eqnarray}
By substituting the equations (\ref{5}), (\ref{6}), and (\ref{7}) into the equation (\ref{4}), we derive our outcome.
\end{proof}
\begin{remark}
From equation (\ref{3}), we have
\begin{eqnarray*}
u^{\prime\prime}(t)&=&\dfrac{1}{\omega}\left(u_1-\int_0^1\Psi(\theta)d\theta\right)+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}\Psi(t)\\
&&\quad +\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu-2)}\int_0^t(t-\theta)^{\mu-3}\Psi(\theta)d\theta.
\end{eqnarray*}
So, for $\Psi\in \mathcal{AC}(I)$, $u^{\prime\prime}\in \mathcal{AC}(I)$, which means that $u\in \mathcal{AC}^{(3)}(I)$.

Thus, $^{\mathcal{ABC}} \mathcal{D}^{\mu}_{0^+}u(t)$ is well defined for $2<\mu <3$. And by performing a simple computation, we arrive at $ ^{\mathcal{ABC}} \mathcal{D}^{\mu}_{0^+}u(t)=\Psi(t)$.

This shows that we have equivalence in the previous lemma.
\end{remark}
Now, we are poised to establish the subsequent solution operator:
\begin{align}\label{8}
\Upsilon u(t)&=u_0+\dfrac{1}{\omega}\left[u_2-\int_0^1(1-\theta)\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)d\theta\right.\nonumber\\
&\quad-\left.\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(u_1-\int_0^1\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)d\theta\right)\right]t\nonumber\\
&\quad +\dfrac{1}{2\omega}\left(u_1-\int_0^1\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)d\theta\right)t^2\nonumber\\
&\quad +\dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^t(t-\theta)\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)d\theta \nonumber\\
&\quad +\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^t(t-\theta)^{\mu-1}\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)d\theta,\qquad \text{for}\quad u\in \mathcal{C}(I).
\end{align}
We take into consideration the subsequent assumptions, which will serve as the foundation for establishing the existence of the solution to our presented problem:
\begin{itemize}
\item[$\left(\mathcal{A}_1\right)$] Let $\psi\left(.,u(.),{^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u}(.)\right)\in\mathcal{AC}(I)$ for $u\in \mathcal{AC}(I)$, and there exists tree functions $\phi_i\in \mathcal{C}(I,\mathbb{R}^{+}),\quad i=1,2,3$  such that
\begin{align*}
\vert \psi(t,y,z)\vert\leq \phi_1(t)+\phi_2(t)\vert y\vert +\phi_3(t)\vert z\vert,\qquad\text{for all}\quad t\in I,\quad y,\,z\in \mathbb{R}.
\end{align*}
\item[$\left(\mathcal{A}_2\right)$] Let $\psi\left(.,u(.),{^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u}(.)\right)\in\mathcal{AC}(I)$ for $u\in \mathcal{AC}(I)$, and there exists two functions $\psi_i\in L^1(I,\mathbb{R}^{+}),\quad i=1,2$  such that
\begin{align*}
\vert \psi(t,y,z)-\psi(t,y',z')\vert\leq \psi_1(t)\vert y-y'\vert +\psi_2(t)\vert z-z'\vert,\qquad
\end{align*}
$\text{for all}\quad t\in I,\quad y,\,z,\,y',\,z'\in \mathbb{R}$.
\end{itemize}
To simplify the relatively complex formulas, we introduce the subsequent notations:
\begin{align*}
\lambda_1&=\dfrac{(1-\nu)\Gamma(\nu)+1}{\mathcal{N}(\nu)\Gamma(\nu)},\\
\lambda_2&=\Vert \phi_2\Vert+\lambda_1\Vert \phi_3\Vert, \\
\lambda_3&= \vert u_0\vert+\dfrac{1}{\omega}\left[\dfrac{1}{2}\vert u_1\vert+\vert u_2\vert+\Vert \phi_1\Vert+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(\vert u_1\vert +\Vert \phi_1\Vert\right)\right]\\
&\quad+\left(\dfrac{3-\mu}{2\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\right)\Vert \phi_1\Vert,\\
\overline{\lambda_3}&=\vert u_0\vert+\dfrac{1}{\omega}\left(\dfrac{1}{2}\vert u_1\vert+\vert u_2\vert+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\vert u_1\vert\right),
\end{align*}
\begin{align*}
\lambda_4&= \dfrac{1}{\omega}\left( 1+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\right)+\dfrac{3-\mu}{2\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)},\\
\overline{\lambda_4}&= \dfrac{1}{\omega}\left(\dfrac{3}{2}+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\right)+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)},\\
\lambda_5&=\int_0^1\psi_1(\theta)d\theta+\lambda_1\int_0^1\psi_2(\theta)d\theta.
\end{align*}
\begin{remark}\label{r1}
\begin{enumerate}
\item For $u\in \mathcal{AC}(I)$, we have
\begin{align*}
\left\vert ^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)\right\vert &=\left\vert\dfrac{1-\nu}{\mathcal{N}(\nu)}u(t)+\dfrac{\nu}{\mathcal{N}(\nu)}  \mathcal{I}^{\nu}_{0^+}u(t)\right\vert\\
&\leq  \dfrac{(1-\nu)\Gamma(\nu)+1}{\mathcal{N}(\nu)\Gamma(\nu)}\Vert u\Vert\\
&\leq \lambda_1\Vert u\Vert, \qquad\text{for all}\quad t\in I.
\end{align*}
\item From $\left(\mathcal{A}_2\right)$, we have
\begin{align*}
\left\vert \psi\left(t,u(t), ^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)\right)\right\vert &\leq \Vert \phi_1\Vert+\left(\Vert \phi_2\Vert +\lambda_1\Vert \phi_3\Vert\right)\Vert u\Vert\\
&\leq \Vert \phi_1\Vert+\lambda_2\Vert u\Vert,\qquad\text{for all}\quad t\in I,\,u\in \mathcal{AC}(I).
\end{align*}
\end{enumerate}
\end{remark}
Now, we have all the necessary data to present our first existence result.
\begin{theorem}\label{th1}
Let $\psi$ be a function satisfying assumption $\left(\mathcal{A}_1\right)$ and $\psi(0,u_0,0)=0$. Suppose in addition that $\lambda_2\lambda_4< 1$. Then, the problem (\ref{1}) possesses at least one solution.
\end{theorem}
\begin{proof}
Let us consider the closed ball
\begin{eqnarray*}
\mathcal{B}_\varrho=\left\{u\in \mathcal{C}(I):\Vert u\Vert\leq \varrho\right\},
\end{eqnarray*}
where
\begin{eqnarray*}
\varrho\geq \dfrac{\lambda_3}{1-\lambda_2\lambda_4}.
\end{eqnarray*}
We transform our problem into a fixed point problem associated with the operator $\Upsilon$, which is defined by (\ref{8}).
\begin{itemize}
\item[(i)] Let us prove that $\Upsilon\left(\mathcal{B}_\varrho\right)\subseteq \mathcal{B}_\varrho$.\\
Using remark \ref{r1}, for $u\in \mathcal{B}_\varrho$,  we have
\begin{align*}
\left\vert \Upsilon u(t)\right\vert &\leq  \vert u_0\vert+\dfrac{1}{\omega}\left[\vert u_2\vert+\dfrac{1}{2}\Vert \phi_1\Vert+\dfrac{1}{2}\lambda_2\Vert u\Vert +\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(\vert u_1\vert +\Vert \phi_1\Vert +\lambda_2\Vert u\Vert\right)\right]\\
&\quad+\dfrac{1}{2\omega}\left(\vert u_1\vert +\Vert \phi_1\Vert+\lambda_2\Vert u\Vert\right)+\dfrac{3-\mu}{2\mathcal{N}(\mu-2)}\left(\Vert \phi_1\Vert +\lambda_2\Vert u\Vert\right)\\
&\quad+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\left(\Vert \phi_1\Vert +\lambda_2\Vert u\Vert\right)\\
&\leq  \vert u_0\vert+\dfrac{1}{\omega}\left[\dfrac{1}{2}\vert u_1\vert+\vert u_2\vert+\Vert \phi_1\Vert+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\left(\vert u_1\vert +\Vert \phi_1\Vert\right)\right]\\
&\quad +\left(\dfrac{3-\mu}{2\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\right)\Vert \phi_1\Vert\\
&\quad+\lambda_2\left[\dfrac{1}{\omega}\left( 1+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\right)+\dfrac{3-\mu}{2\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\right]\Vert u\Vert\\
&\leq \lambda_3+\lambda_2\lambda_4\varrho\\
&\leq \varrho,\qquad \forall t\in I
\end{align*}
which means that $\Upsilon$ maps $\mathcal{B}_\varrho$ into itself.
\item[(ii)]Now, we show that $\Upsilon$ is completely continuous:\\
In $(i)$, we have shown that $\Upsilon\left(\mathcal{B}_\varrho\right)$ is bounded, and note that the continuity of $\psi$ ensures that of $\Upsilon$.

The next step is to show that $\Upsilon$ is equicontinuous:

For $u\in \mathcal{B}_\varrho$, $0\leq t_1<t_2\leq 1$, we have
\begin{align*}
 \left\vert \Upsilon u(t_2)-\Upsilon u(t_1)\right\vert &\leq  \dfrac{1}{\omega}\left[\left\vert u_2\right\vert+\int_0^1(1-\theta)\left\vert \psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta\right.\\
 &\quad\left.+ \dfrac{\mathcal{E}_{\mu-2,3}\left[- \dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[- \dfrac{\mu-2}{3-\mu}\right]}\left(\left\vert u_1\right\vert +\int_0^1\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta\right)\right]\left\vert t_2-t_1\right\vert\\
&\quad+ \dfrac{1}{2\omega}\left(\left\vert u_1\right\vert+\int_0^1\left\vert \psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta\right)\left\vert t_2^2-t_1^2\right\vert
\end{align*}
\begin{align*}
&\quad+ \dfrac{3-\mu}{\mathcal{N}(\mu-2)}\left\vert t_2-t_1\right\vert\int_0^{t_1}\left\vert \psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta \\
&\quad+ \dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_{t_1}^{t_2}(t_2-\theta)\left\vert \psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta \\
&\quad+ \dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^{t_1}\left[(t_2-\theta)^{\mu-1}-(t_1-\theta)^{\mu-1}\right]\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta\\
&\quad+ \dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_{t_1}^{t_2}(t_2-\theta)^{\mu-1}\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)\right\vert d\theta\\
&\leq  \dfrac{1}{\omega}\left[\left\vert u_2\right\vert+ \dfrac{1}{2}\Vert\phi_1\Vert+ \dfrac{1}{2}\lambda_2 \varrho+ \dfrac{\mathcal{E}_{\mu-2,3}\left[- \dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[- \dfrac{\mu-2}{3-\mu}\right]}\left(\left\vert u_1\right\vert +\Vert \phi_1\Vert +\lambda_2\varrho \right)\right]\left\vert t_2-t_1\right\vert\\
&\quad + \dfrac{1}{2\omega}\left(\left\vert u_1\right\vert+\Vert \phi_1\Vert +\lambda_2\varrho\right)\left\vert t_2^2-t_1^2\right\vert+ \dfrac{3-\mu}{\mathcal{N}(\mu-2)}\left(\Vert \phi_1\Vert +\lambda_2\varrho\right)\left\vert t_2-t_1\right\vert \\
&\quad+ \dfrac{3-\mu}{2\mathcal{N}(\mu-2)}\left(\Vert \phi_1\Vert +\lambda_2\varrho\right)(t_2-t_1)^2\\
&\quad + \dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\left(\Vert \phi_1\Vert +\lambda_2\varrho\right)\left\vert t_2^{\mu}-t_1^{\mu}\right\vert .
\end{align*}
So, $$\left\vert \Upsilon u(t_2)-\Upsilon u(t_1)\right\vert\longrightarrow 0\quad \text{as}\quad t_2\rightarrow t_1$$
Then, according to Arzela-Ascoli's Theorem, $\Upsilon$ is relatively compact.\\
Thus, it is completely continuous.
\item[(iii)] The following set
\begin{eqnarray*}
\Theta=\left\{u\in \mathcal{C}(I):u(t)=\kappa\left(\Upsilon u\right)(t)\quad\text{for some}\quad\kappa\in (0,1)\right\},
\end{eqnarray*}
is bounded. Indeed:\\
For $u\in \Theta$ and $t\in I$, we have
\begin{align*}
\left\vert u(t)\right\vert &=\kappa \left\vert \Upsilon u(t)\right\vert\\
&<  \left\vert \Upsilon u(t)\right\vert\\
&< \lambda_3+\lambda_2\lambda_4\Vert u\Vert.
\end{align*}
Hence, we get	
$$\Vert u\Vert\leq \dfrac{\lambda_3}{1-\lambda_2\lambda_4}<\infty.$$
\end{itemize}
Thanks to Lery-Schauder alternative, our problem has at least one solution.
\end{proof}
The theorem presented below establishes the result of uniqueness.
\begin{theorem}\label{th2}
Let $\psi$ be a function satisfying assumption $\left(\mathcal{A}_2\right)$ and $\psi(0,u_0,0)=0$. Suppose in addition that $\overline{\lambda_4}\lambda_5< 1$. Then problem (\ref{1}) has a unique solution.
\end{theorem}
\begin{proof}
We consider the following closed ball:
\begin{eqnarray*}
\mathcal{B}_\rho=\left\{u\in\mathcal{C}(I):\Vert u\Vert\leq \rho\right\},
\end{eqnarray*}
where
\begin{eqnarray*}
\rho\geq \dfrac{\overline{\lambda_3}+\lambda_4\gamma}{1-\overline{\lambda_4}\lambda_5}\quad\text{and }\quad \gamma=\sup_{t\in I}\vert \psi\left(t,0,0\right)\vert .
\end{eqnarray*}
Not that, using $\left(\mathcal{A}_2\right)$, we have
\begin{align*}
\left\vert \psi\left(t,u(t), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)}\right)\right\vert &\leq  \left\vert \psi\left(t,u(t), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)}\right)-\psi\left(t,0,0\right)\right\vert+\left\vert \psi\left(t,0,0\right)\right\vert\\
&\leq  \psi_1(t)\left\vert u(t)\right\vert+\psi_2(t)\left\vert {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)}\right\vert+\gamma\\
&\leq \left(\psi_1(t)+\lambda_1\psi_2(t)\right)\rho+\gamma,\quad\forall t\in I\quad\text{and}\quad u\in \mathcal{B}_\rho .
\end{align*}
Similarly, we have
\begin{align*}
\left\vert \psi\left(t,u(t), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(t)}\right)-\psi\left(t,v(t), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(t)}\right)\right\vert &\leq  \left(\psi_1(t)+\lambda_1\psi_2(t)\right)\Vert u-v\Vert,\\
&\quad\quad\forall t\in I\quad\text{and}\quad u,\,v\in \mathcal{C}(I)
\end{align*}
Firstly, we prove that $\Upsilon$ maps $\mathcal{B}_ \rho$ into itself.\\
For $u\in \mathcal{B}_ \rho$ and $t\in I$, we have
\begin{align*}
 \left\vert \Upsilon u(t)\right\vert &\leq   \vert u_0\vert + \dfrac{1}{\omega}\left[\left\vert u_2\right\vert+\int_0^1(1-\theta)\left[\left(\psi_1(\theta)+\lambda_1\psi_2(\theta)\right)\rho+\gamma\right]d\theta\right.\\
 &\quad +\left.\dfrac{\mathcal{E}_{\mu-2,3}\left[- \dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[- \dfrac{\mu-2}{3-\mu}\right]}\left(\left\vert u_1\right\vert +\int_0^1\left(\left(\psi_1(\theta)+\lambda_1\psi_2(\theta)\right)\rho+\gamma \right) d\theta\right)\right]\\
&\quad + \dfrac{1}{2\omega}\left(\left\vert u_1\right\vert+\int_0^1\left(\left(\psi_1(\theta)+\lambda_1\psi_2(\theta)\right)\rho+\gamma \right) d\theta\right)\\
&\quad+ \dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^{t}(t-\theta)\left(\left(\psi_1(\theta)+\lambda_1\psi_2(\theta)\right)\rho+\gamma \right) d\theta \\
&\quad+ \dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_{0}^{t}(t-\theta)^{\mu-1}\left(\left(\psi_1(\theta)+\lambda_1\psi_2(\theta)\right)\rho+\gamma \right) d\theta
\end{align*}
\begin{align*}
&\leq   \vert u_0\vert + \dfrac{1}{\omega}\left[\left\vert u_2\right\vert+ \dfrac{1}{2}\gamma+\rho\int_0^1\psi_1(\theta)d\theta+\lambda_1\rho\int_0^1\psi_2(\theta)d\theta\right.\\
&\quad +\left.\dfrac{\mathcal{E}_{\mu-2,3}\left[- \dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[- \dfrac{\mu-2}{3-\mu}\right]}\left(\left\vert u_1\right\vert +\gamma +\rho\int_0^1\psi_1(\theta)d\theta+\lambda_1\rho\int_0^1\psi_2(\theta)d\theta\right)\right]\\
&\quad + \dfrac{1}{2\omega}\left(\left\vert u_1\right\vert+\gamma +\rho\int_0^1\psi_1(\theta)d\theta+\lambda_1\rho\int_0^1\psi_2(\theta)d\theta\right)\\
&\quad + \dfrac{3-\mu}{2\mathcal{N}(\mu-2)}\left(\gamma +2\rho\int_0^1\psi_1(\theta)d\theta+2\lambda_1\rho\int_0^1\psi_2(\theta)d\theta\right) \\
&\quad + \dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu+1)}\left(\gamma +\mu\rho\int_0^1\psi_1(\theta)d\theta+\mu\lambda_1\rho\int_0^1\psi_2(\theta)d\theta\right)\\
&\leq \overline{\lambda_3}+\gamma\lambda_4+\overline{\lambda_4}\lambda_5\rho\\
&\leq  \rho
\end{align*}
Now, let us show that $\Upsilon$ is a contraction:\\
For $u,\,v\in \mathcal{B}_\rho$ and $t\in I$, we have
\begin{align*}
\left\vert \Upsilon \right.&\left.u(t)-\Upsilon v(t)\right\vert \\
&\leq \dfrac{1}{\omega}\int_0^1(1-\theta)\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\omega\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\int_0^1 \left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta \\
&\quad+\dfrac{1}{2\omega}\int_0^1 \left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^t(t-\theta)\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^t(t-\theta)^{\mu-1}\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta .
\end{align*}
Using remark \ref{r1}, we obtain then
\begin{align*}
\left\vert \Upsilon u(t)-\Upsilon v(t)\right\vert &\leq  \lambda_5\left(\dfrac{3}{2\omega}+ \dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\omega\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\right)\Vert u-v\Vert\\
&\leq \overline{\lambda_4}\lambda_5 \Vert u-v\Vert.
\end{align*}
Hence, in accordance with the Banach Contraction Principle, our problem (\ref{1}) admits a unique solution.
\end{proof}
\subsection{Continuous dependence of the solution from the initial data}
\hfill \break
Let us denote by $\overline{\lambda_{3,{u_0}}}=\overline{\lambda_3}$ and
\begin{eqnarray*}
\overline{\lambda_{3,{v_0}}}=\vert v_0\vert+\dfrac{1}{\omega}\left(\dfrac{1}{2}\vert u_1\vert+\vert u_2\vert+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\vert u_1\vert\right),
\end{eqnarray*}
where $u_0,\,v_0\in \mathbb{R}$.
\begin{theorem}
Let $\psi$ be a function satisfying assumption $\left(\mathcal{A}_2\right)$ and $\psi(0,u_0,0)=\psi(0,v_0,0)=0$. Suppose in addition that $\overline{\lambda_4}\lambda_5< 1$.\\
 Let $u=u(t,u_0)$ and $v=v(t,v_0)$ be solutions of (\ref{1}) corresponding to $u(0)=u_0$ and $v(0)=v_0$,
respectively. Then
\begin{eqnarray*}
\Vert u-v\Vert \leq \dfrac{1}{1-\overline{\lambda_4}\lambda_5}\vert u_0-v_0\vert.
\end{eqnarray*}
\end{theorem}
\begin{proof}
Note that $u$ and $v$ exist and are well-defined according to the previous theorem, where we can define the operator $\Upsilon $ on the following closed ball:
\begin{eqnarray*}
\mathcal{B}_r=\left\{w\in\mathcal{C}(I):\Vert w\Vert\leq r\right\},
\end{eqnarray*}
where
\begin{eqnarray*}
r\geq \dfrac{max(\overline{\lambda_{3,{u_0}}},\overline{\lambda_{3,{v_0}}})+\lambda_4\gamma}{1-\overline{\lambda_4}\lambda_5}.
\end{eqnarray*}
Furthermore, we have
\begin{align*}
\vert u&(t,u_0)-v(t,v_0)\vert \\
&\leq  \vert u_0-v_0\vert + \dfrac{1}{\omega}\int_0^1(1-\theta)\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{\mathcal{E}_{\mu-2,3}\left[-\dfrac{\mu-2}{3-\mu}\right]}{\omega\mathcal{E}_{\mu-2,2}\left[-\dfrac{\mu-2}{3-\mu}\right]}\int_0^1 \left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta \\
&\quad+\dfrac{1}{2\omega}\int_0^1 \left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{3-\mu}{\mathcal{N}(\mu-2)}\int_0^t(t-\theta)\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\quad+\dfrac{\mu-2}{\mathcal{N}(\mu-2)\Gamma(\mu)}\int_0^t(t-\theta)^{\mu-1}\left\vert\psi\left(\theta,u(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}u(\theta)}\right)-\psi\left(\theta,v(\theta), {^{\mathcal{AB}} \mathcal{I}^{\nu}_{0^+}v(\theta)}\right)\right\vert d\theta\\
&\leq  \vert u_0-v_0\vert +\overline{\lambda_4}\lambda_5 \Vert u-v\Vert,\qquad\text{for all}\quad t\in I.
\end{align*}
Hence, we get
\end{proof}
\begin{eqnarray*}
\Vert u-v\Vert \leq \dfrac{1}{1-\overline{\lambda_4}\lambda_5}\vert u_0-v_0\vert.
\end{eqnarray*}
To better understand obtained results, let us consider the following illustrative example:
\begin{example}
The following problem is under consideration:
\begin{eqnarray}\label{e1}
\begin{cases}
^{\mathcal{ABC}} \mathcal{D}^{\frac{5}{2}}_{0^+}u(t)=\dfrac{t}{e^{\frac{1}{2} t^2}+20}\left(1+\dfrac{\vert u(t)\vert}{\left\vert u(t)\right\vert +1}+\dfrac{\left\vert {^{\mathcal{AB}}\mathcal{I}_{0^+}^{\frac{1}{4}}}u(t)\right\vert}{\left\vert {^{\mathcal{AB}}\mathcal{I}_{0^+}^{\frac{1}{4}}}u(t)\right\vert +1}\right),\quad t\in I=[0,1],\\
u(0)=\dfrac{1}{40},\\
^{\mathcal{ABC}} \mathcal{D}^{\frac{3}{2}}_{0^+}u(1)=\dfrac{1}{12},\\
^{\mathcal{ABC}} \mathcal{D}^{\frac{1}{2}}_{0^+}u(1)=\dfrac{1}{24}.
\end{cases}
\end{eqnarray}
The problem bellow can be expressed as (\ref{1}), where
\begin{align*}
\psi\left(t,u(t),{^{\mathcal{AB}}\mathcal{I}_{0^+}^{\frac{1}{4}}u(t)}\right)&=\dfrac{t}{e^{\frac{1}{2} t^2}+20}\left(1+\dfrac{\vert u(t)\vert}{\left\vert u(t)\right\vert +1}+\dfrac{\left\vert {^{\mathcal{AB}}\mathcal{I}_{0^+}^{\frac{1}{4}}}u(t)\right\vert}{\left\vert {^{\mathcal{AB}}\mathcal{I}_{0^+}^{\frac{1}{4}}}u(t)\right\vert +1}\right),\\
 u_0&=\dfrac{1}{40},\quad u_1=\dfrac{1}{12}\quad\text{and}\quad u_2=\dfrac{1}{24}.
\end{align*}
We consider $\mathcal{N}(x)=1$,  for all $x\in I$.\\
On can verify that assumptions $\left(\mathcal{A}_1\right)$ and  $\left(\mathcal{A}_2\right)$ are satisfied, where
\begin{eqnarray*}
\phi_1(t)=\phi_2(t)=\phi_3(t)=\psi_1(t)=\psi_2(t)=\dfrac{t}{e^{\frac{1}{2} t^2}+20},\qquad\text{for}\quad t\in I.
\end{eqnarray*}
By performing calculations, we obtain
\begin{align*}
&\int_0^1\psi_j(\theta)d\theta=\dfrac{1}{40}\left(1+\ln(441)-2\ln(20+\sqrt{e})\right),\quad j=1,2,\\
&\Vert \phi_i\Vert =\gamma =\dfrac{1}{e^{\frac{1}{2}}+20},\quad\text{for}\quad i=1,2,3,\\
&\omega \simeq 1.1118;\quad \lambda_1\simeq 1.0642;\quad \lambda_2\simeq 0.0953;, 2.2123;\quad\overline{\lambda_4}\simeq 2.6299,\\
&\lambda_2\lambda_4\simeq 0.2108<1; \quad \overline{\lambda_4}\lambda_5\simeq 0.1276<1.
\end{align*}
Therefore, all assumptions of theorems \ref{th1} and \ref{th2} are satisfied, and thus problem (\ref{e1}) has a unique solution.
\end{example}
\section*{Acknowledgements}
 I would like to express our gratitude to the editor for taking time to handle the manuscript. Also, the reviewers for carefully reading the paper and for their valuable comments and suggestions.

\begin{center}
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\end{center}



{\small

\noindent{\bf Abdelati El Allaoui}

\noindent Department of Informatics

\noindent Assistant Professor of Mathematics

\noindent MISCOM, National School of Applied Sciences, Cadi Ayyad University

\noindent Safi, Morocco

\noindent E-mail: a.elallaoui@uca.ma}\\

{\small
\noindent{\bf  Thabet Abdeljawad}

\noindent  Department of Mathematics and Sciences

\noindent Professor of Mathematics

\noindent Prince Sultan University


\noindent Riyadh, Saudi Arabia

\noindent E-mail: tabdeljawad@psu.edu.sa}\\

{\small

\noindent{\bf Youssef Allaoui}

\noindent Department of Mathematics

\noindent Doctor of Mathematics

\noindent LMACS, Faculty of Science and Technics, Sultan Moulay Slimane University

\noindent Beni Mellal, Morocco

\noindent E-mail: youssefbenlarbi1990@gmail.com}\\

{\small

\noindent{\bf  Mohamed Hannabou}

\noindent Department of Mathematics

\noindent Assistant Professor of Mathematics

\noindent LMACS, Faculty of Science and Technics, Sultan Moulay Slimane University

\noindent Beni Mellal, Morocco

\noindent E-mail: hnnabou@gmail.com}\\

\end{document}